The cohomology of lambda-rings and Psi-rings

The cohomology of λ -rings an d Ψ -rings Thesis submitted fo r the degree of Do ctor of Philosoph y at the Univ ersit y of Leicester b y Mic hael Robinson Department of Mathemat ics Univ ersit y of Leicester Marc h 2010 Abstract In this thesis we dev elop the cohomolo g y of diagrams of algebra s and then apply this to the cases of the λ -rings and the Ψ-ring s. A diag r am o f algebr as is a functor from a small category to some category of algebras . F or an appro priate category of algebras w e get a diagra m of groups, a diagra m of Lie alg ebras, a diagr am of commut ative rings, etc. W e deﬁn e the cohomology of diagra ms of alg ebras using comonads. The cohomology of diagrams of algebras classiﬁes extensions in the categor y of functors. Our main result is that there is a sp ectral sequence connecting the cohomolog y of the diag ram of algebras to the cohomology of the mem b ers of the diagram. Ψ-rings can be thought of as functors fro m the category with one ob ject asso ciated to the m ultiplicative monoid of the natur al num ber s to the ca tegory of commutativ e rings. So we can apply the theory we develop ed for the dia grams of algebr as to the cas e of Ψ -rings. Our main result tells us that there is a sp ectra l se q uence connecting the cohomolog y of the Ψ -ring to the Andr´ e-Quillen cohomolog y of the underlying co mm utative ring. The main example of a λ -ring o r a Ψ- ring is the K -theor y o f a top olo gical space. W e lo ok at the e x ample of the K -theory of spher es and use its co homology to give a pro of of the cla ssical result of Adams. W e show that there ar e natural transfor mations connecting the cohomolog y o f the K -theor y of spheres to the homotopy gr oups of spheres. There is a very close connection betw een the cohomolo gy o f the K -theor y of the 4 n -dimensio nal spheres and the homotopy g roups of the (4 n − 1 )-dimensional spheres. Ac kno wledgmen ts Firstly , I would like to thank my sup erviso r, Dr T eim uraz Pir ashvili, for all o f his useful advice and suppor t throughout this pro ject and for in tro ducing me to the sub ject of λ -rings. His passion for maths has b een truly ins pirational and e nc o uraging . I w ould also like to thank my ﬁr st sup e rvisor, P rof. Dietrich Notb ohm, who is now in Ams- terdam. I appre c iate the opp or tunit y he gav e me to carr y out resea rch, for int ro ducing me to the sub ject of homolog ical algebra, and als o for int ro ducing me to T eimuraz. I would like to thank all of the staﬀ in the Department of Mathematics at the Universit y of Leicester. I would a lso like to thank the other res e a rch students that I met. I am esp ecially grateful to Dr. Mark Parsons for his beneﬁcia l advice dur ing the ﬁrst year. I a m gra teful for all the supp or t received from my family; in par ticular my Mum, Da d, and Brothers. I would lik e to say a big thank you to F rieda Mann and Dorothy Kinnear for b eing like Grandmothers to me. A s p ecia l thanks to John F ergus Burns and J ohn Bur n for their inv aluable suppo rt, and fo r allowing me to escap e Leicester from time to time. I would like to acknowledge the EPSRC and the University of Leicester for all o f the ﬁnancial suppo rt they have provided me. i Con ten ts Abstract i Ac kno wledgmen ts i 1 In tro duction 5 1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Homolog ical algebra 8 2.1 Category theor y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Abelia n catego ries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Cohomolog y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Classical derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Pro jective and injectiv e ob jects . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Pro jective and injectiv e resolutio ns . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.3 Right de r ived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.4 Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2.4 Comonad coho mology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Monads and co mo nads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Simplicial metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 Comonad cohomolo g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.4 Andr´ e-Quillen cohomolog y . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2.5 Harrison coho mo logy of commutativ e algebras . . . . . . . . . . . . . . . . . . . . . 1 7 2.5.1 Ho chsc hild cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2.5.2 Harrison Cohomo logy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Baues-Wirsching cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Ψ -rings 24 3.1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Ψ-mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Ψ-deriv ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Ψ-ring extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Crossed Ψ-extens io ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6 Deformation of Ψ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 λ -rings 31 4.1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 λ -mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 λ -deriv ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 λ -ring extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.5 Crossed λ -extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.6 Y au cohomolo gy for λ -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ii 5 Harrison cohomolog y of diagrams of comm utativ e algebras 45 5.1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Natural System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Bicomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Harrison coho mo logy of diagra ms of comm utative algebras . . . . . . . . . . . . . . 47 5.5 Harrison coho mo logy of Ψ-ring s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.6 Harrison coho mo logy and λ -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.7 Gerstenhab er-Schack cohomolo gy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Andr ´ e-Quill en cohomol ogy of diagrams of algebras 56 6.1 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2 Deriv a tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.3 Natural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.4 Bicomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.5 Cohomolog y of diagrams of gro ups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 7 Andr ´ e-Quill en cohomol ogy of Ψ -rings and λ -rings 65 7.1 Cohomolog y of Ψ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Cohomolog y of λ -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8 Applications 69 8.1 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.1.1 V ector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.1.2 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2 Natural transfo rmation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0 8.3 The Hopf inv a r iant of an extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.4 Stable Ex t g roups of spher es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 A Adams Op erations 74 B Universal Polynomials P i , P i,j 75 C Univ ersal P olynomial P artial Deriv atives 76 Bibliog raph y 77 iii Chapter 1 In tro duction λ -rings were ﬁr st in tro duced in an algebraic -geometry setting b y Grothendieck in 19 58, then later used in group theor y by Atiy ah and T all. A λ -ring R is a comm utativ e ring with ident ity , together with ope r ations λ i : R → R , for i ≥ 0. W e require that λ 0 ( r ) = 1 and λ 1 ( r ) = r for all r ∈ R . There are more co mplicated a xioms describing λ i ( r 1 + r 2 ), λ i ( r 1 r 2 ) and λ i ( λ j ( r )). The λ -op erations behave like exter io r powers. The more complicated axioms are diﬃcult to work with, and given a λ -ring R , it is diﬃcult to prov e that it is actually a λ -ring. In 1962 Adams intro duced the o per ations Ψ i to s tudy vector ﬁelds o f spheres. These op erations give us a nother type o f ring, the Ψ -rings, which are r elated to the λ -rings b y the follo wing formula. Ψ i ( r ) − λ 1 ( r )Ψ i − 1 ( r ) + ... + ( − 1 ) i − 1 λ i − 1 ( r )Ψ 1 ( r ) + ( − 1 ) i iλ i ( r ) = 0 . A Ψ -ring is a commutativ e ring R , tog ether with ring homomor phisms Ψ i : R → R , for i ≥ 1. W e only require that Ψ 1 ( r ) = r and Ψ i (Ψ j ( r )) = Ψ ij ( r ) for all r ∈ R . The Ψ -rings are muc h easier to work with, and in several places we will need to pass to Ψ-rings to b e able to c a rry out computations for λ -r ings. Homologica l a lgebra is a r elatively young discipline, whic h a r ose from a lgebraic top olo gy in the early 20 th centu ry . In 19 5 6, Cartan a nd Eilenberg published their b o ok entitled “Homological Algebra” [5], which was the ﬁrst b o ok on homologica l algebra and still r e mains a s ta ndard b o o k of reference today . They found that the cohomology theor ie s for groups, a sso ciative algebra s and Lie algebras could all b e describ ed b y derived functors, deﬁned b y mea ns of pro jectiv e and injective resolutions o f mo dules. How ever the metho d they us ed was not enough to deﬁne the cohomolo gy of commutative algebras. T o o vercome this pro blem, simplicial tech niques w ere develop ed in homologica l a lgebra. In the 195 0 ’s Moo re show ed that every simplicial group K is a Ka n complex who se homotopy groups a re the homolo gy of a c hain co mplex called the Mo ore complex of K . Dold and K an independently found that there is an eq uiv alence b etw een the c a tegory of s implicial ab elian g roups and the categ ory of non- negative c hain complexes of a be lia n groups given b y the Mo ore complex. Using simplicial metho ds Dold and Pupp e showed that one ca n deﬁne the derived functor s of a non-additive fu nctor, since simplicial homotopy doesn’t involv e addition. The notion of a mona d on a category traces back to R. Go dement [9 ]. Aro und 1965, Barr and Beck use d comonads to deﬁne a res olution as a wa y to compute nona be lia n der ived functor s . In 1967, Andr´ e a nd Quillen indep endently develop e d what we now call Andr´ e-Quillen cohomolog y . The Andr´ e- Quillen coho mology is deﬁned in ge neral for alg e br as, using comona ds. The λ -ring s and Ψ-ring s are particula r examples which are included in this scheme, so the Andr´ e-Q uillen cohomolog y is well deﬁned for b oth λ -r ings and Ψ-rings. The main diﬃculty is that the Andr´ e- Quillen cohomolo gy is complicated and diﬃcult to compute. Harriso n had des crib ed a cohomo lo gy for comm utative a lg ebras in 1962 using a subco mplex of the Ho chsc hild complex. The Harr ison cohomolog y coincides with the Andr´ e- Quillen cohomology over a ﬁeld of characteristic zero up to a dimension shift. Our aim is to de velop to ols which aid computation. In 20 0 4, Y au [2 0] deﬁned a cohomology for λ -r ings in order to study deformations of the asso ciated Ψ-o per ations. Ho wev er, Y au’s c o homology for λ -ring s is diﬀerent from the Andr´ e - Quillen cohomolo gy . 5 1.1 Outline of the thesis In Cha pter 2, we give a short overview o f some of the fundamen tal concepts of homolo gical alge br a. W e can trace the ro ots o f these conce pts back to Cartan and E ilenberg in the 1 950’s. W e provide the deﬁnitions of additive ca tegories , abelia n catego ries and short exact sequences in ab elian cate- gories. W e outline the construction of the righ t derived functor Ext i using pro jective a nd injective resolutions. The main r eferences for this part o f the chapter are [19] and [17]. W e sketc h the con- struction o f the cohomology of algebras in general using comonads [19] and w e giv e the example of the Andr´ e-Q uillen cohomo logy for commutativ e rings which are the right derived functors of the deriv ations functor [18]. W e provide an overview of the Ha rrison co homology of commutativ e algebras [10] and the Baues-Wirsching co homology of a small category with c o eﬃcients in a na tural system [4]. Chapter 2 only provides w ell known bac kgro und materia l w hich will b e r equired later . It do es not contain any origina l work. The original resear ch can be found in the remaining chapters of the thesis. In Chapter 3, we turn our attention to Ψ-r ings, which are related to λ -rings via the Adam’s op erations. The ﬁrst sectio n intro duces the bas ic concept of a Ψ -ring which ca n b e found in [14]. W e then develop the Ψ-a nalogue o f mo dules and the semidirect pro duct. These are then used to develop the Ψ-analog ue of deriv a tio ns and extens io ns. The results from this chapter are nee de d in chapter 4 to prove simila r results for λ -rings. In 2005, Donald Y au published a paper entitled, “Cohomolog y of λ -rings” [20]. In the pap er he develops a cohomolog y of λ -rings in order to study the defor mations of the Ψ-ring structure. Y au’s cohomo logy is diﬀerent fro m the Andr´ e-Quillen cohomolog y . In the las t section of Chapter 3 I provide a deﬁnition of the defo rmation o f a Ψ-ring w hich is diﬀerent to Y au’s deﬁnition. This alternative deﬁnition is related to the Andr´ e-Quille n cohomo logy of Ψ-ring s. In Chapter 4, we intro duce λ -rings. The ﬁrst section int ro duces the ba sic notions o f λ -rings which can b e found in [14]. W e then develop the λ -a nalogue of mo dules a nd the semidirect pr o duct. W e then use these to develop the λ -ana logue of deriv ations and extensions. The las t sec tio n of Chapter 4 provides an ov erview of Y au’s coho mology for λ -rings. In Chapter 5, we extend the Ha r rison co chain complex of a comm utative alg ebra to get a bicom- plex whose co homology we deﬁne to be the Harriso n coho mology of a diag ram of a c ommut ative algebra. W e then apply this theory to the case of Ψ-r ings. In Cha pter 6, w e develop a cohomo logy for diagrams of algebra s in general, using co monads. First, w e ﬁx a small ca tegory I . A diagra m of algebra s is a functor I → Alg ( T ), where T is a mona d on sets. F or appr opriate T , we get a diagra m o f groups, a diagram o f Lie alg ebras, a diag ram of commutativ e ring s, etc. The a djoint pair Alg ( T ) / / Sets o o yields a comonad which we denote by G . W e c a n also consider the category I 0 , which ha s the same ob jects as I , but only the iden tit y morphisms. The inclusion I 0 ⊂ I yields the functor Sets I → Sets I 0 which ha s a left adjoin t given by the left Ka n extensio n. W e also have the pair of adjoint functors Alg ( T ) I / / Sets I o o which comes from the adjoint pair Alg ( T ) / / Sets o o . By putting these pa irs together, we g et another adjoint pair Alg ( T ) I / / Sets I 0 o o . This adjoint pair yields a como nad which we denote by G I . W e ca n then take the coho mo logy asso ciated to the como nad G I . Now we hav e b o th a globa l cohomolog y , H ∗ G I ( A, M ), and a lo ca l cohomolog y , H ∗ G ( A ( i ) , M ( i )). Our main result is that there exists a lo cal to global spectr al sequence connecting the tw o: E pq 2 = H p B W ( I , H q ( A, M )) ⇒ H p + q G I ( A, M ) , where H p B W ( I , H q ( A, M )) denotes the B aues-Wirsching cohomolog y of the small categor y I with co eﬃcients in the natural system H q ( A, M ) on I whose v alue on ( α : i → j ) is given b y H q G ( A ( i ) , α ∗ M ( j )). In Cha pter 7, we apply our theory from Chapter 6 to the ca s e of Ψ -rings. A Ψ- ring can be considered as a diagram o f a commutativ e ring, so w e can apply o ur results to g et a cohomolog y for Ψ-ring s. W e also deﬁne the co homology of λ -r ings using c o monads. W e note that there are homomorphisms co nnecting the cohomolog y of λ -rings, the cohomo logy of the asso ciated Ψ-ring s and the Andr´ e-Q uillen cohomology of the under lying commutativ e rings. 6 The la st Cha pter lo ok s at applications of the earlier develop ed theo ry . Our main a pplica tion is in a lgebraic to po logy . F or any top o logical spa ce X such that K 1 ( X ) = 0, there ex ists a homo- morphism natural in X , τ : π 2 n − 1 ( X ) → E xt Ψ ( K ( X ) , e K ( S 2 n )). W e s how that the cohomolo gy of λ -ring s and Ψ- r ings can b e used to prove the classica l result of Adams . W e also show that the Ψ-ring cohomo logy of K ( S 2 n ) is related to the stable homotopy groups of spheres v ia the natur a l transformatio n τ . 7 Chapter 2 Homological a l gebra 2.1 Category theory 2.1.1 Ab elian categories The ma terial in this section can be found in man y textb o oks , including [16] and [19]. Before w e int ro duce ab elian c a tegories , we start by deﬁning the notion of an additive category . An additive c ate go ry A is a category such that the following holds: 1. for every pair of o b jects X and Y in A , the hom-s e t Hom A ( X, Y ) has the structure o f an ab elian gro up such tha t morphism co mpo sition distributes ov er addition. 2. A has a zero ob ject (an ob ject which is b oth initia l and terminal). 3. for every pair of ob jects X and Y in A , their pro duct X × Y exists. An ab elia n catego r y is deﬁned in terms o f kernels and co kernels, so ﬁr st we will r e c all some other basic deﬁnitions from ca tegory theory . In a category C , a mor phism m : X → Y is called a monomorphism if fo r a ll mor phisms f 1 , f 2 : V → X where m ◦ f 1 = m ◦ f 2 we have f 1 = f 2 . A mo rphism e : Y → X is called a n epimorphi sm if for all morphisms g 1 , g 2 : X → V where g 1 ◦ e = g 2 ◦ e we ha v e g 1 = g 2 . In a n additive categ o ry A , a kernel of a morphism f : X → Y is deﬁned to b e a map i : X ′ → X such that f ◦ i = 0 and for any morphism g : Z → X such tha t f ◦ g = 0 there ex ists a unique morphism g ′ : Z → X ′ such that i ◦ g ′ = g . Z g ′ ~ ~ g   0 @ @ @ @ @ @ @ @ X ′ i / / X f / / Y Dually , in an additiv e category A , a c okernel of a morphism f : X → Y is deﬁned to be a map e : Y → Y ′ such that e ◦ f = 0 and for any mo rphism g : Y → Z such tha t g ◦ f = 0 there exists a unique morphism g ′ : Y ′ → Z such that g ′ ◦ e = g . Z X 0 > > ~ ~ ~ ~ ~ ~ ~ ~ f / / Y g O O e / / Y ′ g ′ ` ` An ab elia n c ate gory A is an additive category such that the following holds: 1. every morphism in A has a kernel and cokernel. 2. every monomo r phism in A is the kernel of its cokernel. 3. every epimorphis m in A is the cokernel of its kernel. 8 The basic example of a n abelia n category is the categ ory of abelian gro ups, deno ted b y Ab . In the category Ab , the ob jects are Abe lian groups, and the morphisms ar e abelian gr oup homomor- phisms. In genera l, mo dule categ ories whic h app ear thro ug hout algebra , are ab elian categ o ries. If I is a small catego ry and A is an ab elian category then the categor y of functors A I as a lso an ab elian categor y . The catego ry of sets Sets and the ca tegory of groups Grp ar e not ab elian categorie s. How ev er, if G is a group then the categor y o f left (or r ight) G -mo dules, deno ted b y G − mod , is an a b e lian category . If R is a ring then the categor y of left (or right) R -mo dules, denoted by R − m od , is an ab elian catego ry . If R is a Ψ- ring then the category of Ψ-mo dules ov er R , denoted by R − m od Ψ , is an ab elia n catego ry . If R is a λ -ring then the catego ry of λ -mo dules ov er R , deno ted by R − mo d λ , is an ab elian categ ory . In an ab elia n catego ry A , a short exact se quenc e is a sequence 0 / / A α / / B β / / C / / 0 in which α is a mo nomorphism, β is an e pimo rphism and K er( β ) = Im( α ) . In an ab elia n catego ry A , a sequence . . . / / X n − 1 f n − 1 / / X n f n / / X n +1 / / . . . is said to b e exact at X n if Ker( f n ) = Im( f n − 1 ) . The sequence is said to be exact if it is exact at X n for all n ∈ Z . 2.1.2 Mo dules Let C be a (not necessarily ab elian) categor y with ﬁnite limits, and 1 denote a terminal ob ject in C . An ab elian gr oup obje ct o f C is an o b ject A tog ether with arrows m : A × A → A , i : A → A and z : 1 → A such that the following diagra ms comm ute. (asso ciativity of m ultiplication) A × A × A id A × m   m × id A / / A × A m   A × A m / / A (left and r ight units) A × 1 ∼ = $ $ J J J J J J J J J J id A × z / / A × A m   1 × A z × id A o o ∼ = z z t t t t t t t t t t A (left and r ight in v erses) A ( i,id A ) / /   A × A m   A   ( id A ,i ) o o 1 z / / A 1 z o o (commut ativity) A × A ( p 2 ,p 1 ) / / m " " F F F F F F F F F A × A m | | x x x x x x x x x A These diagra ms say that the arr ows satisfy the equations o f an ab elian g roup. 9 Let A, i, m, z and A ′ , m ′ , i ′ , z ′ be ab elian gr oup ob jects of C , a morphism of a b elia n gr oup obje cts is an arr ow f : A → A ′ such that the following diagram commutes. A × A m / / f × f   A f   A ′ × A ′ m ′ / / A ′ W e denote the categor y of ab e lian group o b jects of C by Ab ( C ). Let A be an y ob ject of the category C . The slic e c ate gory of ob jects of C ov er A , denoted by C / A , has as ob jects the arr ows of C w ith target A . Given tw o o b jects f : B → A and g : C → A of C / A , an a rrow of C / A fro m f to g is a n arrow h : B → C which makes the following diag ram commute. B h / / f   @ @ @ @ @ @ @ C g   ~ ~ ~ ~ ~ ~ ~ A Deﬁnition 2. 1. Let A b e a n ob ject in a categor y C . An A - mo dule is deﬁned to b e an ab elian group ob ject in the category C / A , A − mod := Ab ( C / A ) . The categ ory A − mo d is usually a n ab elian ca tegory . Deﬁnition 2.2. Let p : Y → A be an ob ject and q : Z → A b e a n ab elian gr oup ob ject o f C / A , then we deﬁne the ab elian g roup of p -derivations , denoted Der( Y , Z ), to b e Der( Y , Z ) := Hom C / A ( p, q ) . 