Simplicial Hochschild cochains as an Amitsur complex

Simplicial Ho c hsc hild co c hains as an Amitsur complex Lars Kadison Departmen t of Mathematics, Univ ersit y of P ennsylv ania Philadelphia, P A 19104-6395 E-mail: lk adison@math.up enn.edu Abstract It is demonstr ated that the co c hain complex of rela t ive Ho c hs c hild A - v alued co chains of a depth tw o extension A | B under cup pr o duct is isomorphic as a differ e ntial graded algebr a with the Amitsur co mp lex of the coring S = End B A B ov er the c e n tra lizer R = A B with grouplike elemen t 1 S , which itself is isomorphic to the Cartier co mp lex of S with co efficien ts in the ( S, S )-bicomo dule R e . This specializes to finite dimensional algebras, H-separable extensions and Ho pf-Ga lois extensio ns. 2000 M SC: 18 G25. 1 In tro duction Relativ e Ho c hs child cohomology of a su bring B ⊆ A or ring homomorph i sm B → A is set forth in [4]. The co effici ents of the general f orm of the cohomology theory are ta ken in a bimo dule M o v er A . If M = A ∗ is th e k -dual of the k -algebra A , this gives rise to a cyclic symmetry exploited in cyclic cohomology . If M = A , this has b een sho wn to b e related to th e simplicial cohomology of a fi nitely triangulated space via barycentric sub division, the p oset algebra of incidence relatio n s and the separable subalgebra of s im p lices b y Gerstenhab er and Sc h ac k in a series of p ap ers b eginnin g with [3]. The A -v alued relativ e cohomolog y groups of ( A, B ) are also of int erest in deformation theory . W e r efer to the relativ e Ho c hc hild co chains with cohomo logy groups H n ( A, B ; A ) as simplicial Ho c hsc hild co c hains with cohomolo gy . In this note w e will extend the follo wing algebraic result in [7]: giv en a d epth t w o ring extension A | B with cen tralizer R = A B and endomorphism ring S = End B A B , the simplicial Ho c hsc h ild co c hains und er cup p ro du ct are isomorphic as a graded algebra to the tensor algebra of the ( R, R )-bimo dule S . Since S is a left b ialgebroid o v er R , it is in particular a n R -coring with group lik e elemen t 1 S = id A . The Amitsur complex of suc h a coring is a differenti al graded algebra explained in [2, 29.2]. W e note b elo w that the algebra isomorphism in [7] extends to an isomorphism of differenti al graded algebras. W e also note that the Amitsur complex of the underlying coring of a bialgebroid is a Cartier complex with co efficien ts in a bicomo dule f ormed from source and target homomorph isms. W e r emark on the consequences for relativ e Ho c h s c hild cohomology of v arious types of Galois extensions with bialge b r oid ac tion or coaction. 2 Preliminaries on depth t w o extensions All rings and algebras a re u n ital asso ciativ e; homomorp hisms and mo du les are unital as well. Let R b e a ring, and M R , N R b e righ t R -mo dules. The notation M / N denote s that M is R -mo du le isomorphic to a direct summand of an n -fold dir ect sum p ow er of N : M ⊕ ∗ ∼ = N n . Recall that M and N are similar [1 , p. 268] if M / N and N / M . A ring homomorphism B → A is sometimes called a ring extension A | B (prop er ring extension if B ֒ → A ). Definition 2.1. A r ing homomorph ism B → A is said to b e a right d epth t wo (rD2) exte ns ion if the natural ( A, B )-bimo du les A ⊗ B A and A are similar. Left D2 extension is defined similarly using the n atural ( B , A )-bimo d ule structures: a D2 extension is b oth r D2 and ℓ D2. Note that in either case any r in g extension satisfies A/ A ⊗ B A . Note some ob vious cases of depth t w o: 1) A a finite dimensional algebra, B the ground field. 