Limits over categories of extensions

LIMITS O VER CA TEGORIES OF EXTENSION S R O MAN MIKHAILOV AND INDER BIR S. P ASSI Abstract. W e consider limit s ov er categories of extensions and show ho w cer tain well-known functors on the categor y o f groups turn o ut as suc h limits. W e a lso discuss higher (or derived) limits ov er categor ies of extensio ns . 1. Intr oduction Let k b e a commutativ e ring w ith ide ntit y , a nd let C b e one of the f ollo wing categories: the category Gr of groups, the category Ab of ab elian groups, the category Ass k of asso ciativ e algebras o ve r k . Giv en an o b ject G ∈ C , let Ext C ( G ) b e the category whose ob jects are the extensions H ֌ F ։ G in C with G as t he cok ernel, and the morphisms are the commutativ e diagrams of short exact sequences of the f orm H 1   / /   F 1 / / / /   G H 2   / / F 2 / / / / G. It is clearly nat ura l to consider also the full sub category Fe x t C ( G ) of Ext C ( G ) which consists of the short exact sequences H ֌ F ։ G where F is a free ob ject in C . The category Ex t Gr ( G ) has b een studied extensiv ely fro m the p oin t of view o f the theory of cohomo lo gy of g r oups (see, for example, [6], [7]). The general question whic h w e wish to address here can b e form ulat ed as follows: how can one study the prop erties of ob jects G ∈ Ob ( C ) from the prop erties of the category F ext C ( G )? Let C b e a small category and F : C − → C a cov ariant functor. The in v erse limit lim ← − F of F , by definition, consists of those families ( x c ) c ∈ C in the direct pro duct Q c ∈ C F ( c ) whic h are compatible in the follo wing sense: F or an y tw o ob jects c, c ′ ∈ C and any morphism a ∈ Hom C ( c, c ′ ), w e ha v e F ( a )( x c ) = x c ′ ∈ F ( c ′ ). Let F , G b e t wo functors from C to the category C . Then a na tural transformation η : F − → G induces a homomorphism lim ← − η : lim ← − F − → lim ← − G , b y mapping an y elemen t ( x c ) c ∈ C ∈ lim ← − F on to ( η c ( x c )) c ∈ C ∈ lim ← − G . In this w ay , lim ← − itself b ecomes a functor fro m the functor category C C to the category C . Our aim in this pap er is to consider the catego r ies Ext C ( G ) , F ext C ( G ) and limits lim ← − F for functors F G : Ext C ( G ) → Gr , F G : F ext C ( G ) → Gr . Suppose these functors are natural in the follo wing sense: 1 2 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI Giv en a mor phism α : G 1 → G 2 in C , ev ery comm utat iv e diag ram of the form H 1   / /   F 1 / / / /   G 1 α   H 2   / / F 2 / / / / G 2 induces a ho mo mo r phism of groups F G 1 { H 1 ֌ F 1 ։ G 1 } → F G 2 { H 2 ֌ F 2 ։ G 2 } , compatible with morphisms in Ext C ( G 1 ) and Ext C ( G 2 ), i.e., ev ery comm utative diagram of t he form (1) H ′ 1   / / ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ F ′ 1          / / / / G 1 α ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ H ′ 2     / /   F ′ 2   / / / /   G 2 H 1   ~ ~ | | | | | | | | / / F 1 ~ ~ } } } } } } } / / / / G 1 α ~ ~ | | | | | | | | H 2   / / F 2 / / / / G 2 induces the following comm utat ive diagram o f gro ups (2) F G 1 { H ′ 1 ֌ F ′ 1 ։ G 1 } / /   F G 1 { H 1 ֌ F 1 ։ G 1 }   F G 2 { H ′ 2 ֌ F ′ 2 ։ G 2 } / / F G 2 { H 2 ֌ F 2 ։ G 2 } . In that case the limit of t he functors o v er categories of extensions defines a functor C → Gr b y setting G 7→ lim ← − F G . T o clarify our p oint of view further, let us recall some known examples whic h ha v e motiv ated our presen t in v estigation. Let C = Gr , G a group, Z [ G ] its in tegral group ring and M a Z [ G ]- mo dule. F or n ≥ 1 , define the functor (3) F n : F ext C ( G ) → Ab b y setting F n : { R ֌ F ։ G } 7→ ( R ab ) ⊗ n ⊗ Z [ G ] M , where R ab denotes the a b elianization of R , the action of G on R ⊗ n ab is diag onal and is defined via conjugation in F . It is sho wn b y I. Emmanouil a nd R . Mikhailo v in [4] that lim ← − F n is LIMITS OVER CA TEGORIES OF EXTENSIO NS 3 isomorphic to t he homolog y g r o up H 2 n ( G, M ): (4) lim ← − F n ≃ H 2 n ( G, M ) . F or any group G and field k of c haracteristic 0 view ed as a trivial G -mo dule, the homolog y groups H n ( G, k ), n ≥ 2, app ear as inv erse limits for suitable natural functors define d on the category Ext Gr ( G ) (see [5] for details). Consider the category C = Ass Q of asso ciativ e algebras ov er Q , the field of ra t io nals. F or in tegers n ≥ 1 , and an asso ciativ e a lgebra A ov er Q , consider the functors F n : Ext C ( A ) → Q -mo dules giv en by setting (5) F n : { I ֌ R ։ A } 7→ R/ ( I n +1 + [ R, R ]) , where [ R, R ] is the Q -submodule of R , generated b y the elemen ts r s − sr with r , s ∈ R . It has b een shown b y D. Quillen in [11] that t he in v erse limit lim ← − F n is isomorphic to the ev en cyclic homo lo gy group H C 2 n ( A, Q ): (6) H C 2 n ( A, Q ) ≃ lim ← − F n . F urthermore, the Connes susp ension map S : H C 2 n ( A, Q ) → H C 2 n − 2 ( A, Q ) can b e obtained as follo ws: H C 2 n ( A, Q ) S   lim ← − F n   H C 2 n − 2 ( A, Q ) lim ← − F n − 1 where the rig ht hand side map is induced by the natural pro jection R/ ( I n +1 + [ R, R ]) → R/ ( I n + [ R, R ]) . The motiv a tion for our in v estigation should now b e clear from the ab o ve examples: one takes quite simple functors, lik e (3) and (5), o n the appropriate categories of extensions and asks for the corresp onding in v erse limits. It is then also natural to consider the deriv ed functors lim ← − i : Ab F ext ( G ) → Ab of the limit functor. Ev ery functor F : Fext C ( G ) → Ab whic h is natural in the ab ov e-men tioned sense (see diagrams ( 1) and (2)) determines a series of functors C → Ab by setting G 7→ lim ← − i F ∈ Ab , i ≥ 0. W e no w briefly describ e the con ten ts of the presen t pap er. W e b egin b y recalling in Section 2 t w o prop erties of the limits g iv en in [4] and [5]. The first (Lemma 2.1) pro vides a set of v a nishing conditions for lim ← − F while, the second states that lim ← − F em b eds in F ( c 0 ) for ev ery quasi-initial ob ject c 0 . In Section 3 (Theorems 3.1 and 3 .4) we show how the deriv ed functors in the sense of A. D old and D. Pupp e [3] of certain standard non- additiv e functors on Ab , lik e 4 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI tensor pow er, symmetric p o w er, exterior p ow er, are realized as limits of suitable functors o v er extension categories. In Section 4 w e discuss higher limits and prov e (Theorem 4.4 ) that for the f unctor F : F ext Gr ( G ) → Ab , ( R ֌ F ։ G ) 7→ R / [ F , R ] , lim ← − 1 F is non-trivial if G is not a p erfect group. W e conclude with some remarks and p ossi- bilities for f urther work in Section 5. 