2.2 Cohomology The concepts of complexes and (co)homology b eg an in algebraic topo logy with simplicial and sin- gular (co)homology . The metho ds of algebraic topo logy ha ve b een applied extensiv ely throughout pure algebra , a nd hav e initia ted many developments. Co mplex es a re the bas ic to ols of homolo gical algebra a nd provide us with a wa y of co mputing (co )homology . The following deﬁnitions can b e found in [17] a nd [5]. A c o ch ain c omplex ( C, δ ) of o b jects of an ab elian c a tegory A is a fa mily { C n , δ n } n ∈ Z of ob jects C n ∈ obj ( A ) and morphisms (called the c ob oundary maps or diﬀer ent ial maps ) δ n : C n → C n +1 such that δ n +1 ◦ δ n = 0 for a ll n ∈ Z . · · · / / C n − 2 δ n − 2 / / C n − 1 δ n − 1 / / C n δ n / / C n +1 δ n +1 / / C n +2 / / · · · The last condition is eq uiv ale nt to saying that Im( δ n ) ⊆ Ker( δ n +1 ) for a ll n ∈ Z . Hence, one can deﬁne the c oho molo gy o f C denoted by H ∗ ( C ), H ∗ ( C ) = { H n ( C ) } n ∈ Z where H n ( C ) = Ker( δ n ) Im( δ n − 1 ) . H n ( C ) is called the n th -cohomolo gy of C . An n - c ob oundary is an element of Im( δ n − 1 ). An n -c o cycle is an element of Ker( δ n ). Let ( C, δ ) and ( C ⋄ , δ ⋄ ) b e tw o co chain complexes of a n a b elia n catego r y A . A c o chain map f : ( C , δ ) → ( C ⋄ , δ ⋄ ) is a family of mor phisms { f n : C n → C n ⋄ } n ∈ Z such that δ n ⋄ ◦ f n = f n +1 ◦ δ n for all n ∈ Z . The last condition is eq uiv ale n t to saying the following diagra m commutes. · · · / / C n − 2 f n − 2   δ n − 2 / / C n − 1 f n − 1   δ n − 1 / / C n f n   δ n / / C n +1 f n +1   δ n +1 / / C n +2 f n +2   / / · · · · · · / / C n − 2 ⋄ δ n − 2 ⋄ / / C n − 1 ⋄ δ n − 1 ⋄ / / C n ⋄ δ n ⋄ / / C n +1 ⋄ δ n +1 ⋄ / / C n +2 ⋄ / / · · · 10 A cochain ma p f : ( C, δ ) → ( C ⋄ , δ ⋄ ) induces homomorphisms H n ( f ) : H n ( C ) → H n ( C ⋄ ). This makes e a ch H n int o a functor. A c o chain bic omplex o f ob jects of an ab elian catego ry A is a family { C p,q , δ p,q , ∂ p,q } p,q ∈ Z of o b jects C p,q ∈ obj ( A ) and morphisms δ p,q : C p,q → C p +1 ,q and ∂ p,q : C p,q → C p,q +1 such tha t δ p +1 ,q ◦ δ p,q = 0 a nd ∂ p,q +1 ◦ ∂ p,q = 0 a nd a lso ∂ p +1 ,q δ p,q + δ p,q +1 ∂ p,q = 0 for all p, q ∈ Z . It is useful to visua lise a co chain bicomplex as a la ttice . . . . . . . . . . . . / / C p − 1 ,q +1 O O δ p − 1 ,q +1 / / C p,q +1 O O δ p,q +1 / / C p +1 ,q +1 O O / / . . . . . . / / C p − 1 ,q ∂ p − 1 ,q O O δ p − 1 ,q / / C p,q ∂ p,q O O δ p,q / / C p +1 ,q ∂ p +1 ,q O O / / . . . . . . / / C p − 1 ,q − 1 ∂ p − 1 ,q − 1 O O δ p − 1 ,q − 1 / / C p,q − 1 ∂ p,q − 1 O O δ p,q − 1 / / C p +1 ,q − 1 ∂ p +1 ,q − 1 O O / / . . . . . . O O . . . O O . . . O O where each row ( C ∗ ,q , δ ∗ ,q ) and each column ( C p, ∗ , ∂ p, ∗ ) is a co chain complex and each squar e anticomm utes. The total c omplexes T ot ( C ) = T ot Q ( C ) and T ot L ( C ) of a co chain bicomplex C ar e given b y T ot Q ( C ) n = Y p + q = n C p,q and T ot L ( C ) n = M p + q = n C p,q . The cob oundar y ma ps a re given by d = δ + ∂ . W e note that T ot Q ( C ) = T ot L ( C ) if C is bo unded, esp ecially if C is a ﬁr st quadrant bicomplex. Prop ositio n 2.3. If C is a ﬁrst quadr ant bic omplex then we have t he fo l lowing c onver gent sp e ctr al se quenc e E p,q 2 = H p h H q v ( C ) ⇒ H p + q ( T ot ( C )) , wher e H ∗ h denotes the horizontal c ohomolo gy, and H ∗ v denotes the vertic al c ohomolo gy. 2.3 Classical deriv ed functors A sta ndard method of c omputing classical deriv ed functors betw een abelia n c ategories is to take a resolution, apply the functor, then take the (co)homology of the resulting co mplex. The following material can b e found in [5], [19] a nd [17]. 2.3.1 Pro jective and injectiv e ob ject s An ob ject P of an ab elian categor y A is pr oje ctive if for any epimo rphism e : A ։ B and any morphism f : P → B ther e exists a morphism g : P → A s uch that f = e ◦ g , in other words, if the following diagram commut es. P f   g   A e / / / / B / / 0 An ob ject Q of an abelia n ca tegory A is inje ctive if for any monomorphism m : A ֒ → B and any morphism f : A → Q there exists a mor phism g : B → Q such that f = g ◦ m , in other words, 11 if the following diagram commutes. 0 / / A f     m / / B g   Q An ob ject P is pro jective if and only if Hom A ( P, − ) : A → Ab is an exact functor. In other words, if a nd o nly if for any ex a ct sequence 0 → A → B → C → 0 in A it follows that the following sequence of gr oups 0 / / Hom A ( P, A ) / / Hom A ( P, B ) / / Hom A ( P, C ) / / 0 is also exa ct. An ob ject Q is injective if and only if Hom A ( − , Q ) : A → Ab is an exact functor . In o ther words, if a nd o nly if for any ex a ct sequence 0 → A → B → C → 0 in A it follows that the following sequence of gr oups 0 / / Hom A ( C, Q ) / / Hom A ( B , Q ) / / Hom A ( A, Q ) / / 0 is also exa ct. 2.3.2 Pro jective and injectiv e resolutions Let A b e an o b ject of an ab elian ca tegory A . A pr oje ctive r esolution of A is a complex P , whe r e P i = 0 for all i < 0 and P j is pro jective for all j ≥ 0, to gether with a morphis m ǫ : P 0 → A called the augmentation such that the augmented complex . . . / / P 2 ∂ / / P 1 ∂ / / P 0 ǫ / / A / / 0 is exact. Let A b e an ob ject of an ab elia n catego r y A . An inje ct ive r esolut ion o f A is a co mplex Q , where Q i = 0 fo r all i < 0 and Q j is injective for all j ≥ 0, together with a morphism ǫ : A → Q 0 called the augment ation such that the aug ment ed complex 0 / / A ǫ / / Q 0 δ / / Q 1 δ / / Q 2 / / . . . is exact. An a b elia n ca tegory A is sa id to ha v e enough pr oje ctives if for ev ery ob ject A of A , there exists a pro jective ob ject P o f A and a n epimorphism e : P → A . An ab elian categ ory A is said to have enough inje ctives if for every ob ject A of A , ther e exists an injective ob ject Q of A and a monomo rphism m : A → Q . 2.3.3 Righ t deriv ed functors Let A , B be a be lia n categorie s , where A has enough injectiv es. If F : A → B is a cov a r iant left exact functor, then we can construct the right derive d fun ct ors o f F , denoted by R n F : A → B for n ≥ 0. If A is an ob ject o f A , and Q is an injective reso lution of A , we deﬁne R n F ( A ) := H n ( F ( Q )) . Let A , B be ab elian categ ories, where A has enoug h pro jectives. If G : A → B is a co n trav ar ia nt left exact functor, then w e can construc t the right derive d fun ctors of G , denoted by R n G : A → B for n ≥ 0. If A is an ob ject o f A , and P is a pro jective resolution of A , we deﬁne R n G ( A ) := H n ( G ( P )) . It is known that the functors R n F ( A ) and R n G ( A ) are indep endent of the choice o f pro jec- tive/injectiv e resolution chosen, hence it only dep ends on A . W e alwa ys get R 0 F ( A ) ∼ = F ( A ) and 12 R 0 G ( A ) ∼ = G ( A ). F urthermore, if A is injectiv e then R n F ( A ) = 0 for n > 0, and if A is pro jectiv e then R n G ( A ) = 0 for n > 0. Given a cov ariant left exact functor F : A → B b etw een the ab elian c a tegories A and B and given a shor t exact sequence 0 → A 1 → A 2 → A 3 → 0 in A , then there exists the following long exact sequence. 0 / / R 0 F ( A 1 ) / / R 0 F ( A 2 ) / / R 0 F ( A 3 ) / / R 1 F ( A 1 ) / / . . . . . . / / R n F ( A 1 ) / / R n F ( A 2 ) / / R n F ( A 3 ) / / R n +1 F ( A 1 ) / / . . . 2.3.4 Ext The main exa mple of rig h t derived fun ctors are the functor s Ext n . Let R b e a r ing , and let M , N be left R -mo dules. The functor F ( − ) = Hom R ( M , − ) : R − mod → Ab is a cov arian t additiv e left exac t functor, so w e can deﬁne its rig ht der ived funct ors Ext n R ( M , − ) = R n Hom R ( M , − ) : R − mod → Ab . Given a left R -mo dule M and a shor t exa ct sequence of left R -mo dules 0 → N ′ → N → N ′′ → 0 we obtain the following long exact sequence. 0 → Hom R ( M , N ′ ) → Hom R ( M , N ) → Hom R ( M , N ′′ ) → Ext 1 R ( M , N ′ ) → . . . . . . → Ext n R ( M , N ′ ) → Ext n R ( M , N ) → Ext n R ( M , N ′′ ) → Ext n +1 R ( M , N ′ ) → . . . Similarly Hom R ( − , N ) : R − mod → Ab is also a c o ntra v ar iant additiv e left exact functor, so we can deﬁne its right derived functor s Ext n R ( − , N ) = R n Hom R ( − , N ) : R − mod → Ab . Given a short exact sequence of left R -mo dules 0 → M ′ → M → M ′′ → 0 and a left R - mo dule N we obtain the following long exact sequence. 0 → Hom R ( M ′′ , N ) → Hom R ( M , N ) → Hom R ( M ′ , N ) → Ex t 1 R ( M ′′ , N ) → . . . . . . → Ext n R ( M ′′ , N ) → Ex t n R ( M , N ) → Ext n R ( M ′ , N ) → Ex t n +1 R ( M ′′ , N ) → . . . 2.4 Comonad c ohomology Cartan and Eilenber g uniﬁed the cohomo lo gy theories of gro ups, Lie algebras and ass o ciativ e algebras b y des c ribing them as Ext groups in the appropr iate ab elian categorie s . Unfortunately , this approach do es not work in all ca teg ories, for example in the catego ry of co mmutative algebras . The right appr oach is the co mo nad cohomolo g y using simplicia l methods . This materia l ca n b e found in [3] and [19]. 2.4.1 Monads and comonads A monad T = ( T , η , µ ) in any categor y C cons ists of an endo functor T : C → C tog ether with tw o natural transfor mations: η : I d C → T , µ : T ◦ T = T 2 → T such that the following diagra ms commute. T 3 T µ / / µT   T 2 µ   T 2 µ / / T 13 I d C T ηT / / C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C T 2 µ   T I d C T η o o { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { T The natural tra nsformation η is called the unit, a nd the natura l tra nsformation µ is ca lled the m ultiplication. The diag rams are ca lle d the asso c ia tivity , left unit and r ight unit la ws. A c omonad G = ( G, ε, δ ) in an y categ o ry C consists of an e ndo functor G : C → C together with t wo natural tr ansformatio ns : ε : G → I d C , δ : G → G 2 such that the following diagr ams commute. G δ / / δ   G 2 Gδ   G 2 δG / / G 3 G δ   I d C G { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { G 2 εG o o Gε / / GI d C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A pa ir of functors L : C → B and R : B → C are adjoint if for all ob jects A in C and B in B there exists a na tural bijection Hom B ( L ( A ) , B ) ∼ = Hom C ( A, R ( B )) . Natural means that for all f : A → A ′ in C and g : B → B ′ in B the follo wing diagram commutes. Hom B ( L ( A ′ ) , B ) ∼ =   Lf ∗ / / Hom B ( L ( A ) , B ) ∼ =   g ∗ / / Hom B ( L ( A ) , B ′ ) ∼ =   Hom C ( A ′ , R ( B )) f ∗ / / Hom C ( A, R ( B )) Rg ∗ / / Hom C ( A, R ( B ′ )) W e say that L is the left adjoint o f R , a nd R is the right adj oint of L . Let C L / / B R o o be an adjoint pa ir of functors with adjunction morphisms η : I d → RL and µ : LR → I d . Then T = ( RL , η , RµL ) is a monad on C and G = ( LR, µ, Lη R ) is a comonad on B . Example 2.4 . L et U : Grp → Sets take a gr oup t o the set of its elements for getting the gr oup structur e, and t ake gr oup morphi sms to functions b etwe en sets. The left adjoint functor t o U , is the functor L : Sets → Grp t aking a set to the fr e e gr oup gener ate d by the set. The functor T = U L : Sets → Sets gives rise to a monad and the functor G = LU : Grp → Grp gives rise t o a c omonad . Let G b e a c o monad on C . A morphism f : X → Y in C is called a G -epimorphi sm if the map Hom C ( G ( Z ) , X ) → Hom C ( G ( Z ) , Y ) is surjective for all Z . W e require the f ollowing us eful lemma. Lemma 2.5. F or al l obje cts X in C the morphism GX ε X / / X is a G -epimorphism. 14 Pr o of. F or a ny map h : GZ → X , we wish to ﬁnd a map f : GZ → GX such that f ε X = h . W e deﬁne f via the following comm uting diagra m. G ( GZ ) G ( h ) / / G ( X ) ε X / / X GZ δ ( Z ) O O f ; ; x x x x x x x x x x x x x x x x x x Now we need to chec k that ε X ◦ f = h . By the naturality of ε , the following diagram co mmu tes. GX ε X / / X G ( GZ ) G ( h ) O O ε GZ / / GZ h O O GZ δ ( Z ) O O id GZ < < y y y y y y y y y y y y y y y y y y f J J So ε X is a G -epimorphism. An o b ject P of C is called G -pr oj e ctive if for a ny G - epimorphism f : X → Y , the ma p Hom C ( P, X ) → Hom C ( P, Y ) is surjective. Example 2.6. F or any obje ct X in C the obje ct G ( X ) is G -pr oj e ctive. Lemma 2.7. The c opr o duct of G -pr oj e ctive obje cts is G - pr oje ctive. Pr o of. Let P = ` i P i where P i is G - pro jective for all i . F o r a map f : X → Y , one a pplies the functor s Hom C ( P, − ) and Hom C ( P i , − ) to get the ma ps f ∗ : Hom C ( P, X ) → Hom C ( P, Y ) and f i ∗ : Hom C ( P i , X ) → Hom C ( P i , Y ). If f is a G -epimorphism then f i ∗ is surjective for all i . Using the well-known lemma Hom C ( ` i P i , Z ) ∼ = Q i Hom C ( P i , Z ) one gets that if f is a G -epimorphism then f ∗ ∼ = Q i f i ∗ is surjective. Hence P is G -pro jectiv e if P i is G -pro jective for all i . Lemma 2. 8. An obje ct P is G -pr oj e ctive if and only if P is a r etr ac t of an obje ct of the form G ( Z ) . Pr o of. A r etract of a surjective map is surjective, so it is suﬃcient to consider the case P = G ( Z ), which is clear fro m the deﬁnition o f a G -e pimorphism. 2.4.2 Simplicial metho ds Deﬁnition 2.9. A simplicial obje ct in a categ ory C is a sequence of o b jects X 0 , X 1 , . . . , X n , . . . together with t wo double-indexed families of ar rows in C . The fac e op er ators are a rrows d i n : X n → X n − 1 for 0 ≤ i ≤ n and 1 ≤ n < ∞ . The de gener acy op er ators are ar r ows s i n : X n → X n +1 for 0 ≤ i ≤ n a nd 0 ≤ n < ∞ . The face op era to rs and degenera cy op erator s satisfy the following conditions: d i n ◦ d j n +1 = d j − 1 n ◦ d i n +1 if 0 ≤ i < j ≤ n + 1 s j n ◦ s i n − 1 = s i n ◦ s j − 1 n − 1 if 0 ≤ i < j ≤ n d i n +1 ◦ s j n =    s j − 1 n − 1 ◦ d i n , if 0 ≤ i < j ≤ n ; 1 , if 0 ≤ i = j ≤ n o r 0 ≤ i − 1 = j < n ; s j n − 1 ◦ d i − 1 n , if 0 < j < i − 1 ≤ n . 15 An augmente d simplicia l obje ct in the categ o ry C is a simplicia l ob ject X ∗ together with another ob ject X − 1 and an arr ow ǫ : X 0 → X − 1 such that ǫ ◦ d 0 1 = ǫ ◦ d 1 1 . An a ug ment ed simplicial ob ject X ∗ → X − 1 is called c ontr actible if for each n ≥ − 1 there exists a map s n : X n → X n +1 such that d 0 ◦ s = 1 and d i ◦ s = s ◦ d i − 1 for 0 < i ≤ n a nd s 0 ◦ s = s ◦ s and s i ◦ s = s ◦ s i − 1 for 0 < i ≤ n + 1 . Let X ∗ be a simplicial ob ject in an additive category B . The ass o ciated c hain complex to X ∗ , denoted by C ( X ∗ ), is the complex . . . / / X n +1 d / / X n d / / X n − 1 d / / . . . d / / X 0 / / 0 where the b ounda r y maps d = P n i =0 ( − 1) i d i : X n → X n − 1 . Prop ositio n 2.10. If X ∗ → X − 1 is a c ontr actible augmente d simplicial obje ct in an ab elian c ate gory A , then the asso ciate d chain c omplex C ( X ∗ ) is c ontr actible. 2.4.3 Comonad cohomology Let G b e a co mo nad on a category C . F o r a ny o b ject A in C , we g et a functoria l augmented simplicial ob ject which we denote by G ( A ) ∗ → A . The o b ject of G ∗ ( A ) in degree n is G n +1 ( A ). W e deﬁne the face and degener acy op erators by ϕ i = G i εG n − i : G n +1 ( A ) → G n ( A ) , σ i = G i δ G n − i : G n +1 ( A ) → G n +2 ( A ) , for 0 ≤ i ≤ n . The augmenting ma p is given by ε . . . . / / . . . / / G n A / / . . . / / G n − 1 A / / . . . / / . . . Gε / / εG / / GA ε / / A W e call G ( A ) ∗ the G c omonad r esolution of A . Let E : C → M b e a c ontra v ar iant functor where M is a n ab elian ca tegory . The comonad cohomolog y of an ob ject A with co eﬃcients in E is H ∗ G ( A, E ) where H n G ( A, E ) := H n ( C ( E ( G ∗ ( A )))) . By deﬁnition, H ∗ G ( A, E ) is the cohomolo gy of the asso ciated cochain complex 0 / / E ( G ( A )) / / E ( G 2 ( A )) / / . . . If M ∈ A - mod , then we deﬁne the cohomolo gy of A with coeﬃcie nts in M to b e the comona d cohomolog y of A with co eﬃcients in Der( − , M ) : C → Ab . H n G ( A, M ) := H n G ( A, Der( − , M )) . Lemma 2.11. H 0 G ( A, M ) ∼ = Der( A, M ) for al l A . Lemma 2.12. If A is G -pr oje ctive then H n G ( A, M ) = 0 for n > 0 . Pr o of. F rom lemma 2.8, it is suﬃcient to chec k the case where A = G ( Z ). There exists a co ntract- ing homotopy s n : G n +2 → G n +3 for n ≥ − 1 given by s n = G n +1 δ. W e get that ǫ s − 1 = id , ϕ n +1 s n = id , ϕ 0 s 0 = s − 1 ǫ , and ϕ i s n = s n − 1 ϕ i for a ll 0 ≤ i ≤ n . It follows that H n G ( G ( Z ) , M ) = 0 , for n > 0 . 16 2.4.4 Andr´ e-Quillen cohomology In 1 967, M. Andr´ e and D. Quillen [18] independently intro duced a (co )homology theo r y for c om- m utative algebras. This theory now go es by the name of Andr´ e- Q uillen cohomology . Fix a co mm utative ring k and co nsider the catego ry Commalg of commutativ e k -a lg ebras. The forgetful functor U : Commalg → Sets has a left adjoint which ta kes a se t X to the po lynomial alg e bra k [ X ] on X . This adjoint pair gives us a comonad G on Comma lg . Let R b e a co mm utative k -algebra s, and M ∈ R − mod . W e deﬁne the Andr´ e-Quillen coho mo logy of R with co eﬃcients in M to b e como nad cohomolo gy of R with co eﬃcien ts in Der k ( − , M ), H n AQ ( R/k , M ) := H n G ( R, Der k ( − , M )) . Note that Der k ( − , M )) is a functor fr om the category of co mmutative k -algebras Comma lg to the category of ab elia n groups Ab . An extension of R by M is an exact se q uence 0 / / M α / / X β / / R / / 0 where X is a commutativ e k - a lgebra, the map β is a commut ative k - algebra homomor phism, the map α is a k -mo dule ho momorphism and xα ( m ) = α ( β ( x ) m ) for all x ∈ X and a ll m ∈ M . The map α ident iﬁes M with an ideal of square- zero in X . Two extensio ns X, X ′ with R and M ﬁxed are e quivalent if there ex ists a k -a lgebra homomor- phism φ : X → X such that the following diagram commutes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X / / R / / 0 W e denote the se t of equiv alence classes o f ex tens ions of R by M by E xtalg k ( R, M ). Prop ositio n 2.13 . 1. H 0 AQ ( R/k , M ) ∼ = Der k ( R, M ) . 2. If R is a fr e e c ommu tative algebr a then H n AQ ( R/k , M ) = 0 for n > 0 . 3. H 1 AQ ( R/k , M ) ∼ = E xtalg k ( R, M ) . 2.5 Harrison cohomology of comm utat iv e algebras In 1962, D.K. Harrison in tro duced a cohomology of comm utative alg e bras. The Harr ison co mplex is a subc o mplex of the Ho chsc hild complex in the ca s e of co mm utative algebr as. The Har rison complex consists of the linear functions which v anish o n the shuﬄes. The Harriso n co homology is isomorphic to the comona d cohomology for a commutativ e algebra o ver a ﬁeld of c haracter istic 0 , how ev er, there is a shift of one dimensio n. The following material can b e found in [1 5]. 2.5.1 Ho chsc hild cohomology Let k be a ring, R be an as s o ciative k -algebra and M b e an R − R -bimo dule. All the tensor pro ducts in this section are ov er the g round ring k . The Ho chsc hild co chain complex o f R with co eﬃcients in M is g iven b y C n H H ( R, M ) = Hom R e ( R ⊗ n , M ) , for n ≥ 0 and R e = R ⊗ R op . The cob oundar y maps δ n : C n H H ( R, M ) → C n +1 H H ( R, M ) are given by δ n ( f )( r 0 , . . . , r n ) = r 0 f ( r 1 , . . . , r n ) + n − 1 X i =0 ( − 1) i +1 f ( r 0 , . . . , r i r i +1 , . . . , r n ) + ( − 1) n +1 f ( r 0 , . . . , r n − 1 ) r n . 17 W e can then take the cohomolo gy of the resulting complex to g et the Ho chsc hild cohomolo gy which we denote by H H n ( R, M ). W e get that H H n ( R, M ) ∼ = R n Hom R e ( R, M ) ∼ = Ext n R e ( R, M ) . 2.5.2 Harrison Cohomology Let S n be the sy mmetric group which acts on the s et { 1 , . . . , n } . A (p,q)-shuﬄe is a p ermutation σ in S p + q such that: σ (1) < σ (2) < . . . < σ ( p ) and σ ( p + 1) < σ (2) < . . . < σ ( p + q ) . F or any k -algebra A a nd M ∈ A − m od , we let S n act on the left on C H H n ( A, M ) = M ⊗ A ⊗ n by: σ · ( m, a 1 , . . . , a n ) = ( m, a σ − 1 (1) . . . , a σ − 1 ( n ) ) . Let A ′ be another k -algebr a, M ′ ∈ A ′ − mod . The s hu ﬄe pr o duct : − × − = sh pq : C H H p ( A, M ) ⊗ C H H q ( A ′ , M ′ ) → C H H p + q ( A ⊗ A ′ , M ⊗ M ′ ) , is deﬁned by the following formula: ( m, a 1 , . . . , a p ) × ( m ′ , a ′ 1 , . . . , a ′ q ) = X σ sg n ( σ ) σ · ( m ⊗ m ′ , a 1 ⊗ 1 , . . . , a p ⊗ 1 , 1 ⊗ a ′ 1 , . . . , 1 ⊗ a ′ q ) . Prop ositio n 2.14 . The Ho chschild b oundary b is a gr ade d derivation for the shuﬄe pr o duct b ( x × y ) = b ( x ) × y + ( − 1 ) | x | x × b ( y ) , x ∈ C H H p ( A, M ) , y ∈ C H H q ( A ′ , M ′ ) . wher e the Ho ch schild b oundary b : C H H n ( A, M ) → C H H n − 1 ( A, M ) is given by: b ( m, a 1 , . . . , a n ) =( ma 1 , a 2 , . . . , a n ) + n − 1 X i =1 ( − 1) i ( m, a 1 , . . . , a i a i +1 , . . . , a n ) + ( − 1) n ( a n m, a 1 , . . . , a n − 1 ) . Assume that A is comm utative and M is s ymmetric (symmetric mea ns that am = m a for all a ∈ A a nd m ∈ M ). The pr o duct map µ : A ⊗ A → A is a k - a lgebra homo morphism, and µ ′ : A ⊗ M → M is a homomorphism of bimo dules. Co mpo sition of the shuﬄe ma p with µ ⊗ µ ′ allows us to deﬁne the inn er shuﬄe map − × − = sh pq : C H H p ( A, A ) ⊗ C H H q ( A, M ) → C H H p + q ( A, M ) , given by the for mula ( a 0 , a 1 , . . . , a p ) × ( m, a p +1 , . . . , a p + q ) = X σ =( p,q ) − shuf f le sg n ( σ ) σ · ( a 0 m, a 1 , . . . , a p + q ) . W e let J denote L n> 0 C H H n ( A, A ). Note that J ⊂ C H H ∗ ( A, A ). W e deﬁne the Harr ison chain complex to b e the quotient C H ar r ∗ ( A, M ) = C H H ∗ ( A, M ) /J.C H H ∗ ( A, M ). Note that C n H H ( A, M ) = Hom A e ( A ⊗ n , M ) ∼ = Hom A ⊗ A e ( A ⊗ A ⊗ n , M ) = Hom A ⊗ A e ( C H H n ( A, A ) , M ) . W e deﬁne the Harriso n co chain complex by taking C ∗ H ar r ( A, M ) := Hom A ⊗ A e ( C H ar r ∗ ( A, A ) , M ) . 