2) A | B an H-separable extension. 3) A | B a finite Hopf -Galois extension, since the Galois isomorphism A ⊗ B A ∼ = − → A ⊗ H is an ( A, B )-bim o dule arro w (and its t wist b y the an tip o de sho ws A | B to b e ℓ D2 as w ell). Fix the notatio n S := E n d B A B and R = A B . Equip S with ( R, R )-bimo du le stru cture r · α · s = r α ( − ) s = λ r ◦ ρ s ◦ α where λ, ρ : R → S denote left and righ t m ultiplication of r, s ∈ R on A . Lemma 2.2. [5, 3.11] If A | B is rD2, then the mo dule S R is a pr oje ctive gener ator and f 2 : S ⊗ R S ∼ = − → Hom ( B A ⊗ B A B , B A B ) (1) via f 2 ( α ⊗ R β )( x ⊗ B y ) = α ( x ) β ( y ) for x, y ∈ A . F or example, if A is a finite dimensional algebra o v er ground field B , th en S = End A , the linear endomorphism algebra. If A | B is H-separable, then S ∼ = R ⊗ Z R op , where Z is the cen ter of A [5, 4.8]. If A | B is an H ∗ -Hopf-Galois extension, then S ∼ = R # H , the smash pr o duct where H has d ual action on A restricted to R [5, 4.9]. Recall that a left R -bialgebroid H is a t yp e of bialgebra o v er a p ossibly noncomm utativ e base ring R . More sp ecifically , H and R are rings w ith “target” and “sour ce” r ing ant i-homomorph ism and homomorphism R → H , comm uting at all v alues in H , whic h ind uce an ( R, R )-bim o dule structure on H fr om the left. W.r.t. this stru cture, there is an R -coring s tructure ( H , R, ∆ , ε ) suc h th at 1 H is a group lik e elemen t (see the next section) and the left H -mo dules form a tensor catego ry w ith fib er functor to the category of ( R, R )-bimo dules. One of the main theorems in depth t wo theory is Theorem 2.3. [5, 3.10, 4.1] Supp ose A | B is a lef t or right D 2 ring extension. Then the endomorp hism ring S := End B A B is a left bialgebr oid over the c entr alizer A B := R via the sour c e map λ : R ֒ → S , tar get map ρ : R op ֒ → S , c opr o duct f 2 (∆( α ))( x ⊗ B y ) = X ( α ) f 2 ( α (1) ⊗ R α (2) )( x ⊗ B y ) = α ( xy ) . (2) Also A under the natur al action of S i s a left S -mo dule algebr a with invarian t subring A S ∼ = End E A wher e E := End A B ∼ = ← − A # S via a ⊗ R α 7→ λ a ◦ α . W e n ote in passing the measur ing axiom of mo d u le algebra ac tion from eq. (2 ): in Sweedler notation, P ( α ) α (1) ( x ) α (2) ( y ) = α ( xy ). Note to o that ∆( λ r ) = λ r ⊗ 1 S and ∆( ρ s ) = 1 S ⊗ R ρ s for r, s ∈ R . 2 3 Amitsur complex of a coring with grouplik e An R -coring C has coasso ciativ e copro duct ∆ : C → C ⊗ R C and counit ε : C → R , b oth mapp ings b eing ( R , R )-bimo d ule h omomorphisms. W e assume th at C also has a grouplik e elemen t g ∈ C , whic h means that ∆( g ) = g ⊗ R g and ε ( g ) = 1. The Amitsur complex Ω( C ) of ( C , g ) h as n -co c hain mo dules Ω n ( C ) = C ⊗ R · · · ⊗ R C ( n times C ), the zero’th give n by Ω 0 ( C ) = R . The Amitsur complex is the tensor algebra Ω( C ) = ⊕ ∞ n =0 Ω n ( C ) with a c ompatible differentia l d = { d n } where d n : Ω n ( C ) → Ω n +1 ( C ). These are defin ed by d 0 : R → C , d 0 ( r ) = r g − g r , and d n ( c 1 ⊗ · · · ⊗ c n ) = g ⊗ c 1 ⊗ · · · ⊗ c n + ( − 1) n +1 c 1 ⊗ · · · ⊗ c n ⊗ g (3) + n X i =1 ( − 1) i c 1 ⊗ · · · ⊗ c i − 1 ⊗ ∆( c i ) ⊗ c i +1 ⊗ · · · ⊗ c n Some computations sho w th at Ω( C ) is a differen tial graded algebra [2], with d efining equ ations, d ◦ d = 0 as w ell as the graded Leibniz equatio n on homogeneous elemen ts, d ( ω ω ′ ) = ( dω ) ω ′ + ( − 1) | ω | ω dω ′ . The n ame Amitsur complex comes from the case of a ring homomorphism B → A and A - coring C := A ⊗ B A with copro duct ∆( x ⊗ B y ) = x ⊗ B 1 A ⊗ B y and counit ε ( x ⊗ B