2. Proper t ies of limits recalle d Recall that the copro duct of t wo ob jects a and b in a catego ry C is an ob ject a ⋆ b whic h is endo w ed with t w o morphisms ι a : a − → a ⋆ b and ι b : b − → a ⋆ b having the f o llo wing univ ersal prop ert y: F or any ob j ect c of C and an y pair of morphisms f : a − → c and g : b − → c , there is a unique morphism h : a ⋆ b − → c , suc h that h ◦ ι a = f and h ◦ ι b = g . The morphism h is usually denoted b y ( f , g ). W e recall from [4] the following Lemma whic h prov ides certain conditions whic h imply the trivialit y of the in v erse limit. Lemma 2.1. [4 ] L et C b e a smal l c ate gory and F : C − → Ab a functor to the c ate gory of ab elian gr oups, and supp ose that the fol lowing c onditions a r e satisfie d : ( i ) Any two o b je cts a, b of C hav e a c opr o d uct ( a ⋆ b, ι a , ι b ) a s ab ov e. ( ii ) F or any t wo obje cts a, b of C the mo rp h isms ι a : a − → a ⋆ b and ι b : b − → a ⋆ b induc e a monom orphism ( F ( ι a ) , F ( ι b )) : F ( a ) ⊕ F ( b ) − → F ( a ⋆ b ) of ab elian g r oups. Then, the inverse limi t lim ← − F is the zer o gr oup. Indeed, let ( x c ) c ∈ O b ( C ) ∈ lim ← − F b e a compatible family and fix an o b ject a of C . W e consider the copro duct a ⋆ a of t wo copies of a and the morphisms ι 1 : a − → a ⋆ a and ι 2 : a − → a ⋆ a . Then, w e ha v e F ( ι 1 )( x a ) = x a⋆a = F ( ι 2 )( x a ) and hence t he elemen t ( x a , − x a ) is contained in the k ernel of the additiv e map ( F ( ι 1 ) , F ( ι 2 )) : F ( a ) ⊕ F ( a ) − → F ( a ⋆ a ) . In view of our assumption, this latter ma p is injectiv e and henc e x a = 0 . Since this is the case for an y ob j ect a of C , w e conclude that the compatible family ( x c ) c ∈ O b ( C ) v a nishes, as asserted. W e next men tion another prop erty of the limit functor. Recall that an ob ject c 0 of a category C is called quasi-initial if the set Hom C ( c 0 , c ) is non- empt y for ev ery ob ject c of C . LIMITS OVER CA TEGORIES OF EXTENSIO NS 5 Lemma 2.2. [5] L et C b e a c ate gory with a q uas i - i n itial obje ct c 0 . T hen, for any functor F : C − → Ab , the natur al m a p lim ← − F → F ( c 0 ) is inje ctive, wh e r e as its image c onsists of those e l e m ents x ∈ F ( c 0 ) which e qualize any p air of maps F ( f i ) : F ( c 0 ) → F ( c ) , i = 1 , 2 , wher e f 1 , f 2 ∈ H om C ( c 0 , c ) (i. e ., F ( f 1 )( x ) = F ( f 2 )( x ) ). Observ e that give n the category C and an ob ject G ∈ Ob ( C ), the category Fext C ( G ) consists of quasi-initial ob jects. 3. Derived functors of cer t ain functors on Ab In this section w e study t he limits lim ← − F for certain functors F ∈ Ab F ext Ab ( A ) for ab elian groups A . T o fix notation, let ⊗ n : Ab → Ab , n ≥ 1 , b e the n -th tens o r p ower functor A 7→ A ⊗ n := A ⊗ · · · ⊗ A | {z } n terms . The symmetric group Σ n of degree n acts naturally on A ⊗ n : (7) σ ( x 1 ⊗ · · · ⊗ x n ) = x σ (1) ⊗ · · · ⊗ x σ ( n ) , x i ∈ A, σ ∈ Σ n . W e thus hav e the n -th symmetric p ower functor S P n : Ab → Ab with S P n ( A ) = A ⊗ n mo dulo the subgroup generated b y the elemen ts σ ( x 1 ⊗ · · · ⊗ x n ) − x σ (1) ⊗ · · · ⊗ x σ ( n ) , x i ∈ A, σ ∈ Σ n . The n -th exterior p ower functor Λ n : Ab → Ab is defined by A 7→ A ⊗ n mo dulo the subgroup generated by the elemen ts x 1 ⊗ . . . ⊗ x n with x 1 , . . . , x n ∈ A and x i = x i +1 for some i . The n -th divide d p ower functor Γ n : Ab → Ab is defined, for A ∈ Ab , to the n -th homo g eneous comp onen t of the graded group Γ ∗ ( A ) g enerated by sy mbols γ i ( x ) of degree i ≥ 0 satisfy ing the f ollo wing relations for all x, y ∈ A : ( i ) γ 0 ( x ) = 1 ( ii ) γ 1 ( x ) = x ( iii ) γ s ( x ) γ t ( x ) =  s + t s  γ s + t ( x ) ( iv ) γ n ( x + y ) = X s + t = n γ s ( x ) γ t ( y ) , n ≥ 1 ( v ) γ n ( − x ) = ( − 1) n γ n ( x ) , n ≥ 1 . In particular, t he canonical map A → Γ 1 ( A ) is an isomorphism. It is known that, for a free ab elian group A , t here is a natural isomorphism Γ n ( A ) ≃ ( A ⊗ n ) Σ n , n ≥ 1 where the actio n of the symmetric group Σ n on A ⊗ n is defined as in (7). Let n ≥ 0 b e an in teger, and T an endofunctor on the category Ab of ab elian groups. The doubly indexed family L i T ( − , n ) of deriv ed functors, in the sense of Dold-Pupp e [3], of T are 6 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI defined b y L i T ( A, n ) = π i T N − 1 P ∗ [ n ] , i ≥ 0 , A ∈ Ab , where P ∗ [ n ] → A is a pro jectiv e resolution of A of lev el n , and N − 1 is the Dold-K a n transform, whic h is the t he inv erse of the Mo o r e normalizatio n functor N : S ( Ab ) → C h ( Ab ) from the category of simplicial a b elian groups to the category of chain complexes ( see, for example, [10], pp. 306, 326; o r [12], Section 8.4). F or any functor T , w e set L i T ( A ) := L i T ( A, 0) , i ≥ 0 . F or ab elian groups B 1 , . . . , B n , let the g r oup T or i ( B 1 , . . . , B n ) denote the i -t h homology group of the complex P 1 ⊗ · · · ⊗ P n , where P j is a Z -flat resolution o f B j for j = 1 , . . . , n . W e clearly ha v e T or 0 ( B 1 , . . . , B n ) = B 0 ⊗ · · · ⊗ B n , T or i ( B 1 , . . . , B n ) = 0 , i ≥ n. It turns out from the Eilen b erg-Z ilb er theorem that the deriv ed functors of the n -th tensor p o wer can b e described as L i ⊗ n ( A ) = T or i ( A, · · · , A | {z } n terms ) , 0 ≤ i ≤ n − 1 . W e will use the follo wing notatio n: T or [ n ] ( A ) := T or n − 1 ( A, · · · , A | {z } n terms ) , n ≥ 2 . Theorem 3.1. F or n ≥ 2 , ther e is an isomorphism of ab elian gr oups T or [ n ] ( A ) ≃ lim ← − ( F ⊗ n /H ⊗ n ) , A ∈ Ab , wher e the lim it is taken over the c ate gory Fe x t Ab ( A ) of fr e e extensions H ֌ F ։ A i n the c ate gory Ab . T o pro ceed with the pro of, w e first recall t he following result whic h is w ell-kno wn. Lemma 3.2. L et A = F 1 /H 1 , B = F 2 /H 2 , wher e F 1 , F 2 ar e f r e e a b elian gr oups. Then ther e is a n isomorph i sm of ab elian gr oups T or( A, B ) = ( H 1 ⊗ F 2 ) ∩ ( F 1 ⊗ H 2 ) H 1 ⊗ H 2 , wher e the interse ction is taken in F 1 ⊗ F 2 . Indeed, the ab o ve result follo ws directly from the exact sequence of ab elian groups 0 → T or( A, B ) → H 1 ⊗ B → F 1 ⊗ B → A ⊗ B → 0 and isomorphisms H 1 ⊗ B ≃ ( H 1 ⊗ F 2 ) / ( H 1 ⊗ H 2 ) , F 1 ⊗ B ≃ ( F 1 ⊗ F 2 ) / ( F 1 ⊗ H 2 ). LIMITS OVER CA TEGORIES OF EXTENSIO NS 7 Lemma 3.3. L et A = F /H , w her e F is a fr e e ab elian gr oup. T h en, for every n ≥ 2 , ther e is an isomo rp h ism of ab elian gr oups T or [ n ] ( A ) ≃ T n i =1 ( H ⊗ i − 1 ⊗ F ⊗ H ⊗ n − i ) H ⊗ n wher e the interse ction is taken in F ⊗ n . Pr o of. Observ e t ha t (8) T or [ n ] ( A ) ≃ T or(T or [ n − 1] ( A ) , A ) , n ≥ 3 . T o see (8), one can apply the K¨ unneth formula to the tensor product of the c hain complexes P ⊗ · · · ⊗ P ( n − 1 times) and P , where P is a pro jectiv e resolution of A . The Lemma fo llo ws b y inductiv e ar g umen t and Lemma 3 .2.  