18 F or example C 0 H ar r ( A, M ) = M , C 1 H ar r ( A, M ) = C 1 H H ( A, M ) , C 2 H ar r ( A, M ) = { f ∈ C 2 H H ( A, M ) | f ( x, y ) = f ( y , x ) } , C 3 H ar r ( A, M ) = { f ∈ C 3 H H ( A, M ) | f ( x, y , z ) − f ( y , x, z ) + f ( y , z , x ) = 0 . } W e deﬁne the Harrison cohomology of A with co eﬃcien ts in M to b e the cohomolo gy of the Harrison co chain complex. H ar r n ( A, M ) := H n ( C ∗ H ar r ( A, M )) . Lemma 2.15. H ar r 1 ( A, M ) ∼ = Der( A, M ) . An additively s plit extension o f A by M is a n extensio n of A by M 0 / / M q / / X p / / A / / 0 where there exis ts s : A → X which is an additive section of p . Two a dditively split extensions ( X ) , ( X ) with A, M ﬁx ed are sa id to b e e quivale nt if there exists a homomorphism of commutativ e a lg ebras φ : X → X such that the following diagram comm utes. 0 / / M / / X / / φ   A / / 0 0 / / M / / X / / A / / 0 W e denote the set of equiv alence cla sses of additively split extensio ns o f A by M by AExt( A, M ). Lemma 2.16. H ar r 2 ( A, M ) ∼ = AExt( A, M ) . Pr o of. Giv en an a dditiv ely split ex tension of A by M 0 / / M q / / X p / / A / / 0 there is an additiv e homo morphism s : A → X which is a section of p . The section induces an a dditiv e isomorphism X ≈ A ⊕ M where mult iplication in X is given by ( a, m )( a ′ , m ′ ) = ( aa ′ , ma ′ + am ′ + f ( a, a ′ )) where the bilinear map f : A × A → M is given b y f ( a, a ′ ) = s ( a ) s ( a ′ ) − s ( aa ′ ) . The map f is a 2-cocy cle. Given tw o additively split ex tens ions which are equiv alent, then the t w o 2-co cycles we get diﬀer by a 2-cob oundar y . A 2-co cycle is a map f : A × A → M . W e get an additively split extension of A by M g iven by taking the ex act sequence 0 / / M / / M ⊕ A / / A / / 0 where addition in M ⊕ A is given by ( m, a ) + ( m ′ , a ′ ) = ( m + m ′ , a + a ′ ) and multiplication is given by ( m, a )( m ′ , a ′ ) = ( a ′ m + am ′ + f ( a, a ′ ) , aa ′ ) . Given tw o 2- co cycles which diﬀer by a 2-cob ounda r y , then the tw o additively s plit extensions we get are equiv alent. A cr osse d mo dule co nsists of a commutativ e algebra C 0 , a C 0 -mo dule C 1 and a mo dule homo- morphism C 1 ρ / / C 0 , which s atisﬁes the pr op erty ρ ( c ) c ′ = cρ ( c ′ ) , 19 for c, c ′ ∈ C 1 . In o ther w ords, a crosse d mo dule is a chain algebra which is no n-trivial only in dimensions 0 and 1. Since C 2 = 0 the condition ρ ( c ) c ′ = cρ ( c ′ ) is equiv a le nt to the Leibnitz relation 0 = ρ ( cc ′ ) = ρ ( c ) c ′ − cρ ( c ′ ) . W e can deﬁne a pro duct by c ∗ c ′ := ρ ( c ) c ′ , for c, c ′ ∈ C 1 . This gives us a commutativ e algebra structure on C 1 and ρ : C 1 → C 0 is an algebr a homomorphism. Let ρ : C 1 → C 0 be a cross e d module. W e let M = Ker( ρ ) and A = Coker( ρ ). Then the image Im( ρ ) is a n ideal of C 0 , M C 1 = C 1 M = 0 a nd M ha s a well-deﬁned A -mo dule structure. W e say such a crossed module is a cross ed module ov er A with kernel M . A cr osse d ex t ension o f A by M is an exa ct sequence 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 where ρ : C 1 → C 0 is a crossed module, γ is an algebra homomorphism, and the mo dule s tructure on M coincides with the one induced from the cross e d module. A morphism b etw een tw o crossed extensions consists o f co mm utative algebra homomorphisms h 0 : C 0 → C 0 and h 1 : C 1 → C ′ 1 such that the following diagram comm utes: 0 / / M α / / C 1 h 1   ρ / / C 0 h 0   γ / / A / / 0 0 / / M α ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 Let C ross ( A, M ) denote the categ o ry of crossed modules over A with kernel M , a nd let π 0 C ross ( A, M ) de no te the c onnected comp onents of C r oss ( A, M ). Deﬁnition 2.17. An additively split cr osse d ext ension o f A by M is a cro ssed extension of A by 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 (2.1) such that all the arr ows in the exact sequenc e 2.1 are additively split. W e denote the connected comp onents of the catego ry of additively split crossed ex tensions ov er A with kernel M by π 0 AC ross ( A, M ). Lemma 2.18. If γ : C 0 → A is k -algebr a homomorphism then H ar r 2 ( γ : C 0 → A, M ) ∼ = π 0 AC ross ( γ : C 0 → A, M ) , wher e H ar r ∗ ( γ : C 0 → A, M ) and π 0 AC ross ( γ : C 0 → A, M ) ar e deﬁne d as fol lows. Consider the fol lowing short exact se quenc e of c o chain c omplexes: 0 / / C ∗ H ar r ( A, M ) γ ∗ / / / / C ∗ H ar r ( C 0 , M ) / / κ ∗ / / Coker ( γ ∗ ) / / 0 . We deﬁne the c o chain c omplex C ∗ H ar r ( γ : C 0 → A, M ) := Co ker ( γ ∗ ) . This al lows us t o deﬁne the r elativ e Harrison c ohomolo gy H ar r ∗ ( γ : C 0 → A, M ) := H ∗ ( C ∗ H ar r ( γ : C 0 → A, M )) . We let AC r oss ( γ : C 0 → A, M ) denote the c ate gory whose obje cts ar e the additively split cr osse d extensions of A by M 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 with γ : C 0 → A ﬁxe d. A morphisms b etwe en two of these cr osse d extensions c onsists of a morphism of cr osse d ex tensions with the map h 0 : C 0 → C 0 b ei ng the identity. 0 / / M α / / C 1 h 1   ρ / / C 0 γ / / A / / 0 0 / / M α ′ / / C ′ 1 ρ ′ / / C 0 γ / / A / / 0 Note that AC r oss ( γ : C 0 → A, M ) is a gr oup oid. 20 Pr o of. This pro of is very similiar to a pro o f g iven in [13] for the crosse d mo dules of Lie algebras . Given an y additively split crossed mo dule of A by M , 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 , we let V = Ker γ = Im ρ . There are k -linea r sections s : A → C 0 of γ and σ : V → C 1 of ρ : C 1 → V . W e deﬁne the map g : A ⊗ A → C 1 by: g ( a, b ) = σ ( s ( a ) s ( b ) − s ( ab )) . W e also deﬁne the map ω : C 0 → C 1 by: ω ( c ) = σ ( c − sγ ( c )) . By identifying M with Ker δ , we deﬁne the map f : C 0 ⊗ C 0 → M by: f ( c, c ′ ) = g ( γ ( c ) , γ ( c ′ )) + c ′ ω ( c ) + cω ( c ′ ) − ω ( c ) ∗ ω ( c ′ ) − ω ( cc ′ ) . Since g ( c, c ′ ) = g ( c ′ , c ), it follows that f ( c, c ′ ) = f ( c ′ , c ) a nd s o f ∈ C 2 H ar r ( C 0 , M ). W e deﬁne the map  ∈ C 3 H ar r ( A, M ) by:  ( x, y , z ) = s ( x ) g ( y , z ) − g ( xy , z ) + g ( x, y z ) − g ( y , x ) s ( z ) . Note that  v anishes o n the shuﬄes since g ( x, y ) = g ( y , x ). Consider the following comm uting diagra m. 0 / / C 2 H ar r ( A, M ) γ ∗ / /     C 2 H ar r ( C 0 , M ) κ ∗ / / δ   C 2 H ar r ( γ : C 0 → A, M ) / / δ   0 0 / / C 3 H ar r ( A, M ) γ ∗ / / C 3 H ar r ( C 0 , M ) κ ∗ / / C 3 H ar r ( γ : C 0 → A, M ) / / 0 A direct calculatio n shows that δ f = γ ∗  ∈ C 3 ( C 0 , M ). W e also have that δ κ ∗ f = κ ∗ δ f = κ ∗ γ ∗  = 0, this tells us that κ ∗ f is a co cycle. If we hav e tw o equiv alent a dditively split crosse d mo dules then we can choos e sections in such a wa y that the a sso ciated co cycles are the same. Therefore we ha ve a w ell-deﬁned map: AC ross ( γ : C 0 → A, M ) / / H 3 H ar r ( γ : C 0 → A, M ) . Inv ersely , assume we hav e a co cycle in C 2 H ar r ( γ : C 0 → A, M ) whic h we lift to a cochain f ∈ C 2 H ar r ( C 0 , M ). L et V = Ker γ . W e deﬁne C 1 = M × V as a mo dule over k with the following action of C 0 on C 1 : c ( m, v ) := ( cm + f ( c, v ) , cv ) . It is easy to chec k using the prop erties of f that this ac tio n is well deﬁned and tog ether with the map ρ : C 0 → C 1 given b y ρ ( m, v ) = v , we have an additiv ely split crossed mo dule of A b y M . Lemma 2.19. If k is a ﬁeld of char acteristic 0 then H ar r 3 ( A, M ) ∼ = π 0 AC ross ( A, M ) . Pr o of. F rom the deﬁnition of C ∗ H ar r ( γ : C 0 → A, M ) we get the long exact seq uence: . . . / / H ar r 2 ( A, M ) / / H ar r 2 ( C 0 , M ) / / H ar r 2 ( γ : C 0 → A, M ) / / H ar r 3 ( A, M ) / / . . . (2.2) Given an y additively split crossed mo dule in π 0 AC ross ( A, M ), 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 21 we can lift γ to get a map P 0 → A where P 0 is a p olynomial a lgebra. W e can then use a pullback to construct P 1 to get a crossed mo dule whe r e the following diagram comm utes: 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 0 / / M / / P 1 O O / / P 0 O O / / A / / 0 . Note that these tw o cr ossed mo dules are in the same co nnected comp onent of π 0 AC ross ( A, M ). By considering the s econd cros s ed mo dule in the lo ng exact sequence, we replace C 0 by P 0 to get the new exact s equence: 0 / / H ar r 2 ( γ : P 0 → A, M ) / / H ar r 3 ( A, M ) / / 0 (2.3) since H arr 2 ( P 0 , M ) = 0 and H ar r 3 ( P 0 , M ) = 0. The exact se quence 2.3 tells us that every element in H ar r 3 ( A, M ) comes from an element in H ar r 2 ( γ : P 0 → A, M ) and the previous lemma tells us that this comes from a c rossed mo dule in π 0 AC ross ( A, M ). Therefore the map π 0 AC ross ( A, M ) → H ar r 3 ( A, M ) is surjective. Assuming we hav e t w o crossed mo dules which g o to the same element in H arr 3 ( A, M ), 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 , (2.4) 0 / / M α ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 . (2.5) There exist mor phisms 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 0 / / M / / P 1 O O / / P 0 O O / / A / / 0 0 / / M / / P 2   / / P 0   / / A / / 0 0 / / M α ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 where P 0 is a p olynomial algebra a nd P 1 , P 2 are co nstructed via pullbacks. These give us tw o elements in H ar r 2 ( γ : P 0 → A, M ) whic h go to the same element in H ar r 3 ( A, M ). How ever the exact sequence 2.3 tells us that the t wo crosse d mo dules 2.4 and 2.5 hav e to go to the same element in H ar r 2 ( γ : P 0 → A, M ). The previous lemma tells us that the tw o cros sed mo dules 2 .4 and 2.5 go to the sa me element in AC r oss ( γ : C 0 → A, M ) whic h is a g r oup oid, so there is a map P 2 → P 1 which ma kes the following dia g ram commute: 0 / / M α / / C 1 ρ / / C 0 γ / / A / / 0 0 / / M / / P 1 O O / / P 0 O O / / A / / 0 0 / / M / / P 2 O O   / / P 0   / / A / / 0 0 / / M α ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 Therefore the t wo cr ossed mo dules 2.4 and 2.5 are in the s a me connected comp onent of π 0 AC ross ( A, M ) and the map π 0 AC ross ( A, M ) → H ar r 3 ( A, M ) is injective. 22 2.6 Baues-Wirsc hing cohomology The following materia l can b e found in [4]. A categ ory I is said to b e smal l if the collection of morphisms is a s e t. Consider a sma ll categ o ry I . The c ate gory of factorizations in I , deno ted by F I , is the category whose ob jects are the morphisms f , g , ... in I , a nd morphisms f → g a re pairs ( α, β ) of mor phisms in I such that the following diagram commutes. B α / / B ′ A f O O A ′ β o o g O O Comp osition in F I is given by ( α ′ , β ′ )( α, β ) = ( α ′ α, β β ′ ). A natur al system of ab elian gro ups on I is a functor D : F I → Ab . There exis ts a canonical functor F I → I op × I which takes f : A → B to the pair ( A, B ). This functor allows us to c o nsider a ny bifunctor D : I op × I → Ab as a natural system. Simila r ly , the pro jection I op × I → I gives us the functor F I → I which tak es f : A → B to B . This a llows us to consider a ny f unctor D : I → Ab as a natural system. F ollowing B aues-Wirsching [4], we deﬁne the co homology H ∗ B W ( I , D ) of I with co eﬃcients in the natural system D a s the cohomolog y of the co chain complex C ∗ B W ( I , D ) given by C n B W ( I , D ) = Y α 1 ...α n : i n → ... → i 0 D ( α 1 . . . α n ) , where the pr o duct is indexed over n -tuples of co mp os able morphisms a nd the cob oundar y map d : C n B W ( I , D ) → C n +1 B W ( I , D ) , is given b y ( d f )( α 1 . . . α n +1 ) =( α 1 ) ∗ f ( α 2 , . . . , α n +1 ) + n X j =1 ( − 1) j f ( α 1 , . . . , α j α j +1 , . . . , α n +1 ) + ( − 1) n +1 ( α n +1 ) ∗ f ( α 1 , . . . , α n ) . Lemma 2.20. L et i 0 ∈ I b e an initial obje ct and F : I → Ab a functor t hen H n B W ( I , F ) =  F ( i 0 ) for n = 0 0 for n > 0 . 23 Chapter 3 Ψ -rings 3.1 In tro duction In this c hapter, only the material in this section is already known and everything fr om section 3.2 onw ards is new and original materia l. Note that in a ll of the cited material, including [1], [14] and [20], what the a uthors call a Ψ -ring is what we call a sp ecial Ψ- ring. Also note that in o ur notation N do es no t include 0. λ -rings are co mplicated, and given a λ -ring it is o ften diﬃcult to prove it satisﬁes the λ -ring axioms. W e start by intro ducing a nother kind of ring, the Ψ-r ings, which are closely related to the λ -rings by the Adams op e r ations. The axioms for the Ψ-ring s are a lot simpler than those for the λ -rings. Deﬁnition 3.1. A Ψ -ring is a c ommut ative ring with ide ntit y , R , to gether with a sequence o f ring homomor phisms Ψ i : R → R , for i ∈ N , sa tisfying 1. Ψ 1 ( r ) = r , 2. Ψ i (Ψ j ( r )) = Ψ ij ( r ), for all r ∈ R , and i , j ∈ N . W e say that a Ψ-r ing R is sp e cia l if it also satisﬁe s the prop er t y Ψ p ( r ) ≡ r p mo d pR, for all primes p and r ∈ R . Example 3. 2. Any c ommu tative ring with identity, R , c an b e given a Ψ -ring structu r e by setting Ψ i : R → R to b e Ψ i ( r ) = r for al l r ∈ R and i ∈ N . Let R 1 , R 2 be Ψ -rings. A map of Ψ -rings is a ring ho mo morphism f : R 1 → R 2 , such that Ψ i ( f ( r )) = f (Ψ i ( r )) for all r ∈ R 1 and i ∈ N . The class of all Ψ - rings and maps of Ψ- rings form the catego ry of Ψ-rings , whic h w e deno te b y Ψ − rings . 3.2 Ψ -mo du les F or usual r ings, the modules provide us with the c o eﬃcien ts for the co ho mology . In this section we deﬁne the Ψ-modules for Ψ-rings which provide us with the coeﬃcients for the Ψ-ring cohomolog y . W e then use this to cr eate the Ψ-ana logue of some of the results fo r rings. Deﬁnition 3.3. W e say that M is a Ψ -mo dule ov er the Ψ -ring R if M is an R -mo dule tog ether with a seque nce of ab elian g roup homomorphisms ψ i : M → M , for i ∈ N , satisfying 1. ψ 1 ( m ) = m , 2. ψ i ( rm ) = Ψ i ( r ) ψ i ( m ), 3. ψ i ( ψ j ( m )) = ψ ij ( m ), 24 for all m ∈ M , r ∈ R , and i , j ∈ N . Let M , N b e t w o Ψ-mo dules ov er R . A map of Ψ-mo dules is a mo dule homomorphism f : M → N such that ψ i f ( m ) = f ψ i ( m ) for all m ∈ M and i ∈ N . W e let R − mod Ψ denote the categ o ry of all Ψ-mo dules over R . W e say that M is sp e cial if R is sp e cial and ψ p ( m ) ≡ 0 mo d pM , for all primes p and m ∈ M . Note that any Ψ-ring can b e consider ed a s a Ψ-mo dule ov er itself. Also note that if M is sp ecial, then ψ i ( m ) ≡ 0 mo d iM for all i ∈ N a nd m ∈ M . F or the rest of this c hapter, w e let R de no te a Ψ-ring and M ∈ R − mod Ψ . W e let R denote the underlying commutativ e ring of R , and we let M denote the underlying R -mo dule of M . Lemma 3.4. The set R ⋊ M with ( r , m ) + ( s, n ) = ( r + s, m + n ) , ( r , m )( s, n ) = ( r s, rn + ms ) , to gether with maps Ψ i : R ⋊ M → R ⋊ M for i ∈ N given by Ψ i ( r , m ) = (Ψ i ( r ) , ψ i ( m ) + ε i ( r )) , for a se quenc e of maps ε i : R → M for i ∈ N , is a Ψ -ring if and only if 1. ε 1 ( r ) = 0 , 2. ε i ( r + s ) = ε i ( r ) + ε i ( s ) , 3. ε i ( rs ) = Ψ i ( r ) ε i ( s ) + ε i ( r )Ψ i ( s ) , 4. ε ij ( r ) = ψ i ε j ( r ) + ε i Ψ j ( r ) , for al l r , s ∈ R , and i, j ∈ N . Pr o of of lemma. It is known that R ⋊ M is a commut ative ring with iden tit y . Hence it is suﬃcien t to chec k that Ψ i : R ⋊ M → R ⋊ M sa tisﬁes the Ψ- ring axioms. 1. Ψ 1 ( r , m ) = (Ψ 1 ( r ) , ψ 1 ( m ) + ε 1 ( r )) = ( r , m + ε 1 ( r )) Hence Ψ 1 ( r , m ) = ( r , m ) if a nd only if ε 1 ( r ) = 0. 2. Ψ i (( r , m ) + ( s, n )) = (Ψ i ( r ) + Ψ i ( s ) , ψ i ( m ) + ψ i ( n ) + ε i ( r + s )) Ψ i ( r , m ) + Ψ i ( s, n ) = (Ψ i ( r ) + Ψ i ( s ) , ψ i ( m ) + ψ i ( n ) + ε i ( r ) + ε i ( s )) Hence Ψ i (( r , m ) + ( s, n )) = Ψ i ( r , m ) + Ψ i ( s, n ) if a nd only if ε i ( r + s ) = ε i ( r ) + ε i ( s ). 3. Ψ i (( r , m )( s, n )) = (Ψ i ( rs ) , ψ i ( rn + ms ) + ε i ( rs )) Ψ i ( r , m )Ψ i ( s, n ) = (Ψ i ( rs ) , ψ i ( rn + ms ) + Ψ i ( r ) ε i ( s ) + ε i ( r )Ψ i ( s )) Hence Ψ i (( r , m )( s, n )) = Ψ i ( r , m )Ψ i ( s, n ) if a nd only if ε i ( rs ) = Ψ i ( r ) ε i ( s ) + ε i ( r )Ψ i ( s ). 4. Ψ i Ψ j ( r , m ) = (Ψ i Ψ j ( r ) , ψ i ψ j ( m ) + ψ i ε j ( r ) + ε i Ψ j ( r )) Ψ ij ( r , m ) = (Ψ i Ψ j ( r ) , ψ i ψ j ( m ) + ε ij ( r )) Hence Ψ i Ψ j ( r , m ) = Ψ ij ( r , m ) if a nd only if ε ij ( r ) = ψ i ε j ( r )) + ε i Ψ j ( r ). The maps ε i : R → M g iven by ε i ( r ) = 0, fo r all r ∈ R and i ∈ N , satisfy prop erties 3.4.1-3.4.4 meaning that the maps Ψ i : R ⋊ M → R ⋊ M g iven by Ψ i ( r , m ) = (Ψ i ( r ) , ψ i ( m )) give us a Ψ-ring structure on R ⋊ M . W e call this the semi-dir e ct pr o duct of R and M , denoted by R ⋊ Ψ M . W e note that if R and M ar e both sp ecial, then R ⋊ Ψ M is also sp ecia l. 25 3.3 Ψ -deriv ations The Andr´ e-Quillen co ho mology for commutativ e rings is given by the derived functors of the deriv ations functor. F o r a commutativ e ring S , the deriv ations of S with v alues in a n S -mo dule N are in one- to-one corresp o ndence with the sections of S ⋊ N π / / S . W e deﬁne the Ψ-deriv ations and show that they are in one-to-o ne corresp ondence with the sections of R ⋊ Ψ M π / / R . Deﬁnition 3.5. A Ψ -derivation o f R with v alues in M is an additive homomorphism d : R → M such that 1. d ( r s ) = r d ( s ) + d ( r ) s , 2. ψ i ( d ( r )) = d (Ψ i ( r )), for all r, s ∈ R, a nd i ∈ N . W e let Der Ψ ( R, M ) denote the set of a ll Ψ-deriv ations of R with v alues in M . Example 3.6. L et R and M b e such that Ψ i = I d = ψ i for al l i ∈ N , then Der Ψ ( R, M ) = Der( R , M ) . Theorem 3.7. Ther e is a one-to-one c orr esp ondenc e b etwe en the elements of Der Ψ ( R, M ) and the se ctions of R ⋊ Ψ M π / / R . Pr o of of the or em. Assume we hav e a section of π , then we have the following R ⋊ Ψ M π / / R, σ o o where π σ = I d R . Hence σ ( r ) = ( r, d ( r )) for so me d : R → M . T he prop erties d ( r + s ) = d ( r ) + d ( s ) , d ( rs ) = d ( r ) s + r d ( s ) , follow from σ b eing a ring homo morphism. Howev er σ also preserves the Ψ- ring s tructure, so we get that Ψ i σ ( r ) = σ Ψ i ( r ). W e k now that Ψ i σ ( r ) = Ψ i ( r , d ( r )) = (Ψ i ( r ) , ψ i ( d ( r ))) , σ Ψ i ( r ) = (Ψ i ( r ) , d (Ψ i ( r ))) . Hence Ψ i σ ( r ) = σ Ψ i ( r ) if and only if ψ i d ( r ) = d Ψ i ( r ). This tells us that if σ is a section of π , then we ha ve a Ψ-der iv a tio n d . Conv ersely , if we ha v e a Ψ-der iv a tion d : R → M , then σ ( r ) = ( r, d ( r )) is a section of π . 3.4 Ψ -ring extensions W e hav e seen in prop osition 2 .1 3 that the Andr´ e-Q uille n co homology H 1 AQ ( R , M ) classiﬁes the extensions of R by M. In this section, we develop the Ψ-analog ue o f extensions . Deﬁnition 3.8. A Ψ -ring extension of R by M is a n extension o f R by M 0 / / M α / / X β / / R / / 0 where X is a Ψ - ring, β is a ma p of Ψ-r ings and αψ n = Ψ n α for a ll n ∈ N . Two Ψ-ring extensions ( X ) , ( X ) with R , M ﬁxed are sa id to be e quival ent if there exists a map of Ψ-rings φ : X → X such that the fo llowing diagr am commu tes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X / / R / / 0 W e denote the set of eq uiv ale nce classes of Ψ -ring extensions o f R by M b y E xt Ψ ( R, M ). The Ha r rison coho mology H ar r 2 ( R , M ) clas siﬁes the additively split extensions of R by M . W e can also deﬁne the Ψ-a nalogue of these type s of extensio ns. 26 Deﬁnition 3.9 . An additively split Ψ -ring extension of R b y M is a Ψ-r ing extensio n of R b y M 0 / / M α / / X β / / R / / 0 where β has a s ection which is an additive homomorphism. Multiplication in X = R ⊕ M has the form ( r, m )( r ′ , m ′ ) = ( rr ′ , mr ′ + rm ′ + f ( r, r ′ )), where f : R × R → M is some bilinear map. Asso ciativ ity in X gives us 0 = rf ( r ′ , r ′′ ) − f ( r r ′ , r ′′ ) + f ( r, r ′ r ′′ ) − f ( r, r ′ ) r ′′ . Commutativit y in X gives us f ( r, r ′ ) = f ( r ′ , r ) . The Ψ-op era tions Ψ i : R ⊕ M → R ⊕ M for i ∈ N are given by Ψ i ( r , m ) = (Ψ i ( r ) , ψ i ( m ) + ε i ( r )) for a sequenc e of op era tions ε i : R → M which satisfy the following prop erties 1. ε 1 ( r ) = 0, 2. ε i ( r + s ) = ε i ( r ) + ε i ( s ), 3. ε i ( rs ) = Ψ i ( s ) ε i ( r ) + Ψ i ( r ) ε i ( s ) + f (Ψ i ( r ) , Ψ i ( s )) − ψ i ( f ( r, s )), 4. ε ij ( r ) = ψ i ε j ( r ) + ε i Ψ j ( r ), for all r, s ∈ R and i, j ∈ N . Assuming we hav e tw o Ψ-ring extensions ( X , ε, f ) , ( X , ε, f ) whic h ar e equiv a lent, tog ether with a Ψ-ring map φ : X → X with φ ( r, m ) = ( r, m + g ( r )) for some g : R → M . W e ha ve that φ b eing a homomor phis m tells us that g ( r + r ′ ) = g ( r ) + g ( r ′ ) , f ( r, r ′ ) − f ( r, r ′ ) = rg ( r ′ ) − g ( rr ′ ) + g ( r ) r ′ . W e also hav e φ (Ψ i ) = Ψ i ( φ ) for a ll i ∈ N , which tells us that ε i ( r ) − ε i ( r ) = ψ i ( g ( r )) − g (Ψ i ( r )) . W e denote the s e t o f equiv alence cla sses o f the additively split Ψ-ring extensions of R b y M by AExt Ψ ( R, M ). Deﬁnition 3.10. An additive ly and mu lt iplic atively split Ψ -ring exten s ion of R by M is a Ψ-ring extension of R by M 0 / / M α / / X β / / R / / 0 where β has a s ection which is an additive and mult iplicative homomorphis m. As a comm utativ e ring X = R ⋊ M , i.e. f = 0 ab ov e. The Ψ-op eratio ns Ψ i : X → X for i ∈ N are given b y Ψ i ( r , m ) = (Ψ i ( r ) , ψ i ( m ) + ε i ( r )) for a sequence o f op eratio ns ε i : R → M such that 1. ε 1 ( r ) = 0, 2. ε i ( r + s ) = ε i ( r ) + ε i ( s ), 3. ε i ( rs ) = Ψ i ( s ) ε i ( r ) + Ψ i ( r ) ε i ( s ), 4. ε ij ( r ) = ψ i ε j ( r ) + ε i Ψ j ( r ), for all r, s ∈ R and i , j ∈ N . Note that conditions 2 a nd 3 tell us that ε i ∈ Der( R, M i ) where M i denotes the Ψ-mo dule ov er R with M as an ab elian group and the action of R g iven by ( r , m ) 7→ Ψ i ( r ) m , for r ∈ R, m ∈ M . Assume we have tw o additiv ely and multiplicativ ely split Ψ- r ing extensions ( X , ε ) , ( X , ε ) which are equiv alen t, together with a Ψ-ring map φ : X → X with φ ( r, m ) = ( r , m + g ( r )) for s ome g : R → M . Since φ is a r ing homomor phis m we get that g ∈ Der( R , M ). Since φ is a map of Ψ-rings we get that ε i ( r ) − ε i ( r ) = ψ i ( g ( r )) − g (Ψ i ( r )) , for all i ∈ N . W e denote the set o f equiv alence cla sses of the additively and m ultiplicatively split Ψ-ring extensions of R by M by MExt Ψ ( R, M ). 27 Example 3.11. L et R and M b e such that Ψ i = I d = ψ i for al l i ∈ N , then MExt Ψ ( R, M ) ∼ = Y p prime Der( R , M ) . Lemma 3.12. Ther e exist exact se quenc es 0 / / MExt Ψ ( R, M ) w / / Ext Ψ ( R, M ) u / / H 1 AQ ( R, M ) / / H 1 AQ ( R,M ) I m ( u ) / / 0 0 / / MExt Ψ ( R, M ) w / / AExt Ψ ( R, M ) u / / H ar r 2 ( R, M ) / / H ar r 2 ( R,M ) I m ( u ) / / 0 wher e w is the inclusion, and u maps the class of a Ψ - ring extension to the class of its underlying extension. Pr o of. W e o nly need to chec k exactness at Ext Ψ ( R, M ) and AExt Ψ ( R, M ). A class in Ext Ψ ( R, M ) or AExt Ψ ( R, M ) belo ngs to the kernel of u if the underlying class is the tr ivial class. The additively and m ultiplicatively split extensions ar e precisely the Ψ -ring extensions whose underly ing extension is trivial. Exac tnes s follows. F ro m the deﬁnitions, we see that Ext Ψ ( R, M ) ⊇ AExt Ψ ( R, M ) ⊇ MEx t Ψ ( R, M ). If R and M are b oth sp ecial, then w e say that a Ψ- ring extension 0 / / M α / / X β / / R / / 0 is sp e cial if X is also sp ecial. W e denote the set of equiv alence classes o f the s p ecia l Ψ-r ing extensions of R by M b y Ext Ψ s ( R, M ). Simila r ly , we can deﬁne AExt Ψ s ( R, M ) and MExt Ψ s ( R, M ). 