y ) = xy . The elemen t g = 1 ⊗ B 1 is a group lik e el ement. W e clearly obtain the classical Amitsur complex, whic h is acyclic if A is faithfully flat o ve r B . In general, the Amit su r complex of a Galois A - coring ( C , g ) is acyclic if A is f aithfully flat o v er the g -coin v arian ts B = { b ∈ A | bg = g b } [2, 29.5]. The Amitsur complex of in terest to this note is the follo wing deriv able from the left bial- gebroid S = End B A B of a depth t wo ring extension A | B with cen tralizer A B = R . The underlying R -coring S has grouplike elemen t 1 S = id A , with ( R, R )-bim o dule structure, copr o d- uct and counit d efined in the p r evious section. In Swe edler notation, we m ay summ arize this as follo w s: Ω( S ) = R ⊕ S ⊕ S ⊗ R S ⊕ S ⊗ R S ⊗ R S ⊕ · · · d 0 ( r ) = λ r − ρ r , d 1 ( α ) = 1 S ⊗ R α − α (1) ⊗ R α (2) + α ⊗ R 1 S , . . . It is inte resting to remark th at this particular Amitsur complex is natur ally isomorphic to a Cartier complex of th e R -coring S with co efficien ts in th e ( S, S )-bicomo dule R e [2, 30.3]. The righ t coaction is giv en by ρ R ( r ⊗ s ) = r ⊗ ρ s , left coactio n by ρ L ( r ⊗ s ) = λ r ⊗ s , and we note that Hom R − R ( R e , Ω n ( S )) ∼ = Ω n ( S ), the differenti als b eing preserv ed by the isomorphism. 4 Cup pro duct in simplicial Ho c hsc hild c ohomology Let A | B b e an extension of K -alg ebr as. W e briefly recall the B -relativ e Ho chsc h ild cohomology of A with co efficien ts in A (for co efficien ts in a bimo d ule, see the source [4]). The zero’th co chain group C 0 ( A, B ; A ) = A B = R , while the n ’th co c hain group C n ( A, B ; A ) = Hom B − B ( A ⊗ B · · · ⊗ B A, A ) 3 ( n times A in the domain). In particular, C 1 ( A, B ; A ) = End B A B = S . The cob ound ary δ n : C n ( A, B ; A ) → C n +1 ( A, B ; A ) is giv en b y ( δ n f )( a 1 ⊗ · · · ⊗ a n +1 ) = a 1 f ( a 2 ⊗ · · · ⊗ a n +1 ) + ( − 1) n +1 f ( a 1 ⊗ · · · ⊗ a n ) a n +1 + n X i =1 ( − 1) i f ( a 1 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 ) (4) and δ 0 : R → S is giv en b y δ 0 ( r ) = λ r − ρ r . T he mappings satisfy δ n +1 ◦ δ n = 0 for eac h n ≥ 0. Its cohomolo gy is denoted by H n ( A, B ; A ) = k er δ n / Im δ n − 1 , and might b e referred to as a simplicial Ho c hschild cohomology , since this cohomolog y is isomorp hic to simplicial cohomology if A is the p oset algebra of incidence relation in a fin ite simplicial complex and B is the separable subalgebra of simplices, w here A is embedd able in an u pp er triangular matrix algebra with B the diagonal matrices [3]. The cup pro duct ∪ : C m ( A, B ; A ) ⊗ K C n ( A, B ; A ) → C n + m ( A, B ; A ) make s use of the m ultiplicativ e stucture on A and is giv en by ( f ∪ g )( a 1 ⊗ · · · ⊗ a n + m ) = f ( a 1 ⊗ · · · ⊗ a m ) g ( a m +1 ⊗ · · · ⊗ a n + m ) (5) whic h satisfies the equation δ n + m ( f ∪ g ) = ( δ m f ) ∪ g + ( − 1) m f ∪ δ n g [3]. C u p pr o duct therefore passes to a pr o duct on the cohomolo gy . W e n ote that ( C ∗ ( A, B ; A ) , ∪ , + , δ ) is a differential graded algebra w e d en ote by C ( A, B ). Theorem 4.1. Supp ose A | B is a right or left D2 algebr a extension. Then the r elative H o chschild A -value d c o chains C ( A, B ) is isomorph ic as a differ ential gr ade d algebr a to the Amitsur c omplex Ω( S ) of the R -c oring S . Pr o of. W e define a mapping f b y f 0 = id R , f 1 = id S , and for n > 1, f n : S ⊗ R · · · ⊗ R S ∼ = − → Hom B − B ( A ⊗ B · · · ⊗ B A, A ) . (6) b y f n ( α 1 ⊗ · · · ⊗ α n ) = α 1 ∪ · · · ∪ α n . (Note that f 2 is consistent with our notation in section 2.) W e pro ved by ind uction on n in [7 , T heorem 