Pr o of of T h e or em 3.1. Giv en A = F /H , where F is a free ab elian group, consider the f ollo wing exact sequence o f ab elian groups: (9) 0 → T or [ n ] ( A ) → F ⊗ n /H ⊗ n → F ⊗ n / n \ i =1 ( H ⊗ i − 1 ⊗ F ⊗ H ⊗ n − i ) → 0 . The sequence ( 9) is natural in the f o llo wing sens e: a n y morphism in F ext Ab ( A ), sa y f : ( F 1 , π 1 ) → ( F 2 , π 2 ) (with H 1 = k er ( π 1 ) , H 2 = k er ( π 2 )) implies t he comm utativ e dia- gram 0 / / T or [ n ] ( A ) / / F ⊗ n 1 /H ⊗ n 1 / /   F ⊗ n 1 / T n i =1 ( H ⊗ i − 1 1 ⊗ F 1 ⊗ H ⊗ n − i 1 )   0 / / T or [ n ] ( A ) / / F ⊗ n 2 /H ⊗ n 2 / / F ⊗ n 2 / T n i =1 ( H ⊗ i − 1 2 ⊗ F 2 ⊗ H ⊗ n − i 2 ) Since the in vers e limit functor is left exact in Ab F ext Ab ( A ) , w e obtain a natural monomorphism (10) T or [ n ] ( A ) ֒ → lim ← − F ⊗ n /H ⊗ n , n ≥ 2 . Giv en a free presen tation A = F /H in Ab , consider the tw o morphisms in F ext Ab ( A ) (11) 0 / / H / /   F / / f 1 , f 2   A / / 0 0 / / F ⊕ H / / F ⊕ F / / A / / 0 giv en by setting: f 1 : g 7→ (0 , g ) , g ∈ F , f 2 : g 7→ ( g , g ) , g ∈ F . 8 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI Let α ∈ F ⊗ n /H ⊗ n b e an elemen t whic h b elongs to the equalizer of the ma ps f ∗ 1 , f ∗ 2 : F ⊗ n /H ⊗ n → ( F ⊕ F ) ⊗ n ( F ⊕ H ) ⊗ n induced b y f 1 , f 2 resp ectiv ely . Express α as a coset α = ( X i g i 1 ⊗ · · · ⊗ g i n ) + H ⊗ n , g i j ∈ F . Iden tifying ( F ⊕ F ) ⊗ n ( F ⊕ H ) ⊗ n with L ( i 1 , . . ., i n ) ∈{ 0 , 1 } n F ⊗ n C i 1 ⊗···⊗ C i n where C 0 = F and C 1 = H , we can describe f ∗ i ( α ) , i = 1 , 2, as f ∗ 1 ( α ) = (0 , . . . , 0 , X i g i 1 ⊗ · · · ⊗ g i n ) f ∗ 2 ( α ) = ( X i g i 1 ⊗ · · · ⊗ g i n , . . . , X i g i 1 ⊗ · · · ⊗ g i n ) Since α lies in the equalizer of f ∗ 1 and f ∗ 2 , w e conclude t ha t X i g i 1 ⊗ · · · ⊗ g i n ∈ n \ i =1 ( H ⊗ i − 1 ⊗ F ⊗ H ⊗ n − i ) . The categor y F ext Ab ( A ) clearly consists of quasi-initial ob jects; hence Lemma 2.2 implies that the na tural map (10 ) is an isomorphism and the theorem is prov ed.  Theorem 3.4. F or every ab elian gr oup A and inte ger n ≥ 2 , ther e ar e natur al isomorphisms L n − 1 S P n ( A ) ≃ lim ← − Λ n ( F ) / Λ n ( H ) (12) L n − 1 Λ n ( A ) ≃ lim ← − Γ n ( F ) / Γ n ( H ) (13) wher e the limits ar e taken over the c ate gory Fe x t Ab ( A ) o f fr e e extensions H ֌ F ։ A . Pr o of. Giv en a free extension 0 → H f → F → A → 0 the K oszul complexes (14) 0 → Λ n ( H ) κ n → Λ n − 1 ( H ) ⊗ F κ n − 1 → · · · κ 2 → H ⊗ S P n − 1 ( F ) κ 1 → S P n ( F ) and (15) 0 → Γ n ( H ) κ n → Γ n − 1 ( H ) ⊗ F κ n − 1 → · · · κ 2 → H ⊗ Λ n − 1 ( F ) κ 1 → Λ n ( F ) represen t models of the ob jects LS P n ( A ) and L Λ n ( A ) in the deriv ed category (see [9], Prop o - sition 2.4 and Remark 2.7). In these complexes , the maps κ k +1 : Λ k +1 ( H ) ⊗ S P n − k − 1 ( F ) → Λ k ( H ) ⊗ S P n − k ( F ) , k = 0 , . . . , n − 1 κ k +1 : Γ k +1 ( H ) ⊗ Λ n − k − 1 ( F ) → Γ k ( H ) ⊗ Λ n − k ( F ) , k = 0 , . . . , n − 1 LIMITS OVER CA TEGORIES OF EXTENSIO NS 9 are defined b y setting: κ k +1 : p 1 ∧ · · · ∧ p k +1 ⊗ q k +2 . . . q n 7→ k +1 X i =1 ( − 1) k +1 − i p 1 ∧ · · · ∧ ˆ p i ∧ · · · ∧ p k +1 ⊗ f ( p i ) q k +2 . . . q n p 1 , . . . , p k +1 ∈ H , q k +2 , . . . , q n ∈ F . and κ k +1 : γ r 1 ( p 1 ) . . . γ r k ( p k ) ⊗ q 1 ∧ . . . ∧ q n − k − 1 7→ k X j =1 γ r 1 ( p 1 ) . . . γ r j − 1 ( p j ) . . . γ r k ( p k ) ⊗ f ( p j ) ∧ q 1 ∧ . . . ∧ q n − k − 1 , p 1 , . . . , p k ∈ H , q 1 , . . . , q n − k − 1 ∈ F In pa rticular, the homolog y g r oups of complexes ( 1 4) and (15) are isomorphic to the deriv ed functors L i S P n ( A ) a nd L i Λ n ( A ) respectiv ely . If f : H → F is the iden tity map and A = 0, the complexes (1 4) and (15) are acyclic complexes . The comm utativ e diagram H   f / /  _ f   F F F implies the fo llo wing diagrams with exact columns: (16) Λ n ( H )  _     κ n / / Λ n − 1 ( H ) ⊗ F  _   κ n − 1 / / . . . κ 2 / / H ⊗ S P n − 1 ( F )  _   κ 1 / / S P n ( F ) Λ n ( F )       / / Λ n − 1 ( F ) ⊗ F / /     . . . / / F ⊗ S P n − 1 ( F )     / / S P n ( F ) Λ n ( F ) Λ n ( H ) / / Λ n − 1 ( F ) Λ n − 1 ( H ) ⊗ F / / / / . . . / / A ⊗ S P n − 1 ( F ) , (17) Γ n ( H )  _     κ n / / Γ n − 1 ( H ) ⊗ F  _   κ n − 1 / / . . . κ 2 / / H ⊗ Λ n − 1 ( F )  _   κ 1 / / Λ n ( F ) Γ n ( F )       / / Γ n − 1 ( F ) ⊗ F / /     . . . / / F ⊗ Λ n − 1 ( F )     / / Λ n ( F ) Γ n ( F ) Γ n ( H ) / / Γ n − 1 ( F ) Γ n − 1 ( H ) ⊗ F / / / / . . . / / A ⊗ Λ n − 1 ( F ) . 10 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI Since the middle horizontal sequences in t he diagrams ( 1 6) and ( 17) are exact, w e obtain the follo wing exact sequences : 0 → L n − 1 S P n ( A ) → Λ n ( F ) Λ n ( H ) → Λ n − 1 ( F ) Λ n − 1 ( H ) ⊗ F (18) 0 → L n − 1 Λ n ( A ) → Γ n ( F ) Γ n ( H ) → Γ n − 1 ( F ) Γ n − 1 ( H ) ⊗ F (19) Since the inv erse limit functor is left exact, w e obtain the follow ing natura l sequences: 0 → L n − 1 S P n ( A ) → lim ← − Λ n ( F ) Λ n ( H ) → lim ← − Λ n − 1 ( F ) Λ n − 1 ( H ) ⊗ F (20) 0 → L n − 1 Λ n ( A ) → lim ← − Γ n ( F ) Γ n ( H ) → lim ← − Γ n − 1 ( F ) Γ n − 1 ( H ) ⊗ F (21) where the limits a re t ak en, as usual, o v er the categor y of extensions H ֌ F ։ A . W e claim that fo r n ≥ 2 , and ev ery k ≥ 1, (22) lim ← −  Λ n − 1 ( F ) Λ n − 1 ( H ) ⊗ F ⊗ k  = lim ← −  Γ n − 1 ( F ) Γ n − 1 ( H ) ⊗ F ⊗ k  = 0 . F or n = 2 , the functor { H ֌ F ։ A } 7→ A ⊗ F ⊗ k satisfies the conditio ns of Lemma 2.1 and the assertion follows. No w assume that (22) is pro v ed for a fixed n ≥ 1. Consider the tensor pro ducts of sequences (18) (f o r n + 1) and (19) with F ⊗ k : 0 → L n S P n +1 ( A ) ⊗ F ⊗ k → Λ n +1 ( F ) Λ n +1 ( H ) ⊗ F ⊗ k → Λ n ( F ) Λ n ( H ) ⊗ F ⊗ k +1 0 → L n Λ n +1 ( A ) ⊗ F ⊗ k → Γ n +1 ( F ) Γ n +1 ( H ) ⊗ F ⊗ k → Γ n ( F ) Γ n ( H ) ⊗ F ⊗ k +1 By induction, lim ← −  Λ n ( F ) Λ n ( H ) ⊗ F ⊗ k +1  = lim ← −  Γ n ( F ) Γ n ( H ) ⊗ F ⊗ k +1  = 0 and the f unctors { H ֌ F ։ A } 7→ L n S P n +1 ( A ) ⊗ F ⊗ k { H ֌ F ։ A } 7→ L n Λ n +1 ( A ) ⊗ F ⊗ k satisfy the conditions of Lemma 2.1. Hence (22 ) follo ws. The statemen t of the theorem now follo ws from sequences (20) a nd (2 1).  