3.5 Crossed Ψ - extensions A cr osse d Ψ -mo dule consists of a Ψ-ring C 0 , a Ψ- mo dule C 1 ov er C 0 and a ma p of Ψ-mo dules C 1 ∂ / / C 0 , which s atisﬁes the pr op erty ∂ ( c ) c ′ = c∂ ( c ′ ) , for c, c ′ ∈ C 1 . I n other words, a cr ossed Ψ-mo dule is a chain algebra which is non-trivial o nly in dimensions 0 and 1. Since C 2 = 0 the condition ∂ ( c ) c ′ = c∂ ( c ′ ) is equiv alent to the Leibnitz relation 0 = ∂ ( cc ′ ) = ∂ ( c ) c ′ − c∂ ( c ′ ) . W e can deﬁne a pro duct by c ∗ c ′ := ∂ ( c ) c ′ , for c, c ′ ∈ C 1 . This gives us a Ψ-ring structure on C 1 and ∂ : C 1 → C 0 is a map of Ψ-rings . Let ∂ : C 1 → C 0 be a cros sed Ψ-mo dule. W e let M = Ker( ∂ ) and R = Coker( ∂ ) Then the image Im( ∂ ) is an ideal of C 0 , M C 1 = C 1 M = 0 and M ha s a well-deﬁned Ψ - mo dule structure ov er R . A cr osse d Ψ -extension o f R by M is an exact sequence 0 / / M α / / C 1 ∂ / / C 0 γ / / R / / 0 where ∂ : C 1 → C 0 is a cro ssed Ψ- mo dule, γ is a map o f Ψ- rings, and the Ψ-mo dule structure on M co incides with the one induced from the cro ssed Ψ-mo dule. W e denote the c a tegory of crossed Ψ-extens io ns of R by M by C ross Ψ ( R, M ). W e let π 0 C ross Ψ ( R, M ) denote the connected comp onents of the category C r oss Ψ ( R, M ). An additively s plit cr osse d Ψ -exten s ion of R by M is a cro ssed Ψ-extension 0 / / M ω / / C 1 ρ / / C 0 π / / R / / 0 (3.1) 28 such that all the arr ows in the exa ct seq uence 3.1 are additively split. W e deno te the connected co m- po nent s of the category of additively split crossed Ψ-extensions of R by M by π 0 AC ross Ψ ( R, M ). An additively and multiplic ativel y split cr osse d Ψ -extension of R b y M is a crossed Ψ-extens ion 0 / / M ω / / C 1 ρ / / C 0 π / / R / / 0 such that π is a dditiv ely and multiplicativ ely split. W e de no te the connected comp onents of the cat- egory of a dditively and m ultiplicatively split cros sed Ψ-extensions of R by M b y π 0 M C ross Ψ ( R, M ). 3.6 Deformation of Ψ -rings In this section, we a pply Gerstenhab er and Schac k’s de ﬁnitio n of a deformation of a diagram of algebras [7] to the case o f Ψ-rings. Deﬁnition 3.13. Let α t = α 0 + tα 1 + t 2 α 2 + . . . be a deformation of R , i.e. b e a formal p ower series, in whic h each α k : R × R → R is a bilinear map, α 0 is the m ultiplication in R a nd α t is as s o ciative and commutativ e. F or each i ∈ N , let Ψ i t = ψ i 0 + tψ i 1 + t 2 ψ i 2 + . . . be a for ma l power series, in which eac h ψ i k is a function ψ i k : R → R, satisfying 1. ψ i 0 ( r ) = Ψ i ( r ), 2. ψ 1 k ( r ) = 0, 3. ψ i k ( r + s ) = ψ i k ( r ) + ψ i k ( s ), 4. P k h =0 ψ i h α k − h ( r , s ) = P k h =0 P k − h l =0 α h ( ψ i l ( r ) , ψ i k − h − l ( s )), 5. ψ ij k ( r ) = P k l =0 ψ i l ◦ ψ j k − l ( r ) , for all i, j, k ∈ N and r, s ∈ R . W e ca ll ( α t , Ψ ∗ t ) a Ψ- ring deformation o f R . W e call ( α 1 , ψ ∗ 1 ) the inﬁ nitesimal deformation of ( α t , Ψ ∗ t ). The inﬁnitesimal Ψ-ring defo rmation ( α 1 , ψ ∗ 1 ) is identiﬁed with the additively split Ψ-ring extensions of R by R by setting f = α 1 and ε i = ψ i 1 for all i ∈ N . Deﬁnition 3.14. W e deﬁne a formal automorphism of the Ψ -ring R to b e a for mal power series Φ t = φ 0 + tφ 1 + t 2 φ 2 + . . . where each φ k : R → R s uch that 1. φ 0 ( r ) = r , 2. φ k ( r + s ) = φ k ( r ) + φ k ( s ) . Two Ψ-ring deformations ( α t , Ψ ∗ t ) and ( α t , Ψ ∗ t ) are e quivalent if there exists a for mal automorphism Φ t such that Φ t α t ( r , s ) = α t (Φ t r , Φ t s ) and Φ t Ψ ∗ t = Ψ ∗ t Φ t . If t wo Ψ -ring defo rmations ( α t , Ψ ∗ t ) and ( α t , Ψ ∗ t ) are eq uiv a le n t, then the diﬀer e nc e s satisfy α 1 ( r , s ) − α 1 ( r , s ) = r φ 1 ( s ) − φ 1 ( rs ) + sφ 1 ( r ) and ψ i 1 − ψ i 1 = Ψ i φ 1 − φ 1 Ψ i for all i ∈ N . Hence the equiv a lence classes of the inﬁnit esimal Ψ- r ing deformations are iden tiﬁed with the eq uiv alence classes of the a dditiv ely s plit Ψ-r ing extensio ns, AExt Ψ ( R, R ). Y au [20] deﬁned the co homology of λ -rings in order to study deforma tions with r esp ect to the Ψ-op er ations co rresp onding to the λ -ring . Here, I provide an alternative deﬁnition to Y au’s deﬁnition. A deformation of the Ψ-o per ations should be a Ψ- ring defor mation ( α t , Ψ ∗ t ) wher e α t is the tr ivial defor mation. If we let α k = 0 for all k ≥ 1 in the deﬁnition of a Ψ-ring deforma tion then we get the following deﬁnition. 29 Deﬁnition 3.15. F or ea ch i ∈ N , let Ψ i t = ψ i 0 + tψ i 1 + t 2 ψ i 2 + . . . be a for ma l power series, in which eac h ψ i k is a function ψ i k : R → R, such that 1. ψ i 0 ( r ) = Ψ i ( r ), 2. ψ 1 k ( r ) = 0 for k ≥ 1 . , 3. ψ i k ( r + s ) = ψ i k ( r ) + ψ i k ( s ), 4. ψ i k ( rs ) = P k l =0 ψ i l ( r ) ψ i k − l ( s ), 5. ψ ij k ( r ) = P k l =0 ψ i l ◦ ψ j k − l ( r ), for all i, j, k ∈ N and r, s ∈ R . W e ca ll Ψ ∗ t a Ψ- op er ation deformation o f R . The inﬁnitesimal Ψ-op era tio n deformation ψ ∗ 1 is ident iﬁed with the a dditiv ely and multiplica- tively split Ψ-ring extensions o f R by R by setting ε i = ψ i 1 for a ll i ∈ N . If t wo Ψ-op e ration deformations Ψ ∗ t and Ψ ∗ t are equiv alent, then the diﬀerence satisﬁes ψ i 1 − ψ i 1 = Ψ i φ 1 − φ 1 Ψ i for all i ∈ N . Note that now Φ t ( rs ) = Φ t ( r )Φ t ( s ) so we get that φ 1 ∈ Der( R, R ). Hence the eq uiv a le nc e classes of the inﬁnitesimal Ψ-op era tion deforma tions are identiﬁed with the equiv a lence classes of the additively and mult iplicatively split Ψ- ring extensions, MEx t Ψ ( R, R ). 30 Chapter 4 λ -rings 4.1 In tro duction In this chapter, o nly the materia l in this s ection and section 4.6 is alrea dy known (se e [1], [14] a nd [20]) and everything else is new and orig inal material. Note that in our notation N 0 = N ∪ { 0 } . In this c hapter, we start by in tro ducing the concept of a pr e- λ -ring. After giving the deﬁnition, we will lo ok at so me examples o f pre- λ -r ings. Later , we in tro duce the deﬁnition of λ -rings, which are pre- λ -r ings which satisfy some additional axioms. Then w e will loo k at whic h of the pre- λ -ring structures also g ive us λ -rings. Deﬁnition 4.1 . A pr e- λ -ring is a commutativ e ring R with identit y 1, together with a sequence of op erations λ i : R → R , for i ∈ N 0 , satisfying 1. λ 0 ( r ) = 1, 2. λ 1 ( r ) = r , 3. λ i ( r + s ) = Σ i k =0 λ k ( r ) λ i − k ( s ), for all r, s ∈ R and i ∈ N 0 . T o b e able to describ e examples of pre- λ -rings o r λ -ring s it is o ften useful to co ns ider, for r ∈ R , the formal p ow er series in the v ariable t λ t ( r ) = ∞ X i =0 λ i ( r ) t i = λ 0 ( r ) + λ 1 ( r ) t + λ 2 ( r ) t 2 + . . . Note that λ t ( r + s ) = λ 0 ( r + s ) + λ 1 ( r + s ) t + λ 2 ( r + s ) t 2 + λ 3 ( r + s ) t 3 . . . = 1 + ( r + s ) t + Σ 2 k =0 λ k ( r ) λ 2 − k ( s ) t 2 + Σ 3 k =0 λ k ( r ) λ 3 − k ( s ) t 3 + . . . = (1 + rt + λ 2 ( r ) t 2 + . . . )(1 + st + λ 2 ( s ) t 2 + . . . ) = λ t ( r ) λ t ( s ) . This gives us an equiv alent deﬁnition of a pre- λ -ring. Deﬁnition 4.2 . A pr e- λ -ring is a commutativ e ring R with identit y 1, together with a sequence of op erations λ i : R → R , for i ∈ N 0 , satisfying 1. λ 0 ( r ) = 1, 2. λ 1 ( r ) = r , 3. λ t ( r + s ) = λ t ( r ) λ t ( s ), where λ t ( r ) = P i ≥ 0 λ i ( r ) t i , for all r, s ∈ R and i ∈ N 0 . 31 Example 4.3. We c an get a pr e- λ -ring st r u ctur e on Z by taking λ t ( r ) = (1 + t + n 2 t 2 + n 3 t 3 + . . . ) r , wher e 1 + t + n 2 t 2 + n 3 t 3 + . . . is a p ower series with int e ger c o eﬃcients. We c an get a pr e- λ -ring st ructur e on R by taking either 1. λ t ( r ) = (1 + t + n 2 t 2 + n 3 t 3 + . . . ) r , wher e 1 + t + n 2 t 2 + n 3 t 3 + . . . is a p ower series with inte ger c o eﬃc ients, or 2. λ t ( r ) = e tr . The λ -r ing a xioms inv olve some universal p olyno mials. W e ar e now going to intro duce the elementary symmetric functions in or der to deﬁne these universal po lynomials. Deﬁnition 4.4. Let ξ 1 , ξ 2 , . . . , ξ q ; η 1 , η 2 , . . . , η r be indeterminates. Deﬁne s i and σ j to b e the elementary symmetric functions of the ξ ′ i s, η ′ j s , i.e. (1 + s 1 t + s 2 t 2 + . . . +) = Π i (1 + ξ i t ) , (1 + σ 1 t + σ 2 t 2 + . . . +) = Π j (1 + η j t ) . Let P k ( s 1 , s 2 , . . . , s k ; σ 1 , σ 2 , . . . , σ k ) b e the co eﬃcient of t k in Π i,j (1 + ξ i η j t ). Let P k,l ( s 1 , s 2 , . . . , s kl ) b e the co eﬃcient of t k in Π 1 ≤ i 1 <... 0, there are ma ps Λ i : M → M , for i > 0, whic h ma ke the following dia gram commutes: 0 / / M Λ i   α / / X β / / λ i   R / / λ i   0 0 / / M α / / X β / / R / / 0 . The prop erties o f the Λ- op erations follow from the prop erties of the λ -op erations . F or ex a mple, αλ i ( rm ) = λ i α ( rm ) = λ i ( xα ( m )) , for some x ∈ X with β ( x ) = r . The r efore, αλ i ( rm ) = P i ( λ 1 ( x ) , . . . , λ i ( x ) , λ 1 ( α ( m )) , . . . λ i ( α ( m ))) . How ev er α ( m ) α ( n ) = 0 for a ll m, n ∈ M so mos t of the terms v a nish leaving αλ i ( rm ) = α Ψ i ( r )Λ i ( m ) . F or the rest of this c hapter, w e le t R denote a λ -r ing and M ∈ R − mod λ . W e let R denote the underlying commutativ e ring of R , and M denote the underlying R -mo dule of M . Example 4. 14. In gener al, R is not a λ -mo dule over itself u nless the multiplic ation in R is trivial. However we c an c onsider the se quenc e of op er atio ns Λ i : R → R given by Λ i ( r ) = ( − 1) ( i +1) Ψ i ( r ) . With these Λ -op er ations R is a λ -m o dule over R . Theorem 4.15. The A dams op er ation ψ n : M → M given by ψ n ( m ) = ( − 1) ( n +1) n Λ n ( m ) , give u s a sp e ci al Ψ -mo dule structu r e on M over R Ψ , which we denote by M Ψ . Pr o of. 1. ψ 1 ( m ) = Λ 1 ( m ) = m , 2. ψ i ( m 1 + m 2 ) = ( − 1) i +1 i Λ i ( m 1 + m 2 ) = ( − 1) i +1 i Λ i ( m 1 ) + ( − 1) i +1 i Λ i ( m 2 ) = ψ i ( m 1 ) + ψ i ( m 2 ), 3. ψ i ( rm ) = ( − 1 ) i +1 i Λ i ( rm ) = ( − 1 ) i +1 i Ψ i ( r )Λ i ( m ) = Ψ i ( r ) ψ i ( m ), 4. ψ i ( ψ j ( m )) = ψ i (( − 1) ( j +1) j Λ j ( m )) = ( − 1) ( i + j ) ij Λ i (Λ j ( m )) = ( − 1) ( ij +1) ij Λ ij ( m ) = ψ ( ij ) ( m ) . W e will re q uire the following useful lemma. Lemma 4.16. ν − 1 X i =1 [( − 1) i +1 χ i ( r , m )Ψ ν − i ( r ) + ( − 1 ) ν +1 iλ i ( r )Λ ν − i ( m )] = 0 , for al l r ∈ R, m ∈ M and ν ≥ 2 , wher e χ i ( r , m ) = P i j =1 Λ j ( m ) λ i − j ( r ) . 34 Pr o of. W e a re going to use pro of by induction on ν . Consider the case when ν = 2. LH S =( − 1) 2 χ 1 ( r , m )Ψ 1 ( r ) + ( − 1 ) 3 λ 1 ( r )Λ 1 ( m ) = mr − r m =0 . W e are also g oing to cons ide r the case when ν = 3 . LH S = χ 1 ( r , m )Ψ 2 ( r ) + λ 1 ( r )Λ 2 ( m ) − χ 2 ( r , m )Ψ 1 ( r ) + 2 λ 2 ( r )Λ 1 ( m ) = m [ r 2 − 2 λ 2 ( r )] + r Λ 2 ( m ) − [ mr + Λ 2 ( m )] r + 2 λ 2 ( r ) m =0 . Now assume that ν − k − 1 X i =1 [( − 1) i +1 χ i ( r , m )Ψ ν − k − i ( r ) + ( − 1 ) ν − k +1 iλ i ( r )Λ ν − k − i ( m )] = 0 , for 1 ≤ k ≤ ν − 2. It follows that ν − 1 X i =1 [( − 1) i +1 χ i ( r , m )Ψ ν − i ( r ) + ( − 1 ) ν +1 iλ i ( r )Λ ν − i ( m )] = ν − 1 X i =1 ( − 1) ν ( ν − i ) λ ν − i ( r ) χ i ( r , m ) + ν − 2 X i =1 ( − 1) i +1 χ i ( r , m )[ ν − i − 1 X j =1 ( − 1) j +1 λ j ( r )Ψ ν − i − j ( r )] + ν − 1 X i =1 ( − 1) ν +1 iλ i ( r )Λ ν − i ( m ) = ν − 2 X i =1 ( − 1) ν iλ i ( r )[ ν − i − 1 X j =1 Λ j ( m ) λ ν − i − j ( r )] + ν − 2 X i =1 χ i ( r , m )[ ν − i − 1 X j =1 ( − 1) j + i λ j ( r )Ψ ν − i − j ( r )] = ν − 2 X k =1 λ k ( r )[ ν − k − 1 X i =1 ( − 1) ν iλ i ( r ) λ ν − k − i ( r )] + ν − 2 X k =1 λ k ( r )[ ν − k − 1 X i =1 ( − 1) i + k χ i ( r , m )Ψ ν − k − i ( r )] = ν − 2 X k =1 ( − 1) k +1 λ k ( r )[ ν − k − 1 X i =1 [( − 1) i +1 χ i ( r , m )Ψ ν − k − i ( r ) + ( − 1) ν − k +1 iλ i ( r )Λ ν − k − i ( m )] =0 . as requir ed. Lemma 4.17. The s et R ⋊ M with ( r , m ) + ( s, n ) = ( r + s, m + n ) , ( r , m )( s, n ) = ( r s, rn + ms ) , to gether with maps λ i : R ⋊ M → R ⋊ M for i ∈ N 0 given by λ i ( r , m ) = ( λ i ( r ) , f i ( r , m )) , for a se quenc e of maps f i : R ⋊ M → M , for i ∈ N 0 , is a pr e- λ -ring if and only if 1. f 0 ( r , m ) = 0 , 2. f 1 ( r , m ) = m , 3. f i (( r , m ) + ( s, n )) = P i j =0 ( f j ( r , m ) λ i − j ( s ) + λ j ( r ) f i − j ( s, n )) . Pr o of of lemma. R is a comm utative ring with identit y , and M is an R -mo dule. Then we k now that R ⋊ M is a commutativ e ring with identit y . So we only ha ve to c heck the prop erties of λ i : R ⋊ M → R ⋊ M . 35 1. λ 0 ( r , m ) = ( λ 0 ( r ) , f 0 ( r , m )). Hence λ 0 ( r , m ) = (1 , 0) if a nd only if f 0 ( r , m ) = 0, 2. λ 1 ( r , m ) = ( λ 1 ( r ) , f 1 ( r , m )). Hence λ 1 ( r , m ) = ( r , m ) if a nd only if f 1 ( r , m ) = m , 3. λ i (( r , m ) + ( s, n )) = λ i ( r + s, m + n ) = ( λ i ( r + s ) , f i ( r + s, m + n )) P i j =0 λ j ( r , m ) λ i − j ( s, n ) = P i j =0 ( λ j ( r ) , f j ( r , m ))( λ i − j ( s ) , f i − j ( s, n )) = P i j =0 ( λ j ( r ) λ i − j ( s ) , f j ( r , m ) λ i − j ( s ) + λ j ( r ) f i − j ( s, n )) Hence λ i (( r , m ) + ( s, n )) = P i j =0 λ j ( r , m ) λ i − j ( s, n ) if a nd only if f i (( r , m ) + ( s, n )) = P i j =0 ( f j ( r , m ) λ i − j ( s ) + λ j ( r ) f i − j ( s, n )) . Lemma 4.18. The s et R ⋊ M to gether with maps λ i : R ⋊ M → R ⋊ M , for i ∈ N 0 , given by λ i ( r , m ) =  (1 , 0) for i = 0 , ( λ i ( r ) , P i j =1 Λ j ( m ) λ i − j ( r )) for i ∈ N , gives us a λ -ring. W e call this λ -r ing the semi-dir e ct pr o duct of R a nd M , denoted by R ⋊ λ M . Pr o of. W e s tart by sho wing this is a pre- λ ring by using lemma 4.17 with f i ( r , m ) =  0 for i = 0 , P i j =1 Λ j ( m ) λ i − j ( r ) for i ≥ 1 . Clearly prop erties 1 and 2 hold, so we only hav e to chec k 3. Let i ≥ 2 then f i (( r , m ) + ( s, n )) = f i ( r + s , m + n ) = i X j =1 Λ j ( m + n ) λ i − j ( r + s ) = i X j =1 ((Λ j ( m ) + Λ j ( n )) i − j X k =0 λ k ( r ) λ i − j − k ( s ) = i X j =1 i − j X k =0 (Λ i ( m ) λ k ( r ) λ i − j − k ( s ) + Λ i ( n ) λ k ( r ) λ i − j − k ( s )) = i X j =1 j X k =1 Λ k ( m ) λ j − k ( r ) λ i − j ( s ) + i − 1 X j =1 i − j X k =1 λ j ( r )Λ k ( n ) λ i − j − k ( s ) + i X k =1 Λ k ( n ) λ i − k ( s ) λ 0 ( r ) = i − 1 X j =1 j X k =1 Λ k ( m ) λ j − k ( r ) λ i − j ( s ) + i X k =1 Λ k ( m ) λ i − k ( r ) λ 0 ( s ) + i − 1 X j =1 i − j X k =1 λ j ( r ) λ k ( n ) λ i − j − k ( s ) + i X k =1 Λ k ( n ) λ i − k ( s ) λ 0 ( r ) = i − 1 X j =1 ( f j ( r , m ) λ i − j ( s ) λ j ( r ) f i − j ( s, n )) + f i ( r , m ) λ 0 ( s ) + f i ( s, n ) λ 0 ( r ) + λ i ( s ) f 0 ( r , m ) + λ i ( r ) f 0 ( s, n ) = i X j =0 ( f j ( r , m ) λ i − j ( s ) + λ j ( r ) f i − j ( s, n )) . So w e hav e prov ed that R ⋊ λ M is a pr e- λ -ring. Checking the last tw o ax ioms is reduce d to chec king the following the following universal p olynomial identities hold. 36 • P i ( λ 1 ( r , m ) , . . . , λ i ( r , m ) , λ 1 ( s, n ) , . . . , λ i ( s, n )) = ( P i ( λ 1 ( r ) , . . . , λ i ( r ) , λ 1 ( s ) , . . . , λ i ( s )) , P i − 1 k =1 P i − k ( λ 1 ( r ) , . . . , λ i − k ( r ) , λ 1 ( s ) , . . . , λ i − k ( s ))[Ψ k ( s )Λ k ( m ) + Ψ k ( r )Λ k ( n )]), • P i,j ( λ 1 ( r , m ) , . . . , λ ij ( r , m )) = ( P i,j ( λ 1 ( r ) , . . . , λ ij ( r )) , P i k =1 P j l =1 ( − 1) ( k +1)( l +1) Λ kl ( m )Ψ k ( λ j − l ( r )) P ( i − k ) ,j ( λ 1 ( r ) , . . . , λ ( i − k ) j ( r ))). W e are going to sta r t by considering the case where R is a free λ -ring a nd M is fre e as a λ -mo dule ov er R . Our aim is to show that the Adams o pe r ations give us the Ψ- r ing structure on R ⋊ M with Ψ ν ( r , m ) = (Ψ ν ( r ) , ψ ν ( m )) b y using induction on ν . Then theo r em 4.11 tells us that R ⋊ λ M is a λ -ring and the universal polynomial identities hold. Consider the case when ν = 1 Ψ 1 ( r , m ) = ( r , m ) = (Ψ 1 ( r ) , ψ 1 ( m )) . Consider the case when ν = 2 Ψ 2 ( r , m ) = ( r 2 , 2 rm ) − 2 λ 2 ( r , m ) = ( r 2 − 2 λ 2 ( r ) , − 2 Λ 2 ( m )) = (Ψ 2 ( r ) , ψ 2 ( m )) . Assume that Ψ ν − k ( r , m ) = (Ψ ν − k ( r ) , ψ ν − k ( m )) for 1 ≤ k ≤ ν − 1. It follows that Ψ ν ( r , m ) = ν − 1 X j =1 ( − 1) ν − j +1 ( λ ν − j ( r ) , ν − j X k =1 λ ν − j − k ( r )Λ k ( m ))(Ψ j ( r ) , ψ j ( m )) + ( − 1) ν +1 ν ( λ ν ( r ) , ν X k =1 λ ν − k ( r )Λ k ( m )) = ν − 1 X j =1 ( − 1) ν − j +1 ( λ ν − j ( r )Ψ j ( r ) , Ψ j ( r ) ν − j X k =1 λ ν − j − k ( r )Λ k ( m ) + Ψ j ( m ) λ ν − j ( r )) + ( − 1 ) ν +1 ν ( λ ν ( r ) , ν X k =1 λ ν − k ( r )Λ k ( m )) =(Ψ ν ( r ) , ν − 1 X j =1 ( − 1) ν − j +1 [ λ ν − j ( r )Ψ j ( m ) + Ψ j ( r ) ν − j X k =1 λ ν − j − k ( r )Λ k ( m )] + ( − 1) ν +1 ν [ ν X k =1 λ ν − k ( r )Λ k ( m )]) =(Ψ ν ( r ) , ν − 1 X j =1 [( − 1) j +1 Ψ ν − i ( r ) χ i ( r , m ) + ( − 1) ν +1 j λ j ( r )Λ ν − j ( m )] + ( − 1) ν +1 ν Λ ν ( m )) =(Ψ ν ( r ) , ( − 1) ν +1 ν Λ ν ( m )) = (Ψ ν ( r ) , ψ ν ( m )) , as requir ed. Now c o nsider the case wher e R is a free λ -ring and M is an arbitrary λ -mo dule ov er R . Cho ose P a free λ -mo dule ov er R with a surjective homomorphis m P ։ M , this gives us a s urjective homomorphism R ⋊ λ P ։ R ⋊ λ M . Since the universal p olynomial iden tities hold on R ⋊ λ P they also hold o n R ⋊ λ M . Now we c an consider the cas e when R is an ar bitrary λ -ring a nd M is a λ -mo dule over R . An y λ -ring is the quo tient o f a free λ -r ing, therefore R is the q uotient of a free λ -ring F . There exists a surjective homomor phism F ⋊ λ M ։ R ⋊ λ M . Since the universal po lynomial identities hold on F ⋊ λ M they also hold o n R ⋊ λ M . Hence R ⋊ λ M is a λ -ring. More over we pro ved that ( R ⋊ λ M ) Ψ = R Ψ ⋊ Ψ M Ψ . 4.3 λ -deriv ations Deﬁnition 4. 19. A λ -deriv ation of R with v alues in M is an additive homomor phis m d : R → M such that 37 1. d ( r s ) = r d ( s ) + d ( r ) s , 2. d ( λ i ( r )) = Λ i ( d ( r )) + Λ i − 1 ( d ( r )) λ 1 ( r ) + . . . + Λ 2 ( d ( r )) λ i − 2 ( r ) + Λ 1 ( d ( r )) λ i − 1 ( r ), for all r, s ∈ R , a nd i ∈ N . W e let Der λ ( R, M ) denote the set o f all λ -deriv ations of R with v alues in M . Example 4.20. L et Z λ [ x ] b e t he fr e e λ -ring on one gener ator x , and let M ∈ Z λ [ x ] − mod λ . D er λ ( Z λ [ x ] , M ) ∼ = M . Z λ [ x ] = Z [ x 1 , x 2 , . . . ] t o gether with op er atio ns determine d by λ i ( x 1 ) = x i . F or any λ -derivation, d : Z λ [ x ] → M , we have t hat d ( x 1 ) = m, d ( x i ) = i X j =1 Λ j ( m ) x i − j , wher e m ∈ M and x 0 = 1 . Theorem 4.21. Ther e is a one-to-one c orr esp ondenc e b etwe en t he se ctions of R ⋊ λ M π / / R and the λ -derivations d : R → M . Pr o of of the or em. Assume we hav e a section of π , then we have the following R ⋊ λ M π / / R, σ o o where π σ = I d R . Hence σ ( r ) = ( r, d ( r )) for so me d : R → M . T he prop erties d ( r + s ) = d ( r ) + d ( s ) , d ( rs ) = d ( r ) s + r d ( s ) , follow from σ being a ring homomo rphism. How ev er σ also preser ves the λ -ring structure, meaning that λ i σ ( r ) = σ λ i ( r ). W e k now that λ i σ ( r ) = λ i ( r , d ( r )) = λ i (( r , 0) + (0 , d ( r ))) = Σ i j =0 λ j ( r , 0) λ i − j (0 , d ( r )) = Σ i − 1 j =0 (0 , λ j ( r )Λ i − j ( d ( r ))) + ( λ i ( r ) , 0) = ( λ i ( r ) , Σ i − 1 j =0 λ j ( r )Λ i − j ( d ( r ))) σ λ i ( r ) = ( λ i ( r ) , d ( λ i ( r ))) . Hence λ i σ ( r ) = σ λ i ( r ) if and only if dλ i ( r ) = Σ i − 1 j =0 λ j ( r )Λ i − j ( d ( r )). This tells us that if σ is a section of π , then we hav e a λ -deriv atio n d . Conv ersely , if we ha v e a λ -deriv ation d : R → M , then σ ( r ) = ( r , d ( r )) is a section of π . Theorem 4. 22. The λ -derivations of R wi th values in M ar e also Ψ - derivations of R Ψ with values in M Ψ . Pr o of. Let d : R → M b e a λ -deriv a tion, we ar e going to use induction on ν to show ψ ν ( d ( r )) = d (Ψ ν ( r )) for all ν ≥ 1. Consider the case when ν = 1 ψ 1 ( d ( r )) = d ( r ) = d (Ψ 1 ( r )) . Consider the cas e when ν = 2. d (Ψ 2 ( r )) = d ( r 2 − 2 λ 2 ( r )) = 2 rd ( r ) − 2 [Λ 2 ( d ( r )) + d ( r ) r ] = − 2Λ 2 ( d ( r )) = ψ 2 ( d ( r )) . Also consider the ca se ν = 3. d (Ψ 3 ( r )) = d ( r 3 − 3 r λ 2 ( r ) + 3 λ 3 ( r )) = 3Λ 3 ( d ( r )) = ψ 3 ( d ( r )) . 38 Assume that ψ ν − k ( d ( r )) = d (Ψ ν − k ( r )) for 1 ≤ k ≤ ν − 1. d (Ψ ν ( r )) = ν − 1 X i =1 ( − 1) i +1 d ( λ i ( r ))Ψ ν − i ( r ) + ν − 1 X i =1 ( − 1) i +1 λ i ( r ) d (Ψ ν − i ( r )) + ( − 1) ν +1 ν d ( λ ν ( r )) d (Ψ ν ( r )) − ψ ν ( d ( r )) = ν − 1 X i =1 ( − 1) i +1 [ i X j =1 Λ j ( d ( r )) λ i − j ( r )]Ψ ν − i ( r ) + ν − 1 X i =1 ( − 1) i +1 λ i ( r )[( − 1) ν − i +1 ( ν − i )Λ ν − i ( d ( r ))] + ( − 1) ν +1 ν [ ν − 1 X j =1 Λ j ( d ( r )) λ ν − j ( r )] = ν − 1 X i =1 [( − 1) i +1 χ i ( r , d ( r ))Ψ ν − i ( r ) + ( − 1 ) ν +1 iλ i ( r )Λ ν − i ( d ( r ))] =0 . Hence d (Ψ ν ( r )) = ψ ν ( d ( r )). Theorem 4.23. If M is Z -t orsion-fr e e t hen the Ψ -derivations of R Ψ with values in M Ψ ar e also λ -derivations of R with values in M Der λ ( R, M ) = Der Ψ ( R Ψ , M Ψ ) . Pr o of. Let M b e Z -tor sion-free and d : R Ψ → M Ψ be a Ψ-deriv ation. W e are g o ing to use induction on ν to show d ( λ ν ( r )) = P ν i =1 Λ i ( d ( r )) λ ν − i ( r ) for ν ∈ N . Consider the case when ν = 1. Λ 1 ( d ( r )) = d ( r ) = d ( λ 1 ( r )) . Consider the cas e when ν = 2. d (Ψ 2 ( r )) = ψ 2 ( d ( r )) d ( r 2 − 2 λ 2 ( r )) = − 2Λ 2 ( d ( r )) 2[ d ( λ 2 ( r )) − r d ( r ) − Λ 2 ( d ( r ))] = 0 2[ d ( λ 2 ( r )) − 2 X i =1 Λ i ( d ( r )) λ 2 − i ( r )] = 0 d ( λ 2 ( r )) − 2 X i =1 Λ i ( d ( r )) λ 2 − i ( r ) = 0 . Assume tha t d ( λ ν − k ( r )) = P ν − k i =1 Λ i ( d ( r )) λ ν − i − k ( r ) for 1 ≤ k ≤ ν − 1, we want to show that ν d ( λ ν ( r )) = ν P ν i =1 Λ i ( d ( r )) λ ν − i ( r ). F r om ψ v ( d ( r )) = d (Ψ v ( r )) we get ν (Λ ν ( d ( r )) − d ( λ ν ( r ))) = P ν − 1 i =1 ( − 1) i + v [ d ( λ i ( r ))Ψ ν − i ( r ) + λ i ( r ) d (Ψ ν − i ( r ))] . Therefore we ha ve to sho w that ( − 1) ν ν ν − 1 X i =1 Λ i ( d ( r )) λ ν − i ( r ) = ν − 1 X i =1 ( − 1) i +1 d ( λ i ( r ))Ψ ν − i ( r ) + ν − 1 X i =1 ( − 1) i +1 λ i ( r ) d (Ψ ν − i ( r )) = ν − 1 X i =1 ( − 1) i +1 [ i X j =1 Λ j ( d ( r )) λ i − j ( r )] · [ ν − i − 1 X k =1 ( − 1) k +1 λ k ( r )Ψ ν − i − k ( r ) + ( − 1) ν − i − 1 ( ν − i ) λ ν − i ( r )] + ( − 1 ) ν ν − 1 X i =1 i Λ i ( d ( r )) λ ν − i ( r ) . 