5.1] that f is an isomorphism of graded algebras. W e complete the pro of by noting that f is a co chain morphism, i.e., commutes with d ifferen tials. F or n = 0, w e note that δ 0 ◦ f 0 = f 1 ◦ d 0 , sin ce d 0 = δ 0 . F or n = 1, δ 1 ( f 1 ( α ))( a 1 ⊗ B a 2 ) = a 1 α ( a 2 ) − α ( a 1 a 2 ) + α ( a 1 ) a 2 = f 2 (1 S ⊗ R α − α (1) ⊗ R α (2) + α ⊗ R 1 S )( a 1 ⊗ B a 2 ) = f 2 ( d 1 ( α ))( a 1 ⊗ B a 2 ) using eq. (2). The induction step is carried out in a similar but tedious computation: this completes th e pro of that C ( A, B ) ∼ = Ω( S ). 5 Applications of the theorem W e immediately note th at the cohomology r ings of th e tw o d ifferen tial graded algebras are isomorphic. Corollary 5.1. R elative A -value d Ho chschild c ohomol o g y is isomorphic to the c ohomol o g y of the A B -c oring S = End B A B : H ∗ ( A, B ; A ) ∼ = H ∗ (Ω( S ) , d ) (7) if A | B is a left or right depth two extension. 4 F or example, a depth t w o f.g. p ro jectiv e extension is separable iff its R -coring S is coseparable [6, Theorem 3.1]. Cartier cohomology of a coseparable coring with an y co efficien ts v anishes in p ositiv e d imensions [2, 30.4] as do es Ho c hschild cohomolog y of a separable extension [4]. But cohomology of the Amitsu r complex ab o ve is a particular case of Cartier cohomology as noted at the end of sect ion 3: H ∗ (Ω( S ) , d ) ∼ = H ∗ Ca ( S, R e ) . (8) Corollary 5.2. If the ring extension A | B is H- sep ar able and one-side d faithful ly flat, then the r elative Ho c hschild c ohomolo gy, H n ( A, B ; A ) vanishes in p ositive dimensions. Pr o of. Th e extension is necessarily prop er by faithfu l flatness. Note that S ∼ = R ⊗ Z R is a Galois R -coring, since { r ∈ R | r · 1 S = 1 S · r } = Z , th e cen ter of A and the isomorphism r ⊗ s 7→ λ r ◦ ρ s is clearly an R -coring homomorphism. When ce Ω( S ) is acyclic by [2, 29 .5]. This also follo ws from p ro ving that an H-separable extensions is separable. The next corollary may b e stated more generally for algebras ov er a base ring wh ic h is hereditary , if the univ ersal coefficien t theorem is tak en in to acco unt. L et K b e a Hopf algebra. Corollary 5.3. Supp ose A | B is a finite Hopf- K ∗ -Galois extension of algebr as over a field k . Then r elative Ho chschild A -value d c ohomol o g y i s isomor phic to the Cartier c ohom olo gy of the underlying c o algebr a K with trivial c o effici e nts: for n ≥ 2 , H n ( A, B ; A ) ∼ = A B ⊗ k H n Ca ( K, k ) . (9) Pr o of. Th is follo ws from the determination of R ⊗ k K ∼ = S via r ⊗ h 7→ λ r ◦ ( h ⊲ · ), and that ∆ S = R ⊗ ∆ K in [5]. T he r elation of action of K on A to coactio n A → A ⊗ K ∗ is giv en by h ⊲ a = a (0) h a (1) , h i . The K -bicomo dule structure on k is giv en by the u nit k → K . Note that Ω n ( S ) ∼ = R ⊗ K ⊗ · · · ⊗ K ( n times K ), where d n = R ⊗ d n c and d n c is the differenti al for coalgebra cohomology of K with coefficient s in k [2, 30.3]. F or example, a finite dimensional Hopf alge br a K is Galois o ver k 1 K via its co pr o duct as coacti on, where K ∗ acts on K via h ∗ ⇀ h = h (1) h h ∗ , h (2) i . In t h is case, relativ e cohomology reco v ers absolute cohomolo gy and the corollary states something w ell-kno wn in a somewhat differen t p ersp ectiv e: for n ≥ 2, H n ( K, K ) ∼ = K ⊗ H n Ca ( K ∗ , k ) (also, ∼ = H n Ca ( K ∗ , K ∗ )). Ac kno wledgemen t The author th an k s the organizers and participants of A.G.M .F. in Gothenburg and the Norwe- gian algebra meeting in Oslo (No v. 1-2, 2007 ) for the stim ulating fo cus on cohomolog y . References [1] F .W. 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