LIMITS OVER CA TEGORIES OF EXTENSIO NS 11 4. Higher limits Let C b e a small category . The first deriv ed functor lim ← − 1 : Gr C → p oin ted sets can b e defined via cosimplicial replacemen t in the category Gr C described in [2]. Give n F ∈ Gr C , define a cosimplicial replacemen t Q ∗ F , a cosimplicial gro up, with Y n F = Y u ∈ I n F ( i 0 ) , u = { i 0 α 1 ← · · · α n ← i n } and coface a nd co degeneracy maps induced by d 0 : F ( i 1 ) F ( α 1 ) → F ( i 0 ) , d j : F ( i 0 ) id → F ( i 0 ) , 0 < j ≤ n, s j : F ( i 0 ) id → F ( i 0 ) , 0 ≤ j ≤ n. One can che ck that there is a natural isomorphism (see [2]) lim ← − F = π 0 Y ∗ F . The deriv ed functor of the inv erse limit can b e defined as lim ← − 1 F = π 1 Y ∗ F ∈ p ointing sets . W e then ha ve the following: Prop osition 4.1. L et 1 b e an identity functor in Gr C and let 1 → F 1 → F 2 → F 3 → 1 b e a short exac t se quenc e in Gr C . T her e is a natur al long exa c t s e quenc e of gr oups and p ointe d sp ac es : (23) 1 → lim ← − F 1 → lim ← − F 2 → lim ← − F 3 → lim ← − 1 F 1 → lim ← − 1 F 2 → lim ← − 1 F 3 . In the case of the category Ab C , the functor lim ← − 1 has v alues in Ab and the sequence (23) is a lo ng exact sequence o f ab elian gro ups. In t his case there is a co c hain complex of a b elian groups defined by Y • F : Y 0 F δ 0 → Y 1 F δ 1 → . . . with δ n ( a n ) { i 0 α 1 ← · · · α n +1 ← i n +1 } = F ( i 0 α 1 ← i 1 ) a n { i 1 α 1 ← · · · α n +1 ← i n +1 } + n +1 X j =1 ( − 1) j a n { i 0 α 1 ← · · · ← ˆ i j ← · · · α n +1 ← i n +1 } , a n ∈ Y n F , 12 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI suc h that the derived functors of the in v erse limit are the cohomology groups: lim ← − n F = H n ( Y • F ) , n ≥ 0 (see [8], Theorem 4.1). Clearly , lim ← − F = lim ← − 0 F = k er( δ 0 ) . The question o f v anishing of higher limits of functors defined on small cat ego ries in general reduces to the computatio n of lo cal cohomology of nerv es of these categories. W e give a simple condition for the v anishing of lim ← − 1 . Prop osition 4.2. L et C b e a c ate g ory with a quasi-initial obje ct and F : C − → Ab a functor. Supp ose that every p air of morphisms in C h a s a c o e qualizer, i.e., for every p air of morphisms ε 1 , ε 2 : I 1 → I 0 , I 1 , I 0 ∈ Ob ( C ) ther e is a m orphism ε : I 0 → I ( I 0 , I 1 ) in C such that the fol lowing diag r a m is c om mutative I ( I 0 , I 1 ) I 0 ε o o I 1 ε ◦ ε 1 { { ε 1 x x ε ◦ ε 2 c c ε 2 f f , i.e., ε ◦ ε 1 = ε ◦ ε 2 and the induc e d m ap F ( ε ) : F ( I 0 ) → F ( I ( I 0 , I 1 )) is inje ctive. Then lim ← − 1 F = 0 . Pr o of. Let a 1 ∈ Q 1 F = Q 1 i 0 ← i 1 F b e a 1- co cycle, i.e., (24) δ 1 a 1 ( i 0 ← i 1 ← i 2 ) = F ( i 0 ← i 1 ) a 1 ( i 1 ← i 2 ) + a 1 ( i 0 ← i 1 ) − a 1 ( i 0 ← i 2 ) = 0 for ev ery diagram i 0 ← i 1 ← i 2 . Giv en t w o morphisms ε 1 , ε 2 : i 1 → i 0 , consider a mo r phism ε : i 0 → I ( i 0 , i 1 ) , such that F ( ε ) : F ( i 0 ) → F ( I ( i 0 , i 1 )) is a monomorphism of a b elian g r oups and ε ◦ ε 1 = ε ◦ ε 2 . The co cycle condition (24) implies that F ( ε ) a 1 ( i 0 ε 1 ← i 1 ) = a 1 ( I ( i 0 , i 1 ) ε ◦ ε 1 ← i 1 ) − a 1 ( I ( i 0 , i 1 ) ε ← i 0 ) F ( ε ) a 1 ( i 0 ε 2 ← i 1 ) = a 1 ( I ( i 0 , i 1 ) ε ◦ ε 2 ← i 1 ) − a 1 ( I ( i 0 , i 1 ) ε ← i 0 ) and, therefore, F ( ε ) a 1 ( i 0 ε 1 ← i 1 ) = F ( ε ) a 1 ( i 0 ε 2 ← i 1 ) in F ( I ( i 0 , i 1 )). Since F ( ε ) is a monomorphism, we conclude that (25) a 1 ( i 0 ε 1 ← i 1 ) = a 1 ( i 0 ε 2 ← i 1 ) in F ( i 0 ). No w w e can tak e a quasi-initial ob ject i ∈ Ob ( C ) and define a n eleme nt a 0 ∈ Q 0 F b y setting a 0 ( i 0 ) = a 1 ( i 0 ← i ) for arbitrary map i 0 ← i (suc h a map exists, since i is a quasi-initial ob ject). The equalit y (25) implies tha t this is a w ell-defined elemen t. By definition, we hav e − a 1 ( i 0 ← i 1 ) = F ( i 0 ← i 1 ) a 0 ( i 1 ) − a 0 ( i 0 ) = δ 0 a 0 ( i 0 ← i 1 ) , LIMITS OVER CA TEGORIES OF EXTENSIO NS 13 and the pro of is complete.  