39 Hence it is suﬃcient to show that P ν − 2 i =1 ( − 1) i +1 χ i ( r , d ( r ))[ P ν − i − 1 k =1 ( − 1) k +1 λ k ( r )Ψ ν − i − k ( r )] + P ν − 1 i =1 ( − 1) ν +1 ( i − ν ) λ ν − i ( r ) χ i ( r , d ( r )) + ( − 1 ) ν +1 P ν − 1 i =1 iλ i ( r )Λ ν − i ( d ( r ))] = 0 , with χ i as in lemma 4.16. W e get that ν − 2 X i =1 ( − 1) i +1 χ i ( r , d ( r ))[ ν − i − 1 X k =1 ( − 1) k +1 λ k ( r )Ψ ν − i − k ( r )] + ν − 1 X i =1 ( − 1) ν +1 ( i − ν ) λ ν − i ( r ) χ i ( r , d ( r )) + ( − 1) ν +1 ν − 1 X i =1 iλ i ( r )Λ ν − i ( d ( r ))] = ν − 2 X i =1 χ i ( r , d ( r ))[( − 1) i +1 [ ν − i − 1 X k =1 ( − 1) k +1 λ k ( r )Ψ ν − i − k ( r ) + ( − 1) ν − i − 1 ( ν − i ) λ ν − i ( r ) − Ψ ν − i ( r )]] =0 , as requir ed. 4.4 λ -ring extensions W e hav e seen in prop osition 2 .1 3 that the Andr´ e-Q uille n co homology H 1 AQ ( R , M ) classiﬁes the extensions of R by M. In this section, we develop the λ -analogue of extens io ns. Deﬁnition 4.24. A λ -ring extension o f R by M is an extens ion of R by M 0 / / M α / / X β / / R / / 0 where X is a λ -r ing , β is a ma p of λ -rings and α Λ n = λ n α for a ll n ∈ N . Two λ -ring extensio ns ( X ) , ( X ′ ) with R, M ﬁxe d are said to be e quival ent if there exists a map of λ -rings φ : X → X ′ such that the following diagram commutes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X ′ / / R / / 0 W e denote the set of eq uiv ale nce classes of λ -r ing extensions o f R by M b y E xt λ ( R, M ). The Harriso n cohomolog y H ar r 1 ( R, M ) cla s siﬁes the additiv ely split extensions of R by M. W e can also deﬁne the λ -analog ue of these types of extensio ns. Deﬁnition 4.25. Let R b e a λ -ring and M ∈ R -mo d λ then an additively split λ -ring ex tension o f R by M is a λ -ring ex tension of R by M 0 / / M α / / X β / / R / / 0 where β has a s ection that is an additive homomorphism. Multiplication in X = R ⊕ M has the form ( r, m )( r ′ , m ′ ) = ( rr ′ , mr ′ + rm ′ + f ( r, r ′ )), where f : R × R → M is some bilinear map. Asso ciativ ity in X gives us 0 = rf ( r ′ , r ′′ ) − f ( r r ′ , r ′′ ) + f ( r, r ′ r ′′ ) − f ( r, r ′ ) r ′′ . Commutativit y in X gives us f ( r, r ′ ) = f ( r ′ , r ) . The λ -op er ations λ ν : R ⋊ M → R ⋊ M for ν ∈ N 0 are g iven b y λ ν ( r , m ) = ( λ ν ( r ) , P ν i =1 Λ i ( m ) λ ν − i ( r )+ ǫ ν ( r )) for a s equence of o pe r ations ǫ ν : R → M which satisfy the following prop erties 40 1. ǫ 0 ( r ) = ǫ 1 ( r ) = 0 , 2. ǫ ν ( r + s ) = P ν i =0 [ ǫ i ( r ) λ ν − i ( s ) + ǫ ν − i ( s ) λ i ( r )] , 3. ǫ ν (1) = 0 , 4. P i ( λ 1 ( r , m ) , . . . , λ i ( s, n )) = ( λ i ( rs ) , P i j =1 (Ψ j ( s )Λ j ( m ) + Ψ j ( r )Λ j ( n ) + Λ j ( f ( r, r ′ ))) λ i − j ( rs ) + ǫ j ( rs )) , 5. P i,j ( λ 1 ( r , m ) , . . . , λ ij ( r , m )) = ( λ i ( λ j ( r )) , P i k =1 Λ k ( P j a =1 (Λ a ( m ) λ j − a ( r ) + ǫ j ( r )) λ i − k ( λ j ( r ))) + ǫ i ( λ j ( r ))) . Assuming we ha ve tw o additiv ely split λ -r ing extensions ( X , ε, f ),( X ′ , ε ′ , f ′ ) which a re equiv a- lent , together with a λ -ring map φ : X → X ′ with φ ( r, m ) = ( r, m + g ( r )) for some g : R → M . W e hav e that φ b eing a homomorphism tells us that g ( r + r ′ ) = g ( r ) + g ( r ′ ) , f ( r, r ′ ) − f ′ ( r , r ′ ) = rg ( r ′ ) − g ( rr ′ ) + g ( r ) r ′ . W e also hav e φ ( λ ν ) = λ ν ( φ ) for a ll ν ∈ N 0 , which tells us that ε ν ( r ) − ε ′ ν ( r ) = ν X i =1 Λ i ( g ( r )) λ ν − i ( r ) − g ( λ ν ( r )) . W e denote the set o f equiv alence clas ses of additively split λ -ring extensions of R by M by AE xt λ ( R, M ). In order to describ e the properties of λ -ring extensions w e ne e d to deﬁne the partia l der iv a tives of the universal po ly nomials, see a ppendix C for examples. W e can use the universal poly nomials to deﬁne contin uous functions P i : R 2 i → R , P i,j : R ij → R . F or example P 2 : R 4 → R is given by P 2 ( x 1 , x 2 , x 3 , x 4 ) = x 2 1 x 4 − 2 x 2 x 4 + x 2 x 2 3 . W e can take the par tial deriv atives of these functions which are aga in p olynomials. W e ca ll these new p olyno mials the p artial derivatives of the universal polynomia ls. F or example ∂ P 2 ( x 1 , x 2 , x 3 , x 4 ) ∂ x 1 =2 x 1 x 4 , ∂ P 2 ( x 1 , x 2 , x 3 , x 4 ) ∂ x 2 = x 2 3 − 2 x 4 , F or 1 ≤ j ≤ i , w e let ∂ P i ( r , s ) ∂ λ j ( r ) := ∂ P i ( λ 1 ( r ) , . . . , λ i ( r ) , λ 1 ( s ) , . . . , λ i ( s )) ∂ λ j ( r ) . Since the p o lynomials P i are symmetric, we can let ∂ P i ( r , s ) ∂ λ j ( s ) := ∂ P i ( s, r ) ∂ λ j ( s ) . In our e x amples ∂ P 2 ( r , s ) ∂ λ 1 ( r ) = ∂ P 2 ( λ 1 ( r ) , λ 2 ( r ) , λ 1 ( s ) , λ 2 ( s )) ∂ λ 1 ( r ) = 2 rλ 2 ( s ) , ∂ P 2 ( r , s ) ∂ λ 2 ( r ) = ∂ P 2 ( λ 1 ( r ) , λ 2 ( r ) , λ 1 ( s ) , λ 2 ( s )) ∂ λ 2 ( r ) = s 2 − 2 λ 2 ( s ) . 41 Similarly , for 1 ≤ k ≤ ij , we let ∂ P i,j ( r ) ∂ λ k ( r ) := ∂ P i ( λ 1 ( r ) , . . . , λ ij ( r )) ∂ λ k ( r ) . F or example, ∂ P 2 , 2 ( x 1 , x 2 , x 3 , x 4 ) ∂ x 1 = x 3 . So it follows that ∂ P 2 , 2 ( r ) ∂ λ 1 ( r ) = λ 3 ( r ) . These partial deriv a tives a pp e ar beca use of the multiplication in R ⋊ M . Co nsider the following ( r , m ) 2 = ( r 2 , 2 rm ) , ( r , m ) 3 = ( r 3 , 3 r 2 m ) . Deﬁnition 4.2 6. An additively and multiplic atively split λ -ring extension of R by M is a λ -r ing extension of R by M 0 / / M / / X / / R / / 0 where β has a s ection that is an additive and multiplicativ e homomorphism. As a co mmu tative ring X = R ⋊ M , the sequence of op eratio ns λ ν : R ⋊ M → R ⋊ M for ν ∈ N 0 are given by λ ν ( r , m ) = ( λ ν ( r ) , P ν i =1 Λ i ( m ) λ ν − i ( r ) + ǫ ν ( r )) for a seq ue nc e of op erations ǫ ν : R → M such that 1. ǫ 0 ( r ) = ǫ 1 ( r ) = 0 , 2. ǫ ν ( r + s ) = P ν i =0 [ ǫ i ( r ) λ ν − i ( s ) + ǫ ν − i ( s ) λ i ( r )] , 3. ǫ ν (1) = 0 , 4. ǫ ν ( rs ) = P ν i =1 [ ǫ i ( r ) ∂ P ν ( r,s ) ∂ λ i ( r ) + ǫ i ( s ) ∂ P ν ( r,s ) ∂ λ i ( s ) ] , 5. ǫ k ( λ ν ( r )) = P ν k i =1 ǫ i ( r ) ∂ P ν,k ( r ) ∂ λ i ( r ) − P k j =1 Λ j ( ǫ ν ( r )) λ k − j ( λ ν ( r )) . Two a dditively and m ultiplicatively split λ -ring e x tensions ( X , ǫ ),( X ′ , ǫ ′ ) with R, M ﬁxed a re said to b e e quivalent if ther e exists a map of λ -rings φ : X → X ′ such that the following diagra m commutes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X ′ / / R / / 0 Assuming w e have t wo additively and m ultiplicatively split λ -ring extensions ( X, ǫ ),( X ′ , ǫ ′ ) which a re equiv alent, together with a λ -ring ma p φ : X → X ′ with φ ( r, m ) = ( r, m + g ( r )) for some g : R → M . W e a lso hav e φ be ing a ho momorphism which tells us that g ∈ D er ( R, M ). W e also hav e φ ( λ ν ) = λ ν ( φ ) for a ll ν , which tells us that ε ν ( r ) − ε ′ ν ( r ) = ν X i =1 Λ i ( g ( r )) λ ν − i ( r ) − g ( λ ν ( r )) . W e denote the s et of equiv alence classe s o f additively and m ultiplicatively split λ -ring ex tensions of R b y M by M E xt λ ( R, M ). Theorem 4.27. If ǫ ν : R → M gives us an additively and multiplic atively split λ -ring extension of R by M , then ε ν : R → M with ε ν ( r ) = ν − 1 X i =1 ( − 1) i +1 [ ǫ i ( r )Ψ ν − i ( r ) + λ i ( r ) ε ν − i ( r )] + ( − 1 ) ν +1 ν ǫ ν ( r ) , give u s an additively and mu ltiplic atively split Ψ -ring ext ension of R Ψ by M Ψ . Pr o of. If ǫ ν : R → M gives an additively and m ultiplicatively split λ -ring extension of R b y M , then λ ν : R ⋊ M → R ⋊ M given by λ ν ( r , m ) = ( λ ν ( r ) , P ν i =1 Λ i ( m ) λ ν − i ( r ) + ǫ ν ( r )) is a λ -r ing and hence the Adams op erations g ive the Ψ-r ing with o per ations Ψ ν : R ⋊ M → R ⋊ M given by Ψ ν ( r , m ) = (Ψ ν ( r ) , ψ ν ( m ) + ε ν ( r )) which is an a dditiv ely and multiplicativ ely split Ψ-ring extension of R Ψ by M Ψ . 42 4.5 Crossed λ -extensions A cr osse d λ -mo dule co nsists of a λ -ring C 0 , a λ -mo dule C 1 ov er C 0 and a map of λ -mo dules C 1 ∂ / / C 0 , which s atisﬁes the pr op erty ∂ ( c ) c ′ = c∂ ( c ′ ) , for c, c ′ ∈ C 1 . In other words, a crossed λ -mo dule is a chain algebra w hich is non-tr ivial only in dimensions 0 and 1. Since C 2 = 0 the condition ∂ ( c ) c ′ = c∂ ( c ′ ) is equiv alent to the Leibnitz relation 0 = ∂ ( cc ′ ) = ∂ ( c ) c ′ − c∂ ( c ′ ) . W e can deﬁne a pro duct by c ∗ c ′ := ∂ ( c ) c ′ , for c, c ′ ∈ C 1 . This gives us a λ -ring str ucture on C 1 and ∂ : C 1 → C 0 is a ma p of λ -rings . Let ∂ : C 1 → C 0 be a cr ossed λ -mo dule. W e let M = Ker( ∂ ) and R = Coker( ∂ ) Then the image Im( ∂ ) is a n ideal o f C 0 , M C 1 = C 1 M = 0 and M has a well-deﬁned λ -mo dule structure ov er R . A cr osse d λ -ex t ension o f R by M is an exact sequence 0 / / M α / / C 1 ∂ / / C 0 γ / / R / / 0 where ∂ : C 1 → C 0 is a cr ossed λ -mo dule, γ is a map of λ -rings , and the λ -mo dule str ucture on M co incides with the o ne induced from the cr ossed λ -mo dule. W e denote the categor y o f crossed λ -extensions of R by M by C r oss λ ( R, M ). W e let π 0 C ross λ ( R, M ) denote the connected comp onents of the category C r oss λ ( R, M ). An additively s plit cr osse d λ -extension of R by M is a cross ed λ -extension 0 / / M ω / / C 1 ρ / / C 0 π / / R / / 0 (4.1) such that all the arr ows in the exa ct seq uence 4.1 are additively split. W e deno te the connected co m- po nent s of the catego ry of additively split crossed λ -extensions o f R by M by π 0 AC ross λ ( R, M ). An additively and multiplic atively split cr osse d λ -extension of R by M is a n a dditively split crossed λ -extension 0 / / M ω / / C 1 ρ / / C 0 π / / R / / 0 such that π is a dditiv ely and multiplicativ ely split. W e de no te the connected comp onents of the cat- egory of additively and multiplicativ ely s plit cro ssed λ -extensions of R by M by π 0 M C ross λ ( R, M ). 4.6 Y a u c oh omology for λ -rings In 200 5, Donald Y a u published a pap er entitled, “Cohomolo gy o f λ -ring s” [20], in which he de- veloped a cohomolog y theor y for λ -ring s. In this s ection we describ e Y au’s co chain complex and what it computes. Let R b e a λ -ring. W e let E nd ( R ) deno te the algebra of Z -linear endomo rphisms o f R , wher e the pro duct is given by co mp os ition. W e let E nd ( R ) denote the subalgebra o f E nd ( R ) which consists of the linear endomor phisms f of R which satisfy the condition, f ( r ) p ≡ f ( r p ) mo d pR, for each prime p and every r ∈ R . Y au deﬁned C 0 Y au ( R ) be the underlying g roup of E nd ( R ). He deﬁned C 1 Y au ( R ) be the s e t of functions f : N → E nd ( R ) satisfying the condition f ( p )( R ) ⊂ pR for ea ch pr ime p . Then for ν ≥ 2 he s e t C ν Y au ( R ) to be the set o f functions f : N ν → E nd ( R ). F or ν ∈ N 0 , the cobo undary ma p, δ ν : C ν Y au → C ν +1 Y au , is given by the fo llowing δ ν ( f )( m 0 , . . . , m ν ) =Ψ m 0 ◦ f ( m 1 , . . . , m ν ) + ν X i =1 ( − 1) i f ( m 0 , . . . , m i − 1 m i , . . . , m ν ) + ( − 1) ν +1 f ( m 0 , . . . , m ν − 1 ) ◦ Ψ m ν . 43 W e say that the ν th cohomolog y of the co chain co mplex ( C Y au , δ ) is the ν th Y au cohomolo gy of R , denoted by H ν Y au ( R ) . F ro m the co chain complex it is clear that H 0 Y au ( R ) = { f ∈ E nd ( R ) : f Ψ ν = Ψ ν f for all ν ∈ N } . W e deﬁne the gr oup of Y au derivations of R , denoted by Y De r λ ( R ), to co nsist of the functions f ∈ C 1 Y au ( R ) such that f ( i j ) = Ψ j ◦ f ( i ) + f ( j ) ◦ Ψ i , for all i, j ∈ N . W e deﬁne the group of Y au inner- derivations o f R , denoted by Y I D er λ ( R ), to consist of the functions f : N → E nd ( R ) which are of the form f ( i ) = Ψ i ◦ g − g ◦ Ψ i , for some g ∈ E nd ( R ). The ﬁrst Y au cohomolog y is given by the quo tien t, H 1 Y au ( R ) = Y D er λ ( R ) Y I D er λ ( R ) . Y au tells us that there exis ts a ca no nical surjection, H 2 Y au ( R ) ։ H H 2 ( Z [ N ] , E n d ( R )) , and for ν ≥ 3, there exists a ca nonical isomor phism, H ν Y au ( R ) ∼ = H H ν ( Z [ N ] , E n d ( R )) , where H H ν ( Z [ N ] , E n d ( R )) denotes the ν th Ho chsc hild co homology of Z [ N ] with co eﬃcien ts in E nd ( R ). Y au deﬁned his co homology in order to study deformations of λ -r ings. W e let Ψ ∗ t = ψ ∗ 0 + tψ ∗ 1 + t 2 ψ ∗ 2 + . . . be a for ma l power series, in which eac h ψ ∗ i is a function ψ ∗ i : N → E nd ( R ) , satisfying the following pr op erties. W e let ψ j i denote ψ ∗ i ( j ) . 1. ψ j 0 ( r ) = Ψ j ( r ), 2. ψ 1 i = 0 for i ≥ 1, 3. ψ kl i ( r ) = P i j =0 ψ k j ◦ ψ l i − j ( r ) f or k , l ≥ 1 and i ≥ 0, 4. ψ p i ( r ) ⊂ pR for i ≥ 1 and p prime. Y au ca lls Ψ ∗ t a deformation of R . Note that the Gerstenhab er and Schac k’s deﬁnition we provided in 3.6 is very similar to Y au’s deﬁnition but gives a diﬀerent result. W e w ould like to c o mpare the r esults in the case when α i = 0 fo r i ≥ 1. W e omitted the condition ψ p i ( r ) ⊂ pR for p pr ime, but intro duced the co ndition ψ j i ( rs ) = P i k =0 ψ j k ( r ) ψ j i − k ( s ). This last condition makes things more co mplicated and ma y seem strange, but it is necess ary to ensure that Ψ ∗ t ( rs ) = Ψ ∗ t ( r )Ψ ∗ t ( s ) . Y au’s condition g ives us ψ i 1 ∈ E nd ( R ). Gerstenhaber and Schack’s condition gives us ψ i 1 ∈ Der( R, R i ) where R i is the R -mo dule with R as an abelia n group a nd the following action of R ( r , a ) 7→ Ψ i ( r ) a, for r ∈ R, a ∈ R i . 44 Chapter 5 Harrison cohomology of diagrams of comm utativ e algebras 5.1 In tro duction F or this c hapter we let I denote a sma ll ca tegory . A category I is said to b e smal l if the collection of morphisms is a s et. W e let i, j, k denote o b jects in I and we let α : i → j and β : j → k denote morphisms in I . Deﬁnition 5.1. A diagr am of c ommutative algebr as is a cov a riant functor A : I → Com . alg , where I is a small catego ry , and Co m . alg is some category of comm utativ e algebras . W e ca ll I the shap e o f the diagra m. If A, B are tw o cov ar iant functors from I to Com . alg , then a map o f diag rams is a natura l transformatio n µ : A → B . W e denote the catego ry of diagra ms of c o mm utative algebr as with shap e I by Com . alg I . Deﬁnition 5. 2. An A -mo dule is a functor M : I → Ab such that for all i ∈ I we hav e that M ( i ) ∈ A ( i )- m od and for all α ∈ I we hav e M ( α )( a · m ) = A ( α )( a ) · M ( α )( m ) , for all a ∈ A ( i ) , m ∈ M ( i ). W e let A − mod I denote the categ ory of a ll A - mo dules. 5.2 Natural System Let A : I → Co m . alg be a diagr am of a commutativ e a lgebra, and M b e an A -mo dule. F or any n ≥ 0 there exis ts a natural s ystem on I as follows D α := C n H ar r ( A ( i ) , α ∗ M ( j )) , where ( α : i → j ) ∈ I and M ( j ) is conside r ed an A ( i )-mo dule via α . F o r a n y ( β : j → j ′ ) ∈ I , we hav e β ∗ : D α → D β α which is induced by M ( β ) : M ( j ) → M ( j ′ ). F o r any ( γ : i ′ → i ) ∈ I , we have γ ∗ : D α → D αγ which is induced by A ( γ ) : A ( i ′ ) → A ( i ). 5.3 Bicomplex Let A : I → Com . alg b e a diagram o f a commutativ e alg ebra, and M b e an A -module. F or each i ∈ I we can cons ide r the Ha rrison co chain complex o f the commutativ e alg e bra A ( i ) with co eﬃcients in M ( i ). C 0 H ar r ( A ( i ) , M ( i )) / / C 1 H ar r ( A ( i ) , M ( i )) / / C 2 H ar r ( A ( i ) , M ( i )) / / . . . 45 W e can use this to cons truct the following bicomplex denoted by C ∗ , ∗ H ar r ( I , A, M ): C p,q H ar r ( I , A, M ) = Y α : i 0 → ... → i p C q +1 H ar r ( A ( i 0 )) , α ∗ M ( i p )) , for p, q ≥ 0. The map C p,q H ar r ( I , A, M ) → C p +1 ,q H ar r ( I , A, M ) is the map in the Baues-Wirsching co chain complex, and the map C p,q H ar r ( I , A, M ) → C p,q +1 H ar r ( I , A, M ) is the pr o duct o f the Harr ison cob oundary maps. . . . . . . Q i C 3 H ar r ( A ( i ) , M ( i )) O O δ / / Q α : i → j C 3 H ar r ( A ( i ) , α ∗ M ( j )) / / O O . . . Q i C 2 H ar r ( A ( i ) , M ( i )) δ / / ∂ O O Q α : i → j C 2 H ar r ( A ( i ) , α ∗ M ( j )) / / − ∂ O O . . . Q i C 1 H ar r ( A ( i ) , M ( i )) δ / / ∂ O O Q α : i → j C 1 H ar r ( A ( i ) , α ∗ M ( j )) / / − ∂ O O . . . Let ( α n : i n → i n +1 ) ∈ I , and α = α p . . . α 0 : i 0 → i p +1 . Then the cob oundary ma p δ : C p,q H ar r ( I , A, M ) → C p +1 ,q H ar r ( I , A, M ) is given b y δ ( f ) α p +1 ,...,α 0 ( x 1 , . . . , x q ) = f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x q )) + p X k =0 ( − 1) k +1 f α p +1 ,...,α k +1 α k ,...,α 0 ( x 1 , . . . , x q ) + ( − 1) p +2 M ( α p +1 )( f α p ,...,α 0 ( x 1 , . . . , x q )) . The cob oundary ma p ∂ : C p,q H ar r ( I , A, M ) → C p,q +1 H ar r ( I , A, M ) is given b y ∂ ( f ) α p ,...,α 0 ( x 1 , . . . , x q +1 ) = A ( α )( x 1 ) · f α p ,...,α 0 ( x 2 , . . . , x q +1 ) + q X k =1 ( − 1) k f α p ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q ) + ( − 1) q +1 f α p ,...,α 0 ( x 1 , . . . , x q ) · A ( α )( x q +1 ) . Lemma 5.3. The maps ∂ and δ ar e c ob oundary maps. ∂ 2 = 0 = δ 2 . Pr o of. ∂ ( f ) = P q +1 k =0 ( − 1) k ∂ k ( f ) where ( ∂ k ( f ))( x 1 , . . . , x q +1 ) =    A ( α )( x 1 ) · f α p ,...,α 0 ( x 2 , . . . , x q +1 ) k = 0 , f α p ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q +1 ) 0 < k < q + 1 , f α p ,...,α 0 ( x 1 , . . . , x q ) · A ( α )( x q +1 ) k = q + 1 . δ ( f ) = P p +2 k =0 ( − 1) k δ k ( f ) where ( δ k ( f ))( x 1 , . . . , x q ) =    f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x q )) k = 0 , f α p +1 ,...,α k α k − 1 ,...,α 0 ( x 1 , . . . , x q ) 0 < k < p + 2 , M ( α p +1 )( f α p ,...,α 0 ( x 1 , . . . , x q )) k = p + 2 . ∂ 2 = 0 = δ 2 follows from: ∂ k ∂ l = ∂ l ∂ k − 1 0 ≤ l < k ≤ q + 2 , δ k δ l = δ l δ k − 1 0 ≤ l < k ≤ p + 2 . 46 Lemma 5.4. The c ob ou n dary maps ∂ and δ c ommute. δ ∂ = ∂ δ. The pro o f is given on the next pag e. Pr o of. Let f ∈ C p,q H ar r ( I , A, M ). δ ∂ ( f ) = A ( α )( x 1 ) · f α p +1 ,...,α 1 ( A ( α 0 )( x 2 ) , . . . , A ( α 0 )( x q +1 )) + q X k =1 ( − 1) k f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x k x k +1 ) , . . . , A ( α 0 )( x q +1 )) + ( − 1) q +1 f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x q )) · A ( α )( x q +1 ) + p X l =0 ( − 1) l +1 [ A ( α )( x 1 ) · f α p +1 ,...,α l +1 α l ,...,α 0 ( x 2 , . . . , x q +1 ) + q X k =1 ( − 1) k f α p +1 ,...,α l +1 α l ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q +1 ) + ( − 1) q +1 f α p +1 ,...,α l +1 α l ,...,α 0 ( x 1 , . . . , x q ) · A ( α )( x q +1 )] + ( − 1) p +2 M ( α p +1 )[ A ( α p · · · α 0 )( x 1 ) · f α p ,...,α 0 ( x 2 , . . . , x q +1 ) + q X k =1 ( − 1) k f α p ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q +1 ) + ( − 1) q +1 f α p ,...,α 0 ( x 1 , . . . , x q ) · A ( α p · · · α 0 )( x q +1 )] = A ( α )( x 1 ) · [ f α p +1 ,...,α 1 ( A ( α 0 )( x 2 ) , . . . , A ( α 0 )( x q +1 )) + p X l =0 ( − 1) l +1 f α p +1 ,...,α l +1 α l ,...,α 0 ( x 2 , . . . , x q +1 ) + ( − 1) p +2 M ( α p +1 ) f α p ,...,α 0 ( x 2 , . . . , x q +1 )] + q X k =1 ( − 1) k [ f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x k x k +1 ) , . . . , A ( α 0 )( x q +1 )) + p X l =0 ( − 1) l +1 f α p +1 ,...,α l +1 α l ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q +1 ) + ( − 1) p +2 M ( α p +1 ) f α p ,...,α 0 ( x 1 , . . . , x k x k +1 , . . . , x q +1 )] + ( − 1) q +1 [ f α p +1 ,...,α 1 ( A ( α 0 )( x 1 ) , . . . , A ( α 0 )( x q )) + p X l =0 ( − 1) l +1 f α p +1 ,...,α l +1 α l ,...,α 0 ( x 1 , . . . , x q ) + ( − 1) p +2 M ( α p +1 )( f α p ,...,α 0 ( x 1 , . . . , x q ))] · A ( α )( x q +1 ) = ∂ δ ( f ) . 5.4 Harrison cohomology of diagrams of comm utativ e alge- bras Let A : I → Co m . alg b e a diagram of comm utative algebr as, and M b e an A -mo dule. W e deﬁne the Harrison c ohomo lo gy of A with co eﬃcie nts in M , denoted by H arr ∗ ( I , A, M ), to b e the cohomolog y of the total complex o f C ∗ , ∗ H ar r ( I , A, M ). The sp ectra l sequence of a bicomplex yields the following spec tr al sequence. E p,q 2 = H p B W ( I , H q +1 H ar r ( A, M )) ⇒ H ar r p + q ( I , A, M ) , where H q H ar r ( A, M ) is the na tural system on I who se v alue on ( α : i → j ) is given b y H ar r q ( A ( i ) , α ∗ M ( j )). 47 Deﬁnition 5.5. A derivation d : A → M is of the form d = ( d i ) i ∈ I where ea ch d i : A ( i ) → M ( i ) is a deriv atio n of A ( i ) with v a lues in M ( i ) such that for all ( α : i → j ) ∈ I we hav e that M ( α )( d i ) = d j ( A ( α )). W e denote the set of a ll deriv ations of A with v alues in M by Der ( A, M ). Lemma 5.6. H ar r 0 ( I , A, M ) ∼ = Der ( A, M ) , H 0 B W ( I , H 1 H ar r ( A, M )) ∼ = Der ( A, M ) . Deﬁnition 5.7. An additively split extens ion of A by M is an exact se q uence of functors 0 / / M q / / X p / / A / / 0 where X : I → Co m . alg s uch that for all i ∈ I we get an additively split extens ion of A ( i ) by M ( i ). 0 / / M ( i ) q ( i ) / / X ( i ) p ( i ) / / A ( i ) / / 0 This means tha t there are additive homomorphisms s ( i ) : A ( i ) → X ( i ) for all i ∈ I such that s ( i ) is a section of p ( i ). The sections induce additive isomorphis ms M ( i ) ⊕ A ( i ) ≈ X ( i ) where addition is given b y ( m, a ) + ( m ′ , a ′ ) = ( m + m ′ , a + a ′ ) and multiplication is given b y ( m, a )( m ′ , a ′ ) = ( a ′ m + am ′ + f i ( a, a ′ ) , aa ′ ) , where f i : A ( i ) × A ( i ) → M ( i ) is a bilinear map given b y f i ( a, a ′ ) = s ( i )( a ) s ( i )( a ′ ) − s ( i )( aa ′ ) . Asso ciativity in X ( i ) gives us 0 = af i ( a ′ , a ′′ ) − f i ( aa ′ , a ′′ ) + f i ( a, a ′ a ′′ ) − f i ( a, a ′ ) a ′′ . Commutativit y in X ( i ) gives us f i ( a, a ′ ) = f i ( a ′ , a ) . F or all ( α : i → j ) ∈ I we iden tify M ( j ) with Ker( p ( j )) a nd M ( α ) with the restr ic tion of X ( α ) to get a map ǫ α : A ( i ) → M ( j ) g iven by ǫ α ( a ) = X ( α )( s ( i )( a )) − s ( j )( A ( α )( a )) , which s atisﬁes the following prop er ties: 1. ǫ id ( a ) = 0, 2. ǫ α ( a + a ′ ) = ǫ α ( a ) + ǫ α ( a ′ ), 3. ǫ α ( aa ′ ) = A ( α )( a ) ǫ α ( a ′ ) + A ( α )( a ′ ) ǫ α ( a ) + f j ( A ( α )( a ) , A ( α )( a ′ )) − M ( α )( f i ( a, a ′ )), 4. ǫ β α ( a ) = M ( β )( ǫ α ( a )) + ǫ β ( A ( α )( a )). Two additively split extensions ( X ) , ( X ′ ) with A, M ﬁxed ar e said to b e e qu ivalent if ther e exists a map of diagra ms φ : X → X ′ such that the following diagram commutes. 0 / / M / / X / / φ   A / / 0 0 / / M / / X ′ / / A / / 0 F or all i ∈ I we g et that φ i : X ( i ) → X ′ ( i ) is a homomo rphism of comm utative a lg ebras. Hence φ i ( m, a ) = ( m + g i ( a ) , a ) for so me g i : A → M such that g i ( a + a ′ ) = g i ( a ) + g i ( a ′ ) , f i ( a, a ′ ) − f ′ i ( a, a ′ ) = ag i ( a ) − g i ( aa ′ ) + g i ( a ) a ′ . 48 F or all α ∈ I we get that ǫ α ( a ) − ǫ ′ α ( a ) = M ( α )( g i ( a )) − g j ( A ( α )( a )) . W e denote the set o f equiv alence classes of additively split extensions of A by M by AExt ( A, M ). An additively and multiplic a tively split extension of A b y M is an additively split e x tension of A by M 0 / / M q / / X p / / A / / 0 such that for each i ∈ I the a rrow p ( i ) is additively and m ultiplicatively split. W e denote the set of equiv alence classes of additiv ely and multiplicativ ely s plit extens io ns of A by M by MExt ( A, M ). Lemma 5.8. H ar r 1 ( I , A, M ) ∼ = AExt ( A, M ) . Pr o of. A 1-co cycle is a pair ( f i : A ( i ) × A ( i ) → M ( i )) i ∈ I and ( ǫ α : A ( i ) → M ( j )) ( α : i → j ) ∈ I . W e get an additively split extension of A by M given by taking the exact sequence 0 / / M / / M ⊕ A / / A / / 0 where addition in M ⊕ A is given by ( m, a ) + ( m ′ , a ′ ) = ( m + m ′ , a + a ′ ) and multiplication is given by ( m, a )( m ′ , a ′ ) = ( a ′ m + am ′ + f i ( a, a ′ ) , aa ′ ) . F or all ( α : i → j ) ∈ I set the map ( M ⊕ A )( α ) : ( M ⊕ A )( i ) → ( M ⊕ A )( j ) to b e ( M ⊕ A )( α )( m, a ) = ( M ( α )( m ) + ǫ α ( a ) , A ( α )( a )) . Given tw o 1- co cycles which diﬀer by a 1-cob ounda r y , then the tw o additively s plit extensions we get are equiv alent. Given an additively split extension of A by M 0 / / M q / / X p / / A / / 0 there are additiv e homomorphisms s ( i ) : A ( i ) → X ( i ) for all i ∈ I such that s ( i ) is a section of p ( i ). F or all i ∈ I we deﬁne the maps f i : A ( i ) × A ( i ) → M ( i ) to b e given b y f i ( a, a ′ ) = s ( i )( a ) s ( i )( a ′ ) − s ( i )( aa ′ ) . F or all ( α : i → j ) ∈ I we deﬁne the maps ǫ α : A ( i ) → M ( j ) to b e given by ǫ α ( a ) = X ( α )( s ( i )( a )) − s ( j )( A ( α )( a )) . Then ( f i : A ( i ) × A ( i ) → M ( i )) i ∈ I and ( ǫ α : A ( i ) → M ( j )) ( α : i → j ) ∈ I give us a 1 -co cycle. Given t wo additively split extensions which are equiv alent , then the tw o 1-co cycles we g et diﬀer by a 1-cob oundar y . Corollary 5.9 . H 1 B W ( I , H 1 H ar r ( A, M )) ∼ = MExt ( A, M ) . Deﬁnition 5. 10. An additively split cr osse d ext en sion of A by M is an exact sequence of functors 0 / / M φ / / C 1 ρ / / C 0 π / / A / / 0 such that for all i ∈ I we get an additively split cros sed extension of A ( i ) by M ( i ). 0 / / M ( i ) φ ( i ) / / C 1 ( i ) ρ ( i ) / / C 0 ( i ) γ ( i ) / / A ( i ) / / 0 (5.1) This means that all the arrows in the exact s e quence 5.1 are additively split. W e le t π 0 ACross ( A, M ) denote the connected comp onents of the category of additiv ely split crossed extensions of A by M . 49 An additively and multiplic ativ ely split cr osse d ex tension of A by M is an exact sequence of functors 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 such that for all i ∈ I we get an a dditively and multiplicativ ely split cross ed extension o f A ( i ) by M ( i ), 0 / / M ( i ) φ ( i ) / / C 1 ( i ) ρ ( i ) / / C 0 ( i ) γ ( i ) / / A ( i ) / / 0 (5.2) where γ ( i ) and ρ ( i ) ar e additively a nd multiplicativ ely split. W e let π 0 MCross ( A, M ) deno te the connected comp onents of the categ ory of additively and multiplicativ ely split cross ed ex tensions of A by M . Lemma 5.11. If γ : C 0 → A is a morphism of diagr ams of c ommutative algebr as then H ar r 1 ( I , γ : C 0 → A, M ) ∼ = π 0 ACross ( γ : C 0 → A, M ) , wher e H ar r ∗ ( I , γ : C 0 → A, M ) and π 0 ACross ( γ : C 0 → A, M ) ar e deﬁne d as fol lows. Consider the fol lowing short exact se quenc e of c o cha in c omplexes: 0 / / C ∗ H ar r ( I , A, M ) γ ∗ / / / / C ∗ H ar r ( I , C 0 , M ) / / κ ∗ / / Coker( γ ∗ ) / / 0 , wher e C ∗ H ar r ( I , A, M ) denotes the total c omplex of the bic omplex ( C ∗ , ∗ H ar r ( I , A, M ) . We deﬁne the c o chain c omplex C ∗ H ar r ( I , γ : C 0 → A, M ) := Coker( γ ∗ ) . This al lows u s to deﬁne the r elative Harrison c oho molo gy H ar r ∗ ( I , γ : C 0 → A, M ) := H ∗ ( C ∗ H ar r ( I , γ : C 0 → A, M )) . We let ACross ( γ : C 0 → A, M ) denote the c ate gory whose obje cts ar e the a dditively split cr osse d extensions of A by M 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 with γ : C 0 → A ﬁx e d. A morphism b etwe en two of these cr osse d extensions c onsists of a morphi sm of diagr ams of c ommutative algebr as h 1 : C 1 → C 1 such that the fol lowing diagr am c ommutes. 0 / / M φ / / C 1 h 1   ρ / / C 0 γ / / A / / 0 0 / / M φ ′ / / C ′ 1 ρ ′ / / C 0 γ / / A / / 0 Note that A Cross ( γ : C 0 → A, M ) is a gr oup oid. Pr o of. W e use the metho d used in [1 3] fo r the crossed mo dules o f Lie algebras. Given any additively split cro ssed module of A by M , 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 , we let V = Ker γ = Im ρ . F or all ob jects i ∈ I there a re linear sections s i : A ( i ) → C 0 ( i ) of γ and σ i : V ( i ) → C 1 ( i ) of ρ ( i ) : C 1 ( i ) → V ( i ). W e deﬁne the maps g i : A ( i ) ⊗ A ( i ) → C 1 ( i ) by: g i ( a, b ) = σ i ( s i ( a ) s i ( b ) − s i ( ab )) . W e also deﬁne the maps ω i : C 0 ( i ) → C 1 ( i ) by: ω i ( c ) = σ i ( c − s i γ i ( c )) . By identifying M with Ker ρ , we deﬁne the ma ps f i : C 0 ( i ) ⊗ C 0 ( i ) → M ( i ) b y: f i ( c, c ′ ) = g i ( γ i ( c ) , γ i ( c ′ )) + c ′ ω i ( c ) + cω i ( c ′ ) − ω i ( c ) ∗ ω i ( c ′ ) − ω i ( cc ′ ) . Since g i ( c, c ′ ) = g i ( c ′ , c ), it follows that f i ( c, c ′ ) = f i ( c ′ , c ) and so f i ∈ C 2 H ar r ( C 0 ( i ) , M ( i )). 50 F or all morphisms ( α : i → j ) ∈ I we deﬁne the maps q α : A ( i ) → C 1 ( j ) by: q α ( a ) = σ j ( C 0 ( α )( s i ( a )) − s j ( A ( α )( a ))) . By identifying M with Ker ρ , we deﬁne the ma ps e α : C 0 ( i ) → M ( j ) by: e α ( c ) = ω j ( C 0 ( α )( c )) − C 1 ( α )( ω i ( c )) − q α ( γ i ( c )) . Note that e α ∈ C 1 H ar r ( C 0 ( i ) , α ∗ M ( j )). F or all ob jects i ∈ I we deﬁne the maps θ i ∈ C 3 H ar r ( A ( i ) , M ( i )) by: θ i ( x, y , z ) = s i ( x ) g i ( y , z ) − g i ( xy , z ) + g i ( x, y z ) − g i ( y , x ) s i ( z ) . F or all morphisms ( α : i → j ) ∈ I we deﬁne the maps ϑ α ∈ C 2 H ar r ( A ( i ) , α ∗ M ( j )) by: ϑ α ( x, y ) = g j ( A ( α )( x ) , A ( α )( y )) − C 1 ( α ) g i ( x, y ) + C 0 ( α )( s i ( x )) q α ( y ) − q α ( xy ) + q α ( x ) s j ( A ( α )( y )) . F or all pairs of compo sable morphisms ( β α : i → j → k ) ∈ I we deﬁne the maps η β α ∈ C 1 H ar r ( A ( i ) , ( β α ) ∗ M ( k )) b y: η β α ( x ) = − q β ( A ( α )( x )) + q β α ( x ) − C 1 ( β )( q α ( x )) . W e let f = ( f i ) ( i ∈ I ) and e = ( e α ) ( α : i → j ∈ I ) . W e also let θ = ( θ i ) ( i ∈ I ) , ϑ = ( ϑ α ) ( α : i → j ∈ I ) and η = ( η β α ) ( β α : i → j → k ∈ I ) . Consider the following comm utativ e diagr a m. 0 / / C 1 H ar r ( I , A, M ) γ ∗ / /   C 1 H ar r ( I , C 0 , M ) κ ∗ / / δ   C 1 H ar r ( I , γ : C 0 → A, M ) / / δ   0 0 / / C 2 H ar r ( I , A, M ) γ ∗ / / C 2 H ar r ( I , C 0 , M ) κ ∗ / / C 2 H ar r ( I , γ : C 0 → A, M ) / / 0 Note that ( f , e ) ∈ C 1 H ar r ( I , C 0 , M ) and ( θ , ϑ, η ) ∈ C 2 H ar r ( I , A, M ). A direct calcula tion shows that δ ( f , e ) = γ ∗ ( θ, ϑ, η ). W e a lso hav e that δ κ ∗ ( f , e ) = κ ∗ δ ( f , e ) = κ ∗ γ ∗ ( θ, ϑ, η ) = 0, this tells us tha t κ ∗ ( f , e ) is a co cycle. If w e hav e tw o equiv alent a dditively split crossed mo dules then w e can choose sections in s uch a w ay that the asso ciated co cycles are the same. Therefore we hav e a well-deﬁned map: ACross ( γ : C 0 → A, M ) → H 2 H ar r ( I , γ : C 0 → A, M ) . Inv ersely , assume we hav e a co cycle in C 1 H ar r ( I , γ : C 0 → A, M ) which we lift to a co chain ( f , e ) ∈ C 1 H ar r ( I , C 0 , M ). Let V = Ker γ . F or all ob jects i ∈ I we deﬁne C 1 ( i ) = M ( i ) × V ( i ) a s a mo dule ov er k with the following action of C 0 ( i ) on C 1 ( i ): c ( m, v ) := ( cm + f i ( c, v ) , cv ) . The maps C 1 ( α ) : C 1 ( i ) → C 1 ( j ) are given b y: C 1 ( α )( m, v ) := ( M ( α )( m ) + e α ( v ) , C 0 ( α )( v )) . It is easy to c heck us ing the prop erties of f i and e α that this action is well deﬁned a nd together with the maps ρ i : C 0 ( i ) → C 1 ( i ) given by ρ i ( m, v ) = v , we hav e an additively split crossed mo dule of A by M . Lemma 5.12. If k is a ﬁeld of char acteristic 0 then H ar r 2 ( I , A, M ) ∼ = π 0 ACross ( A , M ) . Pr o of. F rom the deﬁnition of C ∗ H ar r ( I , γ : C 0 → A, M ) we get the lo ng exact seq uence: . . . / / H ar r 1 ( I , A, M ) / / H ar r 1 ( I , C 0 , M ) / / H ar r 1 ( I , γ : C 0 → A, M ) / / H ar r 2 ( I , A, M ) / / . . . (5.3) 51 Given an y additively split crossed mo dule in π 0 ACross ( A, M ), 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 we can lift γ to g et a map P 0 → A where P 0 is free as a diagram o f comm utative algebr a s. W e can then use a pullbac k to co nstruct P 1 to get a crossed mo dule where the following diagra m commutes: 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 0 / / M / / P 1 O O / / P 0 O O / / A / / 0 These tw o crossed mo dules a re in the same connected co mpo nent of π 0 ACross ( A, M ). B y cons id- ering the second cro ssed mo dule in the lo ng exact sequence, we r eplace C 0 by P 0 to get the new exact sequence: 0 / / H ar r 1 ( I , γ : P 0 → A, M ) / / H ar r 2 ( I , A, M ) / / 0 (5.4) since H arr 1 ( I , P 0 , M ) = 0 and H ar r 2 ( I , P 0 , M ) = 0. The exact sequence 5 .4 tells us that every element in H ar r 2 ( I , A, M ) comes from an element in H ar r 1 ( I , γ : P 0 → A, M ) and the previo us lemma tells us that this comes from a cross ed mo dule in π 0 ACross ( A, M ). T he r efore the map π 0 ACross ( A, M ) → H ar r 2 ( I , A, M ) is sur jective. Assume we hav e tw o cro ssed modules which go to the same element in H ar r 2 ( I , A, M ), 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 , (5.5) 0 / / M φ ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 . (5.6) There exist mor phisms 0 / / M φ / / C 1   ρ / / C 0   γ / / A / / 0 0 / / M / / P 1 / / P 0 / / A / / 0 , 0 / / M φ ′ / / C ′ 1   ρ ′ / / C ′ 0   γ ′ / / A / / 0 0 / / M / / P 2 / / P 0 / / A / / 0 , where P 0 is free as a diagram of c o mm utative alg e bras and P 1 , P 2 are constructed via pull- backs. These give us tw o elements in H arr 1 ( I , γ : P 0 → A, M ) which go to the same element in H ar r 2 ( I , A, M ). Howev er the exact sequence 5 .4 tells us that the tw o cross ed modules 5 .5 and 5.6 ha ve to go to the s ame elemen t in H ar r 1 ( I , γ : P 0 → A, M ). The previo us lemma tells us that the t wo cr ossed mo dules 5.5 a nd 5.6 go to the same element in ACross ( γ : C 0 → A, M ) which is a group oid, so there is a map P 2 → P 1 which ma kes the following dia g ram commute: 0 / / M φ / / C 1 ρ / / C 0 γ / / A / / 0 0 / / M / / P 1 O O / / P 0 O O / / A / / 0 0 / / M / / P 2   / / P 0   / / A / / 0 0 / / M φ ′ / / C ′ 1 ρ ′ / / C ′ 0 γ ′ / / A / / 0 52 Therefore the t wo cr ossed mo dules 5.5 and 5.6 are in the s a me connected comp onent of π 0 ACross ( A, M ) and the map π 0 ACross ( A, M ) → H ar r 2 ( I , A, M ) is injective. Corollary 5.1 3. If k is a ﬁ eld of char acteristic 0 then H 2 B W ( I , H 1 H ar r ( A, M )) ∼ = π 0 MCross ( A, M ) . Pr o of. Giv en an additively and multip licatively split cro ssed extension of A by M w e get that (with the notation of lemma 5.11) g i = 0 for all i ∈ I . Since ρ ( i ) is additively and m ultiplicatively split for all i ∈ I it follows that f = 0, θ = 0 and ϑ = 0 . Ther efore η is a cocycle in C 2 B W ( I , H 1 H ar r ( A, M )). Inv ersely , the construction giv en in lemma 5.11 gives us an additiv ely and m ultiplicatively split extension. 5.5 Harrison cohomology of Ψ -rings Let R b e a Ψ-r ing, and M ∈ R − mod Ψ . Let I denote the c a tegory with one o b ject as so ciated to the multiplicativ e monoid of the na tur al num bers N mult . F o r any j ≥ 1, there is a natural system on I as follows: D f := C j H ar r ( R, f ∗ M ) , where f ∗ M is an Ψ-mo dule over R with M as a n ab elian g roup and the following a ction of R ( r , m ) 7→ Ψ f ( r ) m, for r ∈ R, m ∈ M . F or u ∈ F I (the categor y of factoris ations in I ), we have u ∗ : D f → D uf which is induced by Ψ u : f ∗ M → ( u f ) ∗ M . F or v ∈ F I , we ha ve v ∗ : D f → D f v which is induced b y Ψ v : R → R . The bicomplex in section 5.3 b e comes C 1 H ar r ( R, M ) d   b / / Q i ∈ N C 1 H ar r ( R, i ∗ M ) − d   b / / Q i,j ∈ N C 1 H ar r ( R, ( ij ) ∗ M ) d   b / / · · · C 2 H ar r ( R, M ) d   b / / Q i ∈ N C 2 H ar r ( R, i ∗ M ) − d   b / / Q i,j ∈ N C 2 H ar r ( R, ( ij ) ∗ M ) d   b / / · · · C 3 H ar r ( R, M ) d   b / / Q i ∈ N C 3 H ar r ( R, i ∗ M ) − d   b / / Q i,j ∈ N C 3 H ar r ( R, ( ij ) ∗ M ) d   b / / · · · . . . . . . . . . with d : Y t = t 1 ...t i ∈ N C j H ar r ( R, t ∗ M ) → Y t = t 1 ...t i ∈ N C j +1 H ar r ( R, t ∗ M ) , with the pr o duct being ov er i -tuples ( t 1 , . . . , t i ) and t is the comp os ite, is given b y d f t 1 ,...,t i ( x 1 , . . . , x j +1 ) =Ψ t 1 t 2 ...t i ( x 1 ) f t 1 ,...,t i ( x 2 , . . . , x j +1 ) + j X k =1 ( − 1) k f t 1 ,...,t i ( x 1 , . . . , x k x k +1 , . . . , x j +1 ) + ( − 1) j +1 f t 1 ,...,t i ( x 1 , . . . , x j )Ψ t 1 t 2 ...t i ( x j +1 ) . and b : Y t = t 1 ...t i ∈ N C j H ar r ( R, t ∗ M ) → Y t = t 1 ...t i +1 ∈ N C j H ar r ( R, t ∗ M ) , 53 being given b y bf t 1 ,...,t i +1 ( x 1 , . . . , x j ) =Ψ t 1 f t 2 ,...,t i +1 ( x 1 , . . . , x j ) + i X k =1 ( − 1) k f t 1 ,...,t k t k +1 ,...,t i +1 ( x 1 , . . . , x j ) + ( − 1) i +1 f t 1 ,...,t i (Ψ t i +1 ( x 1 ) , . . . , Ψ t i +1 ( x j )) . W e let H ar r i Ψ ( R, M ) deno te the i th cohomolog y o f the total complex of the bicomplex descr ibe d ab ov e. Theorem 5.14. Ther e exists a sp e ctr al se quenc e E p,q 2 = H p B W ( I , H q +1 H ar r ( R, M )) ⇒ H ar r p + q Ψ ( R, M ) . wher e H q H ar r ( R, M ) is the n atur al s yst em on I whose value on ( α : i → j ) is given by H arr q ( R, α ∗ M ) . Theorem 5.15. H ar r 0 Ψ ( R, M ) = Der Ψ ( R, M ) , H ar r 1 Ψ ( R, M ) = AExt Ψ ( R, M ) , H ar r 2 Ψ ( R, M ) = π 0 AC ross Ψ ( R, M ) . 5.6 Harrison cohomology and λ -rings Let R be a λ -ring and M ∈ R − mod λ . Conjecture 5.16. Ther e exists a c o cha in bic omplex which start s : C 1 1 − H ar r ( R , M ) d 1   b 1 1 / / C 1 2 − H ar r ( R, M )) − d 2   b 1 2 / / C 1 3 − H ar r ( R, M ))   / / · · · C 2 1 − H ar r ( R , M ) d 1   b 2 1 / / C 2 2 − H ar r ( R, M )) − d 2   / / . . . C 3 1 − H ar r ( R , M ) d 1   / / . . . . . . wher e the ﬁrst c olumn is the H arrison c o chain c omplex. C i 1 − H ar r ( R , M ) := C i H ar r ( R, M ) . F or al l i ≥ 1 and j ≥ 2 we have that C i j − H ar r ( R , M ) ⊂ Y n 1 ,...,n j − 1 ∈ N M a ps ( R ⊗ i , M ) . F or example, when j = 2 , we have C 1 2 − H ar r ( R , M )) = { f ∈ Y n ∈ N M a ps ( R , M ) | f n ( r + s ) = n X j =1 [ f j ( r ) λ n − j ( s ) + f j ( s ) λ n − j ( r )] } . 54 C 2 2 − H ar r ( R, M )) = { f ∈ Y n ∈ N M a ps ( R ⊗ R, M ) | f n ( r , s ) = f n ( s, r ) , f n (( r , s ) + ( t, u )) = n X j =1 [ f j ( r , s ) λ n − j ( tu + r u + ts ) + f j ( t, u ) λ n − j ( rs + r u + ts ) + f j ( r , u ) λ n − j ( rs + tu + ts )+ f j ( t, s ) λ n − j ( rs + r u + tu )] } . The c ob oundary maps d 2 : C i 2 − H ar r ( R , M ) → C i +1 2 − H ar r ( R, M ) ar e given by ( d 2 ( f )) n ( r 1 , . . . , r i +1 ) = n X j =1 [ ∂ P n ( r 1 , r 2 . . . r i +1 ) ∂ λ j ( r 2 . . . r i +1 ) f j ( r 2 , . . . , r i +1 )] + i X j =1 f n ( r 1 , . . . , r j r j +1 , . . . , r i +1 ) + n X j =1 [ ∂ P n ( r 1 . . . r i , r i +1 ) ∂ λ j ( r 1 . . . r i ) f j ( r 1 , . . . , r i )] . ( b 1 1 ( g )) n ( r ) = g ( λ n ( r )) − n X i =1 Λ i ( g ( r )) λ n − i ( r ) . ( b 1 2 ( f )) n,m ( r ) = f m ( λ n ( r )) − nm X i =1 f i ( r ) ∂ P n,m ( r ) ∂ λ i ( r ) + m X j =1 Λ j ( f n ( r )) λ m − j ( λ n ( r )) . We let H ar r i λ ( R, M ) denote the i th c oh omolo gy of the bic omplex ab ove. Then we get t he fol lowing H ar r 0 λ ( R, M ) ∼ = Der λ ( R, M ) , H ar r 1 λ ( R, M ) ∼ = AExt λ ( R, M ) . 5.7 Gerstenhab er-S chac k cohomology In the pap er [7] Gerstenhab er and Schac k describe a coho mology for diagrams of asso ciative a lge- bras w hich we denote by H ∗ GS ( I , A, M ). Let I = { i, j, k , . . . } be a partially ordered s et. W e can view I as the set of o b jects of a catego ry in which there exists a unique morphism i → j when i ≤ j . They deﬁne a diag ram to b e a co nt rav ar ia nt functor A : I op → Co m . alg . They deﬁne an A -mo dule to b e a co nt rav ar iant f unctor M : I op → Ab s uch that M ( i ) ∈ A ( i ) − mod for a ll i ∈ I and fo r each i ≤ j the map M ( i → j ) is an A ( j )-mo dule homomo r phism where A ( i ) is viewed as an A ( j )-module via the morphism A ( i → j ). If we co nsider A a s a cov a riant functor A : I → Com . alg and M as a cov a riant functor M : I → Ab then we can apply the theor y we dev elope d e a rlier. The bico mplex describ ed by Gerstenhab er and Sc hack coincides with our bico mplex C p,q H ar r ( I , A, M ). Therefore H n GS ( I , A, M ) = H arr n ( I , A, M ) for n ≥ 0. There fore w e g et a new sp ectr a l sequence E p,q 2 = H p B W ( I , H q +1 H ar r ( A, M )) ⇒ H p + q GS ( I , A, M ) , where H q H ar r ( A, M ) is the na tural system on I who se v alue on ( α : i → j ) is given b y H ar r q ( A ( i ) , α ∗ M ( j )). 55 Chapter 6 Andr ´ e-Quillen cohomology of diagrams of algebras In this chapter, let C denote a catego ry with limits, a nd I denote a small categor y . W e hav e a lready seen that for a lgebraic ob jects, we can get co ho mology from mo nads and comonads. In this c hapter, we deﬁne a c o homology fo r diag rams o f a lg ebras. Our approach ca n be describ ed as follows. First, we ﬁx a small category I . A diag ram of algebras is a functor I → Alg ( T ), where T is a monad on sets. F or appro priate T , one gets a diag ram o f groups, a diagram of Lie algebr a s, a diagram of commutativ e r ings, etc. The adjoint pair A lg ( T ) / / Sets o o yields a comonad which we denote by G . W e ca n a ls o consider the catego ry I 0 , which has the same ob jects as I , but only the identit y mor phisms. The inclusion I 0 ⊂ I yields the functor Sets I → S ets I 0 which has a left adjoint given by the left Kan extension. W e also hav e the pair of adjoint functors Alg ( T ) I / / Sets I o o which comes fr om the adjoint pair Alg ( T ) / / Sets o o . By gluing these diagra ms together, one gets a nother adjoint pair Alg ( T ) I / / Sets I 0 . o o This adjoint pair yields a comonad which we denote b y G I . W e will prov e that A lg ( T ) I is monadic in Sets I 0 and the right cohomolo g y theor y of diagr ams of algebras is one whic h is asso ciated to the comona d G I . These cohomo logy theories are deno ted b y H ∗ G I ( A, M ). 6.1 Base change Let C b e a category , and X be an ob ject in C . An X -mo dule in C is an a be lia n g roup o b ject in the catego ry C /X , X − mod := Ab( C /X ) . Theorem 6.1. L et f : X → Y b e a morphism in C , then ther e exist s a b ase-change functor f ∗ : Y − mod → X − mod via pul lb ack s. Pr o of. The pro duct in the slice categ ory is given by pullbacks. The functor we are going to use is f ∗ : C / Y → C /X g iven by pullbacks. f ∗ ( M ) / /   M p   X f / / Y If M ∈ Y − m od then f ∗ ( M ) has a ca nonical X - mo dule structure. In set-theo retic notation, f ∗ ( M ) = { ( x, m ) | x ∈ X , m ∈ M , f ( x ) = p ( m ) } , f ∗ ( M ) × X f ∗ ( M ) = { ( x, m, m ′ ) | x ∈ X , m, m ′ ∈ M , f ( x ) = p ( m ) = p ( m ′ ) } , f ∗ ( M ) × X f ∗ ( M ) ≃ f ∗ ( M × Y M ) . 56 Consider the following commuting diagram. f ∗ ( M × Y M ) / /   ∃ ! M × Y M   mult   6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 X f / / @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Y 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 f ∗ ( M ) / /   M   X f / / Y The unique morphism f ∗ ( mul t ) : f ∗ ( M × Y M ) → f ∗ ( M ) exists by the universal prop erty of pullbacks. The isomorphism f ∗ ( M ) × X f ∗ ( M ) ≃ f ∗ ( M × Y M ) and this unique mor phism yie ld m ultiplication f ∗ ( mul t ) : f ∗ ( M ) × X f ∗ ( M ) → f ∗ ( M ) , which g ives an abelia n group ob ject structur e on f ∗ ( M ). 6.2 Deriv ations F or M ∈ X − m od , one deﬁnes a derivation from X to M to b e a morphism d : X → M which is a se ction o f the ca nonical mor phism M → X . Let Der( X , M ) denote the set of deriv ations d : X → M . This is a sp ecial case o f 2.2 and ther e is an ab elian group structure. W e will require the following useful theorem later. Theorem 6.2. If X = ` α ∈ I X α and M ∈ X − mod , then Der( X , M ) ∼ = Y α ∈ I Der( X α , M α ) , wher e M α is the X α -mo dule pr o duc e d fr om M by t he b ase-change functor fr om the morphism i α : X α → X . Pr o of. F rom the deﬁnition of the copro duct one has a mo rphism i α : X α → X . Using this one gets M α ∈ X α - mod via the following pullback diagram. M p   M α j α o o p α   X X α i α o o Let f b e a section of p , this means that pf = id X . Consider the following diagram. X α f α } } f i α v v m m m m m m m m m m m m m m m m id X α                  M p   M α j α o o p α   X f O O X α i α o o The diagr am commutes since pf i α = id X i α = i α id X α . By the universal pro per ty of pullbacks p α f α = id X α . So if f is a section of p then f α is a s e ction of p α . 57 Conv ersely , le t f α be a section of p α , this means that p α f α = id X α . By the deﬁnition of the copro duct there exis ts a unique mo rphism f such that the following diagram commutes. M X f O O X α i α o o j α f α a a C C C C C C C C This mea ns that f i α = j α f α . Comp osing with p on the left gives us that pf i α = pj α f α = i α p α f α = i α id X α = i α Thu s the following diagram comm utes. X id X / / pf / / X X α i α O O i α = = | | | | | | | | The universal prop erty o f the co pro duct says that pf = id X . Hence f is a s ection of p . W e will re q uire the following useful lemma la ter. Lemma 6.3 . F or al l obje cts Z ∈ C I , for M ∈ G I ( Z ) − mod , and α : i → j in the smal l c ate gory I , one has Der( G ( Z ( i )) , α ∗ M ( j )) = Y m ∈ U Z ( i ) p − 1 j α ∗ γ i ( m ) , wher e p j is the c anonic al morphism p j : M ( j ) → GZ ( j ) and γ i is the inclusion γ i : U Z ( i ) → GZ ( i ) . Pr o of. The deriv a tions Der( G ( Z ( i )) , α ∗ M ( j )) a re the sections of p α in the following pullbac k dia- gram. α ∗ M ( j ) / / p α   M ( j ) p j   U Z ( i )   γ i / / GZ ( i ) α ∗ / / GZ ( j ) By deﬁnition, U Z ( i ) is the basis of the free ob ject GZ ( i ). Der( G ( Z ( i )) , α ∗ M ( j )) = { s : U Z ( i ) → M ( j ) | α ∗ γ i = p j s, s is a set map. } = Y m ∈ U Z ( i ) p − 1 j α ∗ γ i ( m ) . 6.3 Natural system W e require the following useful theo rem. Theorem 6. 4. L et A ∈ C I and M ∈ A - mod . If α : i → j is a morphism in I then M ( j ) ∈ A ( j ) − mo d and Der ( A, M )( α ) = Der( A ( i ) , α ∗ M ( j )) , deﬁnes a natur al system on I . Pr o of. Start b y ﬁxing A and M , then let D ( α ) denote Der ( A, M )( α ). Let γ , α, β ∈ I such that i ′ γ / / i α / / j β / / j ′ . W e are going to show that we have induced maps as follows. D ( αγ ) D ( α ) γ ∗ o o β ∗ / / D ( β α ) . 58 Let s ∈ D ( α ), then the following diagra m commutes with ps = id A ( i ) , and α ∗ M ( j ) is a pullback, α ∗ M ( j ) ∈ A ( i ) − mod . α ∗ M ( j ) / / p   M ( j )   A ( i ) s O O A ( α ) / / A ( j ) Consider the following comm uting diagra m. M ( i )   ∃ !   M ( α ) $ $ I I I I I I I I I I I I I I I I I I I I α ∗ M ( j ) p   ∃ ! τ   / / M ( j )   ∃ !   M ( β ) # # G G G G G G G G G G G G G G G G G G α ∗ β ∗ M ( j ′ ) / / p ′   β ∗ M ( j ′ )   / / M ( j ′ )   A ( i ) s E E A ( α ) / / A ( j ) A ( β ) / / A ( j ′ ) Let s ′ : A ( i ) → α ∗ β ∗ M ( j ′ ) b e the map s ′ = τ s . Hence p ′ τ s = ps = id A ( i ) . So deﬁne β ∗ ( s ) = s ′ . Hence s ′ ∈ Der( A ( i ) , α ∗ β ∗ M ( j ′ )) = Der( A ( i ) , ( β α ) ∗ M ( j ′ )). Consider the following comm utativ e diagra m, with s a sectio n of p . ( αγ ) ∗ M ( j ) / / p ′   α ∗ M ( j ) / / p   M ( j )   A ( i ′ ) A ( γ ) / / A ( i ) s O O A ( α ) / / A ( j ) There exists a unique s ′ : A ( i ′ ) → ( αγ ) ∗ M ( j ) which is a s e ction of p ′ which would make the ab ov e diagram still co mmu te. So deﬁne γ ∗ ( s ) = s ′ . Therefore s ′ ∈ D er ( A ( i ′ ) , ( αγ ) ∗ M ( j )). Corollary 6. 5. F or q ≥ 0 ther e exists a natur al system H q ( A, M ) on I whose value on ( α : i → j ) is given by H q G ( A ( i ) , α ∗ M ( j )) . This co rollar y allows us to deﬁne, for ﬁxed q ≥ 0, the Ba ues-Wirsching co homology H ∗ B W ( I , H q ( A, M )) of I with co eﬃcients in the na tur al system H q ( A, M ). F urther more, we can consider a natura l system on the ca tegory of chain complexes C haincomplex as follows. T o e ach morphism α : i → j ∈ I we assig n the ch ain complex Der( G ∗ ( A ( i )) , α ∗ M ( j )). This gives us a functor, D : F I → Chainco mplex , 59 where F I denotes the c ate gory of factori zations in I . This natura l system gives rise to a co simplicial ob ject in Cha incomplex : Q i D ( id i ) / / / / Q α : i → j D ( α ) / / / / / / . . . which g ives rise to a bicomplex describ ed in the next in the next sec tio n. 6.4 Bicomplex Let G be a comona d in C , let A ∈ C I and M ∈ A − mod . Then we can construct the following bicomplex denoted b y C ∗ , ∗ ( I , A, M ). C p,q ( I , A, M ) = Y α : i 0 → ... → i p Der( G q +1 ( A ( i 0 )) , α ∗ M ( i p )) . The map C p,q ( I , A, M ) → C p +1 ,q ( I , A, M ) is the map in the Baues- Wirsching co chain complex, and the map C p,q ( I , A, M ) → C p,q +1 ( I , A, M ) is the pro duct of maps in the comona d co chain complex. . . . . . . Q i Der( G 3 ( A ( i )) , M ( i )) ∂ O O δ / / Q α : i → j Der( G 3 ( A ( i )) , α ∗ M ( j )) δ / / − ∂ O O . . . Q i Der( G 2 ( A ( i )) , M ( i )) δ / / ∂ O O Q α : i → j Der( G 2 ( A ( i )) , α ∗ M ( j )) δ / / − ∂ O O . . . Q i Der( G ( A ( i )) , M ( i )) δ / / ∂ O O Q α : i → j Der( G ( A ( i )) , α ∗ M ( j )) δ / / − ∂ O O . . . This bic o mplex lives in the c a tegory of abe lian gr o ups. W e let H ∗ ( I , A, M ) denote the co ho- mology of the tota l complex o f C ∗ , ∗ ( I , A, M ). W e will need the following useful lemmas. Lemma 6.6. If A is G I -pr oj e ctive, then A ( i ) is G -pr oje ctive for al l i ∈ I . Pr o of. Consider A = G I ( Z ) : I → C where G I ( Z )( i ) = ` x → i G ( Z ( x )). Since G ( Z ( x )) is G - pro jective, it follows that ` x → i G ( Z ( x )) is G -pro jective for all i ∈ I . Lemma 6. 7. H 0 ( I , A, M ) ∼ = Der ( A, M ) , furthermor e, if A is G I -pr oj e ctive then H n ( I , A, M ) = 0 for n > 0 . Pr o of. It is suﬃcient to co nsider the cas e when A = G I ( Z ). When A = G I ( Z ), it is k nown that A is G I -pro jective. B y lemma 6.6 and lemma 2 .1 2, one gets that the vertical columns in o ur bicomplex ar e exact ex c ept in dimension 0. There is a well known lemma for bicomplexes which tells us the co homology o f the total co mplex is isomorphic to the co homology o f the following chain complex. Q i Der( A ( i ) , M ( i )) / / Q α : i → j Der( A ( i ) , α ∗ M ( j )) / / . . . It is known that the co homology of this co chain co mplex is just H ∗ B W ( I , Der ( A, M )). T o prov e the ﬁrst sta tement it is eno ugh to show that 0 → Der ( A, M ) → Y i Der( A ( i ) , M ( i )) → Y α : i → j Der( A ( i ) , α ∗ M ( j )) 60 is exa ct. Let ψ ∈ Q i Der( A ( i ) , M ( i )) and ( α : i → j ) ∈ I , then dψ ( α : i → j ) = α ∗ ψ ( i ) − α ∗ ψ ( j ) . Therefore dψ ( α : i → j ) = 0 if and only if α ∗ ψ ( i ) = α ∗ ψ ( j ). How ev er α ∗ ψ ( i ) = α ∗ ψ ( j ) if and o nly if M ( α ) ψ ( i ) = ψ ( j ) A ( α ) , i.e . the following diagram commutes. A ( i ) A ( α )   ψ ( i ) / / M ( i ) M ( α )   A ( j ) ψ ( j ) / / M ( j ) Hence ψ ∈ Der ( A, M ). This tells us that the sequence above is exact. Hence H 0 ( I , A, M ) = Der ( A, M ). T o prov e the second statement, let us cons ider D ( α : i → j ) : = Der ( A ( i ) , α ∗ M ( j )) = Der( a β : y → i GZ ( y ) , α ∗ M ( j )) = Y β : y → i Der( GZ ( y ) , β ∗ α ∗ M ( j )) , b y lemma 6.2. Deﬁne D y for a ﬁxed ob ject y ∈ I to be a natur a l system o n I (using theorem 6 .4) given b y: D y ( α : i → j ) = Y β : y → i Der( GZ ( y ) , β ∗ α ∗ M ( j )) . So one ha s that D ( i → j ) = Y y D y ( i → j ) . Hence, H ∗ B W ( I , D ) = Y y ∈ I H ∗ B W ( I , D y ) . Now consider the co chain complex C ∗ B W ( I , D y ). C ∗ B W ( I , D y ) = Q i D y ( i → i ) / / Q α : i → j D y ( i → j ) / / . . . = Q i Q β : y → i Der( GZ ( y ) , β ∗ M ( i )) / / Q α : i → j Q β : y → i Der( GZ ( y ) , β ∗ α ∗ M ( j )) / / . . . U Z ( y ) forms a basis o f the free ob ject GZ ( y ), a pplying lemma 6.3, one can rewr ite the co chain complex as C ∗ B W ( I , D y ) = Q y → i Q m ∈ U Z ( y ) A β j ( m ) / / Q α : i → j Q β : y → i Q m ∈ U Z ( y ) A αβ j ( m ) / / . . . , where A β j ( m ) = preimage of β γ ( m ) in the pro jection M ( j ) → GZ ( j ). This a llows us to re wr ite the co chain complex as C ∗ B W ( I , D y ) = Y m ∈ U Z ( y ) C ∗ B W ( y /I , F m ) where F m : y /I → Ab is a functor de ﬁned by F m ( β : y → i ) = A β j ( m ) . Since the categor y y /I co nt ains an initial ob ject i d y : y → y , by lemma 2 .20 the cohomology v a nishes in p ositive dimensions. Theorem 6.8. H ∗ G I ( A, M ) ∼ = H ∗ ( I , A, M ) . 61 Pr o of. Consider the bicomplex C ∗ ( I , G I ( A ) ∗ , M ) shown below. . . . . . . . . . C 2 ( I , G I ( A ) , M ) O O / / C 2 ( I , G 2 I ( A ) , M ) O O / / C 2 ( I , G 3 I ( A ) , M ) O O / / · · · C 1 ( I , G I ( A ) , M ) O O / / C 1 ( I , G 2 I ( A ) , M ) O O / / C 1 ( I , G 3 I ( A ) , M ) O O / / · · · C 0 ( I , G I ( A ) , M ) O O / / C 0 ( I , G 2 I ( A ) , M ) O O / / C 0 ( I , G 3 I ( A ) , M ) O O / / · · · W e are go ing to show that H ∗ ( I , A, M ) ∼ = H ∗ ( T ot ( C • ( I , G I ( A ) • , M ))) ∼ = H n G I ( A, M ) . Since G p I ( A ) is G I -pro jective, le mma 6.7 tells us that the vertical cohomology H n ( C ∗ ( I , G p I ( A ) , M )) ∼ =  Der ( G p I ( A ) , M ) , n = 0, 0 , otherwise, so eac h column of the bico mplex C ∗ ( I , G I ( A ) ∗ , M ) is exac t except at C 0 ( I , G p I ( A ) , M ). There fore by the spectra l sequence ar gument H n ( T ot ( C • ( I , G I ( A ) • , M ))) ∼ = H n ( Der ( G I ( A ) , M ) → Der ( G 2 I ( A ) , M ) → · · · ) = H n G I ( A, M ) . W e are now going to c ompute the hor izontal cohomolog y . F rom the de ﬁnitio n of C ∗ ( I , A, M ) we see that eac h row of the bicomplex C ∗ ( I , G I ( A ) ∗ , M ) is a pro duct o f co chain complexes of the form Der( G p G I ( A ) ∗ ( i ) , α ∗ M ( j )). Consider G I ( A ) ∗ → A which is an augmented simplicial ob ject. F o r all ob jects i ∈ I we hav e G I ( A ) ∗ ( i ) → A ( i ) which is a lso a n augmented simplicial ob ject. Applying the forgetful functor U : Alg ( T ) → Sets we get U G I ( A ) ∗ ( i ) → U A ( i ) which is contractible in the c a tegory Sets . The n applying the free functor F : Sets → Alg ( T ) w e get G G I ( A ) ∗ ( i ) → GA ( i ) which is contractible in the categ ory Alg ( T ). Rep eated applications o f the functor s U and F g ive us G p G I ( A ) ∗ ( i ) → G p A ( i ) which is contractible in the catego ry Alg ( T ). F or any arrow α : i → j in I we can apply the functor Der( − , α ∗ M ( j )) to get a contractible cosimplicial ab elia n g roup Der( G p A ( i ) , α ∗ M ( j )) → Der( G p G I ( A ) ∗ ( i ) , α ∗ M ( j )). Ther e fore each r ow of the bicomplex C ∗ ( I , G I ( A ) ∗ , M ) is exact except at C p ( I , G I ( A ) , M ). Therefore H n ( C p ( I , G I ( A ) ∗ , M )) ∼ =  C p ( I , A, M ) , n = 0, 0 , otherwise. Therefore by the spe c tral sequence a rgument H n ( T ot ( C • ( I , G I ( A ) • , M ))) ∼ = H n ( C ∗ ( I , A, M )) = H n ( I , A, M ) . Now one ha s b oth a glo bal cohomolo gy , H ∗ G I ( A, M ), a nd a lo ca l cohomo logy , H ∗ G ( A ( i ) , M ( i )). One can ask how these tw o a re related; the a ns wer is given by the lo c a l to global sp ectra l sequence. Theorem 6.9. Ther e exists a sp e ct ra l se quenc e E pq 2 = H p B W ( I , H q ( A, M )) ⇒ H p + q G I ( A, M ) , wher e H q ( A, M ) is a natur al system on I whose value on ( α : i → j ) is given by H q G ( A ( i ) , α ∗ M ( j )) . 62 Deﬁnition 6.10. An ex tension o f A by M is an exa ct sequence of functors 0 / / M q / / X p / / A / / 0 where X : I → Com . alg such that for all i ∈ I we get an extens io n of A ( i ) b y M ( i ) 0 / / M ( i ) q ( i ) / / X ( i ) p ( i ) / / A ( i ) / / 0 Two extensions ( X ) , ( X ′ ) with A, M ﬁxed are sa id to b e e quivalent if ther e e xists a map of diagrams φ : X → X ′ such that the following diagram commutes. 0 / / M / / X / / φ   A / / 0 0 / / M / / X ′ / / A / / 0 W e denote the set of eq uiv ale nce classes of e xtensions of A by M by Ext ( A, M ). Theorem 6.11. H 1 G I ( A, M ) ∼ = Ext ( A, M ) . Pr o of. Suppose we hav e a free resolution P ∗ of A and an extension repres e n ting a class in Ext ( A, M ). 0 / / M i / / X u / / A / / 0 The ma p u is a surjection and P 0 is fr ee, so there exists a lift h : P 0 → X which makes the following diagram commute. 0 / / M i / / X u / / A / / 0 . . . ϕ 2 0 / / ϕ 2 2 / / / / P 1 ϕ 1 0 / / ϕ 1 1 / / P 0 h O O ε / / A / / 0 Then we can get a map d = i − 1 ( hϕ 1 0 − hϕ 1 1 ) : P 1 → M . d is a der iv a tion, and d is also a 1-co cycle in Der ( P ∗ , M ) and deﬁnes a class in H 1 G I ( A, M ). This class is indep endent of the c hoice of lifting h . This gives a map Φ : Ext ( A, M ) → H 1 G I ( A, M ). Conv ersely , given a deriv ation D : P 1 → M we let X = Coker( P 1 ( ϕ 1 0 , 0) / / ( ϕ 1 1 , 0) / / P 0 ⊕ M ) . The co kernel is in the categor y A − mo d , and we let p : P 0 ⊕ M → X b e the canonica l pro jection. If D is a 1 -co cycle in Der ( P ∗ , M ) then we obtain a n extension in Ext ( A, M ) where i : M → X is given by i ( m ) = p (0 ⊕ m ) and u : X → A is g iven b y u ( p ( y ⊕ m )) = ε ( y ). . . . ϕ 2 0 / / ϕ 2 2 / / / / P 1 D   ϕ 1 0 / / ϕ 1 1 / / P 0   ε / / A / / 0 0 / / M i / / X u / / A / / 0 This pro c e dur e gives us an inv erse to Φ. Deﬁnition 6.12. A cr osse d extension of A by M is an exac t sequence of functor s 0 / / M ω / / C 1 ρ / / C 0 π / / A / / 0 such that for all i ∈ I we get a cro ssed extension of A ( i ) by M ( i ) 0 / / M ( i ) ω ( i ) / / C 1 ( i ) ρ ( i ) / / C 0 ( i ) π ( i ) / / A ( i ) / / 0 W e let π 0 ACross ( A, M ) denote the c onnected comp onents of the catego ry of a dditiv ely split crossed extensions of A by M . 63 Lemma 6.13. H 2 G I ( A, M ) ∼ = π 0 Cross ( A, M ) . Pr o of. W e a r e going to show that the crossed extensions are equiv alent to the simplicial g roups whose Mo or e complex is of length one. Given a crossed extension we hav e a crosse d mo dule C 1 ∂ / / C 0 . Let X 0 = C 0 and X 1 = C 1 ⊕ C 0 where a ddition is given by ( c 1 , c 0 ) + ( d 1 , d 0 ) = ( c 1 + d 1 , c 0 + d 0 ) and multip lication is g iven b y ( c 1 , c 0 )( d 1 , d 0 ) = (0 , c 0 d 0 + ∂ ( c 1 ) d 1 + c 0 d 1 + d 0 c 1 ). F o r all α : i → j then we ha ve X 1 ( α )( c 0 , d 0 ) = ( C 1 ( α )( c 1 ) , C 0 ( α )( c 0 )). This g ives us that X 1 is a diagram of algebras . W e s e t d 1 : X 1 → X 0 to b e d 1 ( c 1 , c 0 ) = c 0 and d 0 : X 1 → X 0 to b e d 0 ( c 1 , c 0 ) = ∂ ( c 1 ) + c 0 . Then d 0 is a natural tr ansformatio n. W e deﬁne the c ategory C to b e the category whos e ob jects a re the elements of X 0 and whose morphisms are the elemen ts of X 1 . The sourc e of the morphism ( c 1 , c 0 ) ∈ C is given by d 0 ( c 1 , c 0 ) = ∂ ( c 1 ) + c 0 and the target of ( c 1 , c 0 ) ∈ C is given by d 1 ( c 1 , c 0 ) = c 0 . The comp osable morphisms in C are pairs o f morphisms ( c 1 , c 0 ) , ( c ′ 1 , c ′ 0 ) suc h that c ′ 0 = ∂ c 1 + c 0 . The ner ve of the category C is a simplicial gro up whose Mo ore complex is . . . / / 0 / / Ker d 1 / / C 0 , which is of length one. Let K ∗ be a simplicial ob ject whose Mo or e complex is of length one. Then the Moore complex Ker d 1 d 0 / / K 0 , is a cros sed module. The categ ory of diagrams of alg ebras is exact and s o the results of Glenn [8] tell us that H 2 G I ( A, M ) classiﬁes the simplicial gro ups whose Mo ore co mplexes are of length one. 6.5 Cohomology of diagrams of group s In the pap er by Cegarra [6], the coho mology of diagra ms of g r oups is describ ed, which we denote by H ∗ C g ( G, M ). A diagr am of gr oups is a functor G : I → Gp where I is a small categor y and Grp is the categ o ry o f groups. A G -mo dule is a functor M : I → Ab such that for a ll ob jects i ∈ I w e hav e that M ( i ) ∈ A ( i ) − mod and fo r all morphisms ( α : i → j ) ∈ I we have that M ( α )( g m ) = G ( α ) ( g ) · M ( α )( m ) for a ll g ∈ G ( i ) and m ∈ M ( i ). A deriv a tion of G in to M is a natur a l transformation d : G → M s uch that d ( i ) : G ( i ) → M ( i ) is a deriv a tio n of the gr oup G ( i ) in to M ( i ). W e deno te the ab elian g roup of all deriv ations of G int o M by Der I ( G, M ). When G is lo cally co nstant then H n +1 C g ( G, M ) = R n Der I ( G, M ) and the following s pe c tral sequence ex is ts. E p,q 2 = H p B W ( I , H q +1 ( G, M )) ⇒ H p + q +1 C g ( G, M ) , where H q ( G, M ) is a natural system on I whose v a lue o n ( α : i → j ) is g iven by H q ( G ( i ) , α ∗ M ( j )). So when G is lo c a lly consta n t the co ho mology describ ed by Cegar ra co incides with the Andr´ e- Quillen cohomolo gy describ ed ab ov e with a dimension shift. 64 Chapter 7 Andr ´ e-Quillen cohomology of Ψ -rings and λ -rings 7.1 Cohomology of Ψ -rings Let I denote the categor y with one ob ject asso ciated to the multiplicativ e monoid of the nonzero natural n um bers . W e can consider Ψ-r ings as diagrams of comm utative r ings; Ψ-rings are functors from I to the c a tegory of commutativ e rings R : I → Com . rings . Therefore we can use the theo r y we dev elop ed in the pr evious chapter. W e are now going to co nstruct the fr ee Ψ -ring on one genera tor a . Let A b e the free commutative ring gener a ted by { a i | i ∈ N } . Let the op eratio ns Ψ i : A → A b e g iven by Ψ i ( a j ) = a ij , fo r i, j ∈ N . Then A is the fr e e Ψ -ring on one gener ator . Lemma 7.1. If R and S ar e Ψ -rings, then R ⊗ S with Ψ i : R ⊗ S → R ⊗ S given by Ψ i ( r , s ) = (Ψ i ( r ) , Ψ i ( s )) is the c opr o duct in t he c ate gory Ψ − rings . Pr o of. The copr o duct o f tw o comm utative r ing s is given by the tensor pro duct, so w e o nly ne e d to chec k the Ψ-op er ations. Ther e is a unique Ψ -ring structure o n R ⊗ S such that R → R ⊗ S, r 7→ r ⊗ 1 , S → R ⊗ S, s 7→ 1 ⊗ s, are homomorphisms of Ψ-rings given by Ψ i ( r ⊗ s ) = Ψ i (( r ⊗ 1)(1 ⊗ s )) = Ψ i ( r ⊗ 1)Ψ i (1 ⊗ s ) = (Ψ i ( r ) ⊗ 1 )(1 ⊗ Ψ i ( s )) = Ψ i ( r ) ⊗ Ψ i ( s ) . Corollary 7.2. L et A b e the fr e e c ommutative ring gener ate d by { a i , b i , . . . , x i | i ∈ N } . Le t the op er ations Ψ i : A → A b e given by Ψ i ( a j ) = a ij , Ψ i ( b j ) = b ij , . . . , Ψ i ( x j ) = x ij for i, j ∈ N . Then A is the fr e e Ψ - ring gener ate d by { a, b, . . . , x } . It is well kno wn that there is an adjoint pair of functors Sets F / / Com . rings U o o , where U is the forgetful functor and F takes a set S to the free comm utative ring genera ted by S . The adjoint pair gives rise to a comonad G on Com . rings whic h is monadic a nd the cohomology with resp ect to this comonad is the Andr´ e-Quillen c o homology of commutative rings . 65 The adjoint pair gives rise to another adjoint pair Sets F I / / Com . rings I U I o o , where U I is the for getful functor and F I takes a set S to the free Ψ -ring gener ated by S . This adjoint pair yields a comonad G I on Com . rings I = Ψ − rings which is monadic. Note that for any R ∈ Ψ − rings , we get that G I ( R ) = F i ∈ N G ( R ). W e deﬁne the cohomolo g y of a Ψ- ring R with co eﬃcients in M ∈ R − mod Ψ to b e H ∗ Ψ ( R, M ) := H ∗ G I ( R, M ) = H ∗ G I ( R, Der Ψ ( − , M )) . F ro m theorem 6.4 it follows that for any n ≥ 0, there is a natural sy stem on I as follows D f := H n AQ ( R, M f ) , where M f is an R -mo dule with M as an ab elia n group with the following action of R ( r , m ) 7→ Ψ f ( r ) m, for r ∈ R, m ∈ M . F or any morphism u ∈ I , we hav e u ∗ : D f → D uf which is induced by Ψ u : M f → M uf . F or any morphism v ∈ I , we hav e v ∗ : D f → D f v which is induced by Ψ v : R → R . Therefore theor em 6.9 gives us the following theorem. Theorem 7.3. Ther e exists a sp e ct ra l se quenc e E p,q 2 = H p B W ( I , H q ( R, M )) ⇒ H p + q Ψ ( R, M ) , wher e H q ( R, M ) is the natu r al system on I whose value on a morphism α in I is given by H q AQ ( R, M α ) . Theorem 7.4. L et R b e a Ψ -ring and M ∈ R − mod Ψ , then 1. H 0 Ψ ( R, M ) ∼ = Der Ψ ( R, M ) , 2. H 1 Ψ ( R, M ) ∼ = Ext Ψ ( R, M ) , 3. H 2 Ψ ( R, M ) ∼ = π 0 C ross Ψ ( R, M ) , 4. If R is a fr e e Ψ -ring, then H n Ψ ( R, M ) = 0 for n ≥ 1 . 7.2 Cohomology of λ -rings W e ar e now going to co nstruct the free λ -ring on one gener a tor a . Let A be the free commutativ e ring generated b y { a i | i ∈ N } . Let the op erations λ i : A → A b e given by λ i ( a j ) = P i,j ( a 1 , . . . , a ij ) for i, j ∈ N . Then A is the fr e e λ -ring on one gener a tor . Lemma 7 .5. If R and S ar e λ -rings, then R ⊗ S with λ i : R ⊗ S → R ⊗ S given by λ i ( r , s ) = P i (( λ 1 ( r ) , 1) , . . . , ( λ i ( r ) , 1) , (1 , λ 1 ( s )) , . . . , (1 , λ i ( s ))) is the c opr o duct in the c ate gory λ − rings . It is known that there is an a djoint pa ir of functors Sets F / / λ − ring s U o o , where U is the forgetful functor and F takes a set S to the free λ -ring generated by S . The a djoint pair g ives rise to a como na d G on λ − rings which is mona dic. W e deﬁne the cohomo lo gy of a λ -ring R with co eﬃcients in M ∈ R − mod λ to b e H ∗ λ ( R, M ) := H ∗ G ( R, M ) = H ∗ G ( R, Der λ ( − , M )) . Theorem 7.6. L et R b e a λ -ring and M ∈ R − mod λ , then 66 1. H 0 λ ( R, M ) ∼ = Der λ ( R, M ) , 2. H 1 λ ( R, M ) ∼ = Ext λ ( R, M ) , 3. H 2 λ ( R, M ) ∼ = π 0 C ross λ ( R, M ) , 4. If R is a fr e e λ -ring, then H n λ ( R, M ) = 0 for n ≥ 1 . Pr o of. Prop erty 1 follows from lemma 2.12, and proper ty 4 follo ws from lemma 2.11. W e a re now going to pr ov e pro pe r ty 2. Suppo se we hav e a free r esolution P ∗ of R as a λ -ring and an extension r epresenting a class in Ext λ ( R, M ). 0 / / M i / / X u / / R / / 0 The ma p u is a surjection and P 0 is fr ee, so there exists a lift h : P 0 → X which makes the following diagram commute. 0 / / M i / / X u / / R / / 0 . . . ϕ 2 0 / / ϕ 2 2 / / / / P 1 ϕ 1 0 / / ϕ 1 1 / / P 0 h O O ε / / R / / 0 Then we can get a map d = i − 1 ( hϕ 1 0 − hϕ 1 1 ) : P 1 → M d is a Ψ-der iv atio n, and d is also a 1-co cyc le in Der Ψ ( P ∗ , M ) a nd deﬁnes a class in H 1 Ψ ( R, M ). This c la ss is indep endent of the c hoice o f lifting h . This g ives a map Φ : Ext Ψ ( R, M ) → H 1 Ψ ( R, M ). Conv ersely , given a λ -deriv ation D : P 1 → M we let X = Coker( P 1 ( ϕ 1 0 , 0) / / ( ϕ 1 1 , 0) / / P 0 ⊕ M ) . The cokernel is in the ca tegory R − mod λ , a nd we let p : P 0 ⊕ M → X be the canonica l pr o jection. If D is a 1-c o cycle in Der λ ( P ∗ , M ) then we obtain a n extension in Ex t λ ( R, M ) where i : M → X is given b y i ( m ) = p (0 ⊕ m ) and u : X → R is given by u ( p ( y ⊕ m )) = ε ( y ). . . . ϕ 2 0 / / ϕ 2 2 / / / / P 1 D   ϕ 1 0 / / ϕ 1 1 / / P 0   ε / / R / / 0 0 / / M i / / X u / / R / / 0 This pro c e dur e gives us an inv erse to Φ. W e ar e now going to prov e proper t y 3 b y sho wing that the crossed λ -extensions are equiv alent to the simplicial gr oups whose Mo ore complex is of leng th one. Given a crossed λ -extension we hav e a crossed λ -mo dule C 1 ∂ / / C 0 . Let X 0 = C 0 and X 1 = C 1 ⊕ C 0 where a ddition is given by ( c 1 , c 0 ) + ( d 1 , d 0 ) = ( c 1 + d 1 , c 0 + d 0 ) and multiplication is given by ( c 1 , c 0 )( d 1 , d 0 ) = (0 , c 0 d 0 + ∂ ( c 1 ) d 1 + c 0 d 1 + d 0 c 1 ). W e let λ n ( c 0 , d 0 ) = ( P i j =1 Λ j ( c 1 ) λ i − j ( c 0 ) , λ i ( c 0 ). This g ives us tha t X 1 is a λ -ring . W e set d 1 : X 1 → X 0 to b e d 1 ( c 1 , c 0 ) = c 0 and d 0 : X 1 → X 0 to b e d 0 ( c 1 , c 0 ) = ∂ ( c 1 ) + c 0 . Then d 0 is a λ -ring ma p. W e deﬁne the c ategory C to b e the category whos e ob jects a re the elements of X 0 and whose morphisms are the elemen ts of X 1 . The sourc e of the morphism ( c 1 , c 0 ) ∈ C is given by d 0 ( c 1 , c 0 ) = ∂ ( c 1 ) + c 0 and the target of ( c 1 , c 0 ) ∈ C is given by d 1 ( c 1 , c 0 ) = c 0 . The comp osable morphisms in C are pairs of morphisms ( c 1 , c 0 ) , ( c ′ 1 , c ′ 0 ) such tha t c ′ 0 = ∂ c 1 + c 0 . Hence the nerve of the categ ory C is a simplicial gro up who se Mo ore complex is . . . / / 0 / / Ker d 1 / / C 0 , which is of length one. 67 Let K ∗ be a simplicial ob ject whose Mo or e complex is of length one. Then the Moore complex yields Ker d 1 d 0 / / K 0 , which is a cr ossed λ -mo dule. The categor y of λ -ring s is exact and so the results of Glenn [8] tell us that H 2 λ ( R, M ) classiﬁes the simplicial g r oups whose Mo ore complexes ar e of length one. Lemma 7.7. L et R b e a λ -ring and let M ∈ R -mo d λ . Then t her e exist homomorphisms, for n ≥ 0 , ς n : H n λ ( R, M ) → H n Ψ ( R Ψ , M Ψ ) , ρ n : H n λ ( R, M ) → H n AQ ( R, M ) ,  n : H n Ψ ( R Ψ , M Ψ ) → H n AQ ( R, M ) . Pr o of. Let P ∗ be a pro jective r esolution of R in the category of λ -ring s. Then a pplying the Adams op erations we get that ( P ∗ ) Ψ is a (not necessarily pro jective) r esolution of R Ψ in the catego ry of Ψ -rings. W e let L ∗ be a pro jective resolution of R Ψ in the categor y of Ψ-r ing s. Since L ∗ is pro jective, we c a n use the lifting pro pe r ty to get a map α : L ∗ → ( P ∗ ) Ψ , such that the following diagram commutes. ( P ∗ ) Ψ / / R Ψ L ∗ ∃ α O O / / R Ψ W e then apply the functor Der Ψ ( − , M Ψ ) to get the commutativ e diagr a m. Der Ψ (( P ∗ ) Ψ , M Ψ ) / / α ∗   Der Ψ ( R Ψ , M Ψ ) Der Ψ ( L ∗ , M Ψ ) / / Der Ψ ( R Ψ , M Ψ ) The inclusion i : Der λ ( R, M ) ֒ → Der Ψ ( R Ψ , M Ψ ) g ives us maps whic h ma ke the following diagram commute. Der λ ( P ∗ , M ) / / i   Der λ ( R, M ) i   Der Ψ (( P ∗ ) Ψ , M Ψ ) / / α ∗   Der Ψ ( R Ψ , M Ψ ) Der Ψ ( L ∗ , M Ψ ) / / Der Ψ ( R Ψ , M Ψ ) . This gives us homomor phisms ς n : H n λ ( R, M ) = H n (Der λ ( P ∗ , M )) ( α ∗ i ) ∗ / / H n (Der Ψ ( L ∗ , M Ψ )) = H n Ψ ( R Ψ , M Ψ ) . The homomorphisms ρ n and  n are induced by the forgetful functors fr om λ − rings and Ψ − rings resp ectively to Com . rings . 68 Chapter 8 Applications 8.1 K-theory The material covered in this s e c tion can b e found in [2] a nd [11]. 8.1.1 V ector bundles In this section we will develop the notion of co mplex vector bundles. A lot of the basic theory for real vector bundles is the same as for co mplex vector bundles, how ever we will o nly b e concerned with complex vector bundles in this chapter. Deﬁnition 8.1. A c omplex ve ctor bund le consists o f 1. top o lo gical spaces X (calle d the base spa ce) and E (ca lled the total spa ce.) 2. a contin uous map p : E → X (calle d the pro jection.) 3. a ﬁnite dimensio nal complex vector space structure o n each E x = p − 1 ( x ) for x ∈ X , (w e ca ll the p − 1 ( x ) the ﬁbres) such that the following lo cal triviality condition is satisﬁed. There ex is ts a n o pen cov er o f X b y op en sets U α and for each there exists a homeomorphism ϕ α : p − 1 ( U α ) → U α × C d which takes p − 1 ( b ) to { b } × C d via a vector space isomorphism for each b ∈ U α . Example 8.2 . L et E = X × C d , and p b e the pr oje ction ont o the ﬁ rst factor. We c al l t his t he pr o duct or t rivial bund le. A homomorphism from a complex vector bundle p : E → X to a nother complex vector bundle q : F → X is a contin uous map ϕ : E → F such that 1. q ϕ = p , 2. ϕ : E x → F x is a linear ma p of vector spaces for a ll x ∈ X . If ϕ is a bijection and ϕ − 1 is contin uous, then we say that ϕ is an isomorphism a nd that E and F are isomorph ic . W e will let V ect ( X ) denote the set of isomorphism classes o f complex v ector bundles on X . Let E b e a complex vector bundle over X . W e get that dim ( E x ) is lo ca lly co nstant on X , furthermore it is a consta nt function on each of the c onnected comp onents of X . F or vector bundles E , F w e can deﬁne the following corresp onding bundles • E ⊕ F , the direct sum o f E and F , • E ⊗ F , the tensor pro duct o f E and F , • λ k ( E ), the k th exterior p ower o f E . 69 There exist the following natural isomorphis ms • E ⊕ F ∼ = F ⊕ E , • E ⊗ F ∼ = F ⊗ E , • E ⊗ ( F ⊕ F ′ ) ∼ = ( E ⊗ F ) ⊕ ( E ⊗ F ′ ), • λ k ( E ⊕ F ) ∼ = L i + j = k ( λ i ( E ) ⊗ λ j ( F )). 8.1.2 K-theory F or any s pa ce X , we can co nsider the set V ect ( X ) whic h has an ab elian semigr oup structure wher e addition is g iven b y the dire c t sum. There is also a multiplication, given b y tensor pro ducts, which is distributive ov er the addition of V ect ( X ) (this makes V ect ( X ) into a se mir ing.) If A is an ab elian semigroup, we can asso cia te a n ab elian g r oup K ( A ) to A . Let F ( A ) b e the free ab elia n gro up gener ated by A , and let E ( A ) b e the subgr oup of F ( A ) gener ated by element s of them fo r m a + a ′ − ( a ⊕ a ′ ), where a, a ′ ∈ A and ⊕ is the addition in A . W e deﬁne the a b elia n group K ( A ) = F ( A ) /E ( A ). If A is a semiring, then K ( A ) is a ring. If X is a space, then we will write K ( X ) for the ring K ( V ect ( X )). Let f : X → Y b e a contin uous map. Then f ∗ : V ect ( Y ) → V ect ( X ) induces a ring homomorphism f ∗ : K ( Y ) → K ( X ) which o nly dep ends on the homo topy clas s of f . W e can deﬁne op er ations λ k : K ( X ) → K ( X ) us ing the exterior powers. These ma ke K ( X ) int o a λ -ring. W e can then us e these to deﬁne the Adams ope rations Ψ k : K ( X ) → K ( X ) which makes K ( X ) into a Ψ-ring. If X is a compact space with distinguished basep oint, then we deﬁne e K ( X ) to be the kernel of i ∗ : K ( X ) → K ( x 0 ) wher e i : x 0 → X is the inclusion of the basep oint. Let c : X → x 0 be the collapsing map, then c ∗ induces a na tural splitting K ( X ) ∼ = e K ( X ) ⊕ K ( x 0 ). Example 8. 3. e K ( S 2 n ) ∼ = Z [ y ] / ( y ) 2 , wher e y is the n - fold ext ernal pr o du ct ( H − 1) ∗ . . . ∗ ( H − 1) and H is the c anonic al line bund le of S 2 = C P 1 . Multiplic ation in e K ( S 2 n ) is trivial, and the λ -op er ations λ k : e K ( S 2 n ) → e K ( S 2 n ) ar e given by λ k ( x ) = ( − 1) k − 1 k n − 1 x. Henc e the Ψ - op er ations Ψ k : e K ( S 2 n ) → e K ( S 2 n ) ar e given by Ψ k ( x ) = k n x. 8.2 Natural transformation Let X , Y b e topo logical spaces such that e K ( Y ) = 0 a nd e K (Σ X ) = 0. Let f : Y → X be a contin uous map, then we can consider the P uppe exact se quence Y f / / X / / C f / / Σ Y / / Σ X / / Σ C f / / . . . where C f is the mapping cone of f , a nd Σ X is the susp ension o f X . After applying the functor e K ( − ) we get the lo ng exact sequence. . . . / / e K (Σ X ) / / e K (Σ Y ) / / e K ( C f ) / / e K ( X ) / / e K ( Y ) How ev er, since e K (Σ X ) = 0 and e K ( Y ) = 0 w e obtain the sho rt exact sequence. 0 / / e K (Σ Y ) / / K ( C f ) / / K ( X ) / / 0 This gives us the following prop osition. Prop ositio n 8.4. If X and Y ar e top olo gic al sp ac es as ab ove then ther e exist natura l tr ansforma- tions τ λ : [ Y , X ] → E xt λ ( K ( X ) , e K (Σ Y )) and τ Ψ : [ Y , X ] → E xt Ψ ( K ( X ) , e K (Σ Y )) . Corollary 8.5. I f X is a top olo gic al sp ac e such that e K (Σ X ) = 0 then ther e ex ist natur al t r ansfor- mations τ λ,n : π 2 n − 1 ( X ) → E xt λ ( K ( X ) , e K ( S 2 n )) and τ Ψ ,n : π 2 n − 1 ( X ) → E xt Ψ ( K ( X ) , e K ( S 2 n )) . 70 8.3 The Hopf in v arian t of an extension W e are going to give a pro of of the classica l r esult of Adams which was ﬁrst prov ed b y Adams, and subsequen tly by Ada ms - A tiyah [1]. W e are going to use the same approach a s Adams-A tiy ah; using Ψ-ring s. Deﬁnition 8 .6. Consider the commut ative ring R which is free as an ab elia n gr oup with generators x and y , R ∼ = Z x ⊕ Z y , wher e x is the unit of the ring a nd y 2 = 0. Let M ∼ = Z z b e the R -mo dule such that y · z = 0. W e ca n co nsider the s quare zero extensions of R by M in the catego r y of commutativ e rings. All the squar e zero extensions have the following for m 0 / / M / / X ⊕ Z γ / / R / / 0 (8.1) where X ∼ = Z α ⊕ Z β as an a b elia n group with α b eing the ima g e of the genera tor z , the image of the unit γ is the unit x a nd the image of β b eing the generator y . Since M 2 = 0 we g et tha t α 2 = 0. Since y 2 = 0, we get that αβ = 0 and β 2 = hα for some integer h . W e deﬁne h to b e the Hopf invariant o f the extension (8.1). Let f : S 4 n − 1 → S 2 n be a contin uous map. W e deﬁne the Hopf invariant of the map f to b e the Hopf inv ar iant of the shor t exact sequence 0 / / e K ( S 4 n ) / / K ( C f ) / / K ( S 2 n ) / / 0 obtained from applying the natura l transformatio n τ Ψ to f . W e are going to consider the extensio ns of K ( S 2 n ) by e K ( S 2 n ′ ) in the categor y of Ψ-rings. W e are going to pr ov e the following theorem. Theorem 8.7. E xt Ψ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ =  Z ⊕ Z G n,n ′ if n 6 = n ′ ; Z ⊕ Q p prime Z if n = n ′ . wher e G n,n ′ denotes the gre atest c ommon divisor of al l the inte gers in the set { l n − l n ′ | l ∈ Z , l ≥ 2 } . Corollary 8.8 . If n 6 = n ′ then, E xt λ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ = { ( h, ν ) ∈ Z ⊕ Z G n,n ′ | h ≡ ν (2 n − 2 n ′ ) G n,n ′ mo d 2 } . If n = n ′ , then E xt λ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ = { ( h, ν 2 , ν 3 , . . . ) ∈ Z ⊕ Y p prime Z | h ≡ ν 2 mo d 2 , ν p ≡ 0 mo d p, p > 2 } , All the Ψ-ring extensions of K ( S 2 n ) by e K ( S 2 n ′ ) hav e the form (8.1). The Ψ- o p erations on Ψ k : X → X are given b y ψ k ( m, r ) = ( k n ′ m + ν k r , k n r ) , for some ν k ∈ Z . Ψ k (Ψ l ( m, r )) = ( k n ′ l n ′ m + k n ′ ν l r + ν k l n r , k n l n r ) , Ψ l (Ψ k ( m, r )) = ( l n ′ k n ′ m + l n ′ ν k r + ν l k n r , l n k n r ) . Since the Ψ-o pe r ations commute, w e get that ν l r ( k n ′ − k n ) = ν k r ( l n ′ − l n ) . If n = n ′ then ther e is no restriction o n the choice of ν p for p pr ime. Otherwise we can rearr ange the ab ov e to get that ν l = ν k ( l n ′ − l n ) ( k n ′ − k n ) . 71 By setting k = 2 we get that for all l ≥ 2 ν l = ν 2 ( l n ′ − l n ) (2 n ′ − 2 n ) . W e can write all the ν l ’s as multiples of ν 2 since ν l = ν 2 ( l n ′ − l n ) (2 n ′ − 2 n ) = ν 2 ( k n ′ − k n ) (2 n ′ − 2 n ) ( l n ′ − l n ) ( k n ′ − k n ) = ν k ( l n ′ − l n ) ( k n ′ − k n ) . Since ν 2 is an integer, w e g et that ν 2 = z (2 n ′ − 2 n ) G n,n ′ for some integer z. If we r eplace the genera tor β by β + N α , no te that ( β + N α ) 2 = hα , then we hav e to replace ν k by ν k + N ( k n ′ − k n ). W e get that ν k + N ( k n ′ − k n ) = ν 2 k n ′ − k n 2 n ′ − 2 n + N ( k n ′ − k n ) = ( ν 2 + N (2 n ′ − 2 n ))( k n ′ − k n ) (2 n ′ − 2 n ) . So we only hav e to b e concerne d with replacing ν 2 by ν 2 + N (2 n ′ − 2 n ), then our usual f ormula for ν k holds. Hence we are replacing z (2 n ′ − 2 n ) G n,n ′ by z (2 n ′ − 2 n ) G n,n ′ + N (2 n ′ − 2 n ) = ( z + N G n,n ′ )(2 n ′ − 2 n ) G n,n ′ . This pr ov es theor em 8.7. The isomorphism dep ends o n n and n ′ . If we now introduce the prop erty that Ψ p ( x ) ≡ x p mo d p , w e get that ν 2 r ≡ hr 2 mo d 2 and ν p r ≡ 0 mod p for p ≥ 3. This prov es corollary 8.8. Prop ositio n 8.9. If t her e exists an extension in E xt λ ( K ( S 2 n ) , e K ( S 2 n ′ )) whose H opf invariant is o dd, then either n = n ′ or min ( n, n ′ ) ≤ g 2 | n − n ′ | , wher e g p j denotes the m ultiplicity of the prime p in t he prime factorisation of t he gr e atest c ommon divisor of the set of inte gers { ( k j − 1) | k ∈ N − { 1 , q p |∀ q ∈ N }} . Pr o of. The cas e when n = n ′ is c lear. Assume that n 6 = n ′ , then the sp ecial Ψ -ring extensions are given by a pair ( h, ν ) where h is the Hopf inv ar iant. By 8.8, h ca n only b e o dd if 2 n divides G n,n ′ . Assume tha t n < n ′ , sinc e the other cas e is analogo us. The multiplicit y of 2 in the prime factorisatio n of G n,n ′ is n if n ≤ g 2 | n − n ′ | or g 2 | n − n ′ | if g 2 | n − n ′ | < n . It follo ws that if n ≤ g 2 | n − n ′ | then 2 n divides G n,n ′ . Note that g 2 2 n − 1 = 1 for all n ∈ N . Since ( k 2 n − 1) = ( k n + 1)( k n − 1) it follows that g 2 2 n =  3 , n o dd g 2 n + 1 , n even. Theorem 8.1 0. If ther e exists an ext ension in E xt λ ( K ( S 2 n ) , e K ( S 2 n ′ )) whose Hopf invariant is o dd, then one of the fol lowing is satisﬁe d 1. n = n ′ , 2. n = 1 or n ′ = 1 , 3. n ′ − n is even and either n = 2 or n ′ = 2 , 4. n ′ > n ≥ 3 and n ′ = n + 2 n − 2 b for some b ∈ N 0 , 5. n > n ′ ≥ 3 and n = n ′ + 2 n ′ − 2 b for some b ∈ N 0 . Pr o of. 1. is clear. 2. follows from g 2 n ≥ 1 for a ll N . 3. follows from g 2 2 n ≥ 3 for a ll n ∈ N . 4. and 5 . follows from g 2 | n − n ′ | being 2 plus the m ultiplicity of 2 in the prime factor isation of | n − n ′ | . 72 Corollary 8.11. If t her e exists an extension in E xt λ ( K ( S 2 n ) , e K ( S 2( n + k ) )) whose Hopf invariant is o dd, then one of the fol lowing is satisﬁe d 1. k = 0 , 2. n = 1 , 3. k is even and n = 2 , 4. n ≥ 3 and k = n + 2 n − 2 b for some b ∈ N 0 . Lemma 8.12. If t her e exists an ex tension in E xt λ ( K ( S 2 n ) , e K ( S 2 an )) for a ∈ N who se Hopf invariant is o dd, then one of t he fol lowing is satisﬁe d 1. n = 1 , 2 or 4 , 2. n = 3 and a is even, 3. n ≥ 5 and an = 2 n + 2 n − 2 b for some b ∈ N 0 . Corollary 8.13. If t her e exists an extension in E xt λ ( K ( S 2 n ) , e K ( S 4 n )) whose Hopf invariant is o dd, then n = 1 , 2 or 4 . Corollary 8.1 4 (Adams) . If f : S 4 n − 1 → S 2 n is a c ontinuous map whose Hopf invariant is o dd, then n = 1 , 2 or 4 . 8.4 Stable Ext groups of spheres Prop ositio n 8.15 . If n > k + 1 t hen G n,n + k = G n +1 ,n + k +1 . Pr o of. Let n > k + 1. W e know that G n,n + k = G n +1 ,n + k +1 if and only if the m ultiplicit y of a ny prime p in the prime factorization o f G n,n + k is g p k . F or all primes p > 2 we get that p n > 2 k − 1 , s o the multiplicit y of p in the prime factor isation of G n,n + k is g p k . W e can easily see that g 2 k ≤ k + 1 for all k . It follows that the multiplicit y of 2 in the prime factorisa tion of G n,n + k is g 2 k . Corollary 8.1 6. If n > k + 1 then Ext λ ( K ( S 2 n ) , e K ( S 2( n + k ) )) ∼ = Ext λ ( K ( S 2( n +1) ) , e K ( S 2( n + k +1) )) . The groups Ext λ ( K ( S 2 n ) , e K ( S 2( n + k ) )) are indep endent of n for n > k + 1, we ca ll these the stable Ext gr oups of spher es which we denote b y Ext s 2 k . Prop ositio n 8.17 . Ther e ar e natur al tr ansformations Υ k : π s 2 k − 1 → Ext s 2 k , wher e π s 2 k − 1 denotes the stable homotopy gr oups of spher es. F or small k these gro ups loo k as follows. k π s 2 k − 1 Ext s 2 k 1 Z 2 2 Z ⊕ Z 2 2 Z 24 ⊕ Z 3 2 Z ⊕ Z 24 3 0 2 Z ⊕ Z 2 4 Z 240 2 Z ⊕ Z 240 5 Z 2 ⊕ Z 2 ⊕ Z 2 2 Z ⊕ Z 2 6 Z 504 2 Z ⊕ Z 504 7 Z 3 2 Z ⊕ Z 2 8 Z 480 ⊕ Z 2 2 Z ⊕ Z 480 73 App endix A Adams O p erations Ψ k ( r ) = k − 1 X i =1 ( − 1) i +1 λ i ( r )Ψ k − i ( r ) + ( − 1 ) k +1 k λ k ( r ) Ψ 1 ( r ) = r Ψ 2 ( r ) = r 2 − 2 λ 2 ( r ) Ψ 3 ( r ) = r 3 − 3 r λ 2 ( r ) + 3 λ 3 ( r ) Ψ 4 ( r ) = r 4 − 4 r 2 λ 2 ( r ) + 4 rλ 3 ( r ) + 2( λ 2 ( r )) 2 − 4 λ 4 ( r ) Ψ 5 ( r ) = r 5 − 5 r 3 λ 2 ( r ) + 5 r 2 λ 3 ( r ) + 5 r ( λ 2 ( r )) 2 − 5 r λ 4 ( r ) − 5 λ 2 ( r ) λ 3 ( r ) + 5 λ 5 ( r ) Ψ 6 ( r ) = r 6 − 6 r 4 λ 2 ( r ) + 6 r 3 λ 3 ( r ) + 9 r 2 ( λ 2 ) 2 − 6 r 2 λ 4 − 12 r λ 2 ( r ) λ 3 ( r ) + 6 r λ 5 ( r ) − 2( λ 2 ( r )) 3 + 3( λ 3 ( r )) 2 + 6 λ 2 ( r ) λ 4 ( r ) − 6 λ 6 ( r ) Ψ 7 ( r ) = r 7 − 7 r 5 λ 2 ( r ) + 7 r 4 λ 3 ( r ) + 1 4 r 3 ( λ 2 ( r )) 2 − 7 r 3 λ 4 ( r ) − 2 1 r 2 λ 2 ( r ) λ 3 ( r ) + 7 r 2 λ 5 ( r ) − 7 r ( λ 2 ( r )) 3 + 7 r ( λ 3 ( r )) 2 + 14 r λ 2 ( r ) λ 4 ( r ) − 7 rλ 6 ( r ) + 7( λ 2 ) 2 λ 3 ( r ) − 7 λ 3 ( r ) λ 4 ( r ) − 7 λ 2 ( r ) λ 5 ( r ) + 7 λ 7 ( r ) Ψ 8 ( r ) = r 8 − 8 r 6 λ 2 ( r ) + 8 r 5 λ 3 ( r ) + 2 0 r 4 ( λ 2 ( r )) 2 − 8 r 4 λ 4 ( r ) − 3 2 r 3 λ 2 ( r ) λ 3 ( r ) + 8 r 3 λ 5 ( r ) − 1 6 r 2 ( λ 2 ( r )) 3 + 12 r 2 ( λ 3 ( r )) 2 + 24 r 2 λ 2 ( r ) λ 4 ( r ) − 8 r 2 λ 6 ( r ) + 24 r ( λ 2 ( r )) 2 λ 3 ( r ) − 1 6 rλ 3 ( r ) λ 4 ( r ) − 1 6 rλ 2 ( r ) λ 5 ( r ) + 8 rλ 7 ( r ) + 2 ( λ 2 ( r )) 4 − 8 λ 2 ( r )( λ 3 ( r )) 2 + 4( λ 4 ( r )) 2 − 8( λ 2 ( r )) 2 λ 4 ( r ) + 8 λ 3 ( r ) λ 5 ( r ) + 8 λ 2 ( r ) λ 6 ( r ) − 8 λ 8 ( r ) 74 App endix B Univ er s al P olynomi a l s P i , P i,j F or more informa tion on the universal p olyno mials, refer to the thesis of Hopkinson [12]. He has several results a nd gives the p olynomial P i upto i = 10, a s well a s giving several formulas for the po lynomial P i,j . • P 1 ( s 1 ; σ 1 ) = s 1 σ 1 • P 2 ( s 1 , s 2 ; σ 1 , σ 2 ) = s 2 1 σ 2 − 2 s 2 σ 2 + s 2 σ 2 1 • P 3 ( s 1 , s 2 , s 3 ; σ 1 , σ 2 , σ 3 ) = s 3 1 σ 3 + s 1 s 2 σ 1 σ 2 − 3 s 1 s 2 σ 3 + s 3 σ 3 1 − 3 s 3 σ 1 σ 2 + 3 s 3 σ 3 • P 4 ( s 1 , s 2 , s 3 , s 4 ; σ 1 , σ 2 , σ 3 , σ 4 ) = − 2 s 1 s 3 σ 2 2 + 2 s 4 σ 2 2 + 4 s 4 σ 1 σ 3 − 4 s 2 1 s 2 σ 4 − 2 s 2 2 σ 1 σ 3 − 4 s 4 σ 2 1 σ 2 + 4 s 1 s 3 σ 4 + s 2 1 s 2 σ 1 σ 3 + s 1 s 3 σ 2 1 σ 2 − s 1 s 3 σ 1 σ 3 + s 4 1 σ 4 + s 2 2 σ 2 2 + 2 s 2 2 σ 4 + s 4 σ 4 1 − 4 s 4 σ 4 • P 1 , 1 ( s 1 ) = s 1 • P 1 ,j ( s 1 , . . . , s j ) = s j • P i, 1 ( s 1 , . . . , s i ) = s i • P 2 ,j ( s 1 , . . . , s 2 j ) = P j − 1 k =1 ( − 1) k +1 s j − k s j + k + ( − 1) j +1 s 2 j Consider the po lynomials • P 2 , 4 ( s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , s 8 ) = s 3 s 5 − s 2 s 6 + s 1 s 7 − s 8 • P 4 , 2 ( s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , s 8 ) = s 1 s 3 s 4 − 3 s 1 s 2 s 5 + s 3 1 s 5 − s 2 4 + s 3 s 5 − s 2 1 s 6 + s 1 s 7 + 2 s 2 s 6 − s 8 • P 5 , 2 ( s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , s 8 , s 9 , s 10 ) = s 4 1 s 6 + s 2 s 2 4 + 3 s 1 s 2 s 7 + 3 s 1 s 3 s 6 − 4 s 2 1 s 2 s 6 − 2 s 1 s 4 s 5 − 2 s 2 s 3 s 5 + s 2 1 s 3 s 5 + s 10 − s 3 s 7 + 2 s 2 5 − s 3 1 s 7 − 2 s 4 s 6 + 2 s 2 2 s 6 + s 2 1 s 8 − s 1 s 9 − 2 s 2 s 8 So we can see that in g eneral P i,j 6 = P j,i . 75 App endix C Univ er s al P olynomi a l P artial Deriv ativ es k ∂ P 1 ( r,s ) ∂ λ k ( r ) ∂ P 2 ( r,s ) ∂ λ k ( r ) ∂ P 3 ( r,s ) ∂ λ k ( r ) ∂ P 4 ( r,s ) ∂ λ k ( r ) 1 Ψ 1 ( s ) r ( s 2 − Ψ 2 ( s )) . . . . . . 2 0 Ψ 2 ( s ) r ( s 3 − Ψ 3 ( s )) . . . 3 0 0 Ψ 3 ( s ) r ( s 4 − Ψ 4 ( s )) 4 0 0 0 Ψ 4 ( s ) Conjecture C.1. F or al l i ∈ N ∂ P i ( r , s ) ∂ λ i ( r ) = Ψ i ( s ) , ∂ P i +1 ( r , s ) ∂ λ i ( r ) = r ( s i +1 − Ψ i +1 ( s )) F ro m the other universal p olyno mial, we get ∂ P 1 ,n ( r ) ∂ λ k ( r ) =  1 k = n 0 otherwis e ∂ P 2 ,n ( r ) ∂ λ k ( r ) =  0 k = n , or k > 2 n ( − 1) k +1 λ 2 n − k ( r ) otherw is e ∂ P i,j ( r ) ∂ λ ij ( r ) = ( − 1) ( i +1)( j +1) k ∂ P 4 , 2 ( r ) ∂ λ k ( r ) ∂ P 5 , 2 ( r ) ∂ λ k ( r ) 3 rλ 4 ( r ) + λ 5 ( r ) . . . 4 rλ 3 ( r ) − 2 λ 4 ( r ) . . . 5 Ψ 3 ( r ) − 2 λ 3 ( r ) . . . 6 − Ψ 2 ( r ) Ψ 4 ( r ) − r λ 3 ( r ) + 2 λ 4 ( r ) 7 r − Ψ 3 ( r ) + 2 λ 3 ( r ) 8 -1 Ψ 2 ( r ) 9 0 − r 10 0 1 76 Bibliograph y [1] J . Adams and M. Atiy ah. K-the ory and the Hopf invariant . Quant. J Math Oxford (2) 17, 1966. pages 3 1 -38. [2] M. Atiy ah. K-the ory . Addison W esley Longman Publishing Co, 1989 . ISB N 020 1 0939 44. [3] M. Barr. A cyclic Mo dels , volume V o l. 1 7 o f CRM Mono gr aph Series . American Mathematical So ciety , 20 0 2. ISBN 08 21828 770. [4] H. B aues a nd G. Wirsching. Coho mology of small catego r ies. Journal of pur e and applie d algebr a 38 , 198 4. [5] H. Car tan and S. Eilenberg. H omolo gic al A lgebr a . Princeto n University P ress, Princeton, 1 3th edition, 19 99. ISBN 0-69 1 -0797 7-3. [6] A. Ceg arra . Cohomolog y o f diag rams of groups. the cla ssiﬁcation of (co )ﬁbr ed ca tegorical groups. Int. Math. J . vol. 3 no. 7 , pages 64 3–680 , 200 3. [7] M. Ger stenhab er and S. D. Schac k. On the deformation of diagr ams of algebras. Algebr aists’ Homage: Pap ers in R ing The ory and R elate d T opics, Contemp or ary Mathematics , Amer. Math. So c., Pr ov idenc e, V olume 13 , pages 193 –197 , 1982. [8] P . Glenn. Realisation of c o homology class es in arbitrar y exact categor ies. Jour n al of pur e and applie d algebr a 25 , pages 3 3–10 5 , 19 82. [9] R. Go dement. T op olo gie alg ´ ebrique et th´ eorie des faisc e aux . Her mann, 1958 . [10] D. Harriso n. Commutativ e alg ebras and cohomolo g y . T r ans. Amer. Math. So c. 104 , pag es 191–2 04, 1962. [11] A. Ha tc her. V e ctor bund les and K-the ory , volume v2.2. Online at author’s website, 2001 . [12] J. R. Hopkinson. Universal p olynomials in lamb da rings and the K- the ory of t he inﬁnite lo op sp ac e t mf . PhD thesis, Ma s sach usetts Institute of T echnology , 200 6. [13] C. Kassell and J.-L. Lo day . Extensions cen trales d’alg` ebres de lie . Ann. Inst. F ourie r, Gr eno- ble , pag es 119–14 2, 1 9 82. [14] D. Knutson. λ -Rings and t he R epr esentation The ory of t he Symmetric Gr oup , volume 308. Springer, 19 73. ISBN 3-54 0-061 84-3. [15] J. Lo day . Cyclic Homolo gy , volume 301 . Springer , Berlin, 2nd edition, 1998. ISBN 3-54 0- 63074 -0. [16] S. MacLane. Cate gories for the Working Mathematician . Springe r-V erlag, Berlin, Heidelberg , New Y ork., 19 71. [17] S. Ma cLane. Homolo gy . Spr inger-V erlag , Berlin, Heidelb erg, New Y o rk, 4th edition, 1 994. ISBN 3-5 40-58 662-8 . [18] D. Q uillen. O n the (co)ho mology of comm utative rings. Pr o c. Symp. Pur e Math. XVII. , pages 65–87 , 1970. [19] C. W eib el. An intr o duction to homolo gic al algebr a . Ca mbridge Universit y Pre s s, ha rdback edition, 19 97. ISBN 0-52 1 -4350 0-5. [20] D. Y au. Co homology of λ -ring s. J ournal of algebr a , (284):37–5 1, 20 05. 77

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