A t the momen t w e are not able to compute hig her limits ov er categories of free extensions. W e presen t here an approac h tow ar ds this problem and illustrate it with an application. In particular, w e sho w that higher limits ‘co v er’ certain homolo gy functors. Giv en a category C , ob ject G ∈ Ob ( C ), and the category of free extensions F ext C ( G ) , suppose w e hav e t w o pairs of functors H 1 , H 2 : C → Ab F 1 , F 2 : Fext ( G ) → Ab suc h that, for ev ery α ∈ Fext C ( G ) , there is a natural 4-term exact sequence (26) 0 → H 2 ( G ) → F 1 ( α ) → F 2 ( α ) → H 1 ( G ) → 0 whic h is natural in the sense that ev ery morphism β → α in Fe xt C ( G ) induces the comm utative diagram H 2 ( G )   / / F 1 ( β ) / /   F 2 ( β ) / / / /   H 1 ( G ) H 2 ( G )   / / F 1 ( α ) / / F 2 ( α ) / / / / H 1 ( G ) . Supp ose further t ha t (27) lim ← − F 2 = 0 . The condition (27) implies the follow ing exact sequences of ab elian groups: (28) lim ← − 1 F 1   H 1 ( G ) f & & L L L L L L L L L L L L   / / lim ← − 1 α ∈ F ext C ( G ) C ( α ) / /   lim ← − 1 F 2 / / lim ← − 1 H 1 ( G ) lim ← − 2 H 2 ( G ) where C ( α ) = coker {H 2 ( G ) → F 1 ( α ) } = ker {F 2 ( α ) → H 1 ( G ) } , α ∈ F ext C ( G ) . F or ev ery α ∈ F ext C ( G ), fix sections s α : H 1 ( G ) → F 2 ( α ) and t α : C ( α ) → F 1 ( α ) . T o describe the map f , let a ∈ H 1 ( G ) and let γ → β → α be a diag ram in F ext C ( G ) . Consider 14 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI the f ollo wing diagram H 2 ( G )   / / F 1 ( γ ) / / F 1 ( γ → β )   F 2 ( γ ) / / / / F 2 ( γ → β )   H 1 ( G ) H 2 ( G )   / / F 1 ( β ) / / F 1 ( β → α )   F 2 ( β ) / / / / F 2 ( β → α )   H 1 ( G ) H 2 ( G )   / / F 1 ( α ) / / F 2 ( α ) / / / / H 1 ( G ) Define a 2 ( γ → β → α ) := F 1 ( β → α ) t β ( F 2 ( γ → β ) s γ ( a ) − s β ( a )) − t γ F 2 ( β → α )( F 2 ( γ → β ) s γ ( a ) − s β ( a )) The 2-co cycle condition can b e c hec k ed directly; moreov er, note that the elemen t ξ ∈ lim ← − 2 H 2 ( G ) defined b y the cocycle a 2 ( γ → β → α ) do es not dep end on the choice of sections s α , t α . The map f : H 1 ( G ) → lim ← − 2 H 2 ( G ) is thus the one giv en by a 7→ ξ . W e are in terested in finding conditions whic h imply the t r iviality o f the map f . Prop osition 4.3. Supp ose we have functors F 3 , F 4 , F : F ext C ( G ) → Gr such that the fol lowing c onditions ar e satisfie d: (1) Ther e is a na tur al diagr am (29) H 2 ( G )   / / F 1 ( α ) / / F 3 ( α )     / / / / F 4 ( α )     H 2 ( G )   / / F 1 ( α ) / / F 2 ( α ) / / / / H 1 ( G ) . (2) The natur a l map lim ← − F 4 → H 1 ( G ) is a n epimorph ism. (3) F o r every α ∈ F ext C ( G ) , ther e is a n atur al monomorphism F 1 ( α ) → F ( α ) and natur al s h ort exac t se quenc es 1 → F 1 ( α ) → F ( α ) → F 4 ( α ) → 1 1 → H 2 ( G ) → F ( α ) → F 3 ( α ) → 1 . LIMITS OVER CA TEGORIES OF EXTENSIO NS 15 Then the natur al map lim ← − F 4 → lim ← − 2 H 2 ( G ) is the trivial map and ther efor e the map f : H 1 ( G ) → lim ← − 2 H 2 ( G ) is the zer o map. Pr o of. The pro of follow s from the functoriality o f the considered constructions and the fol- lo wing natural comm utat ive diagr a m: H 2 ( G ) H 2 ( G )  _   0 / / F 4 ( α )  _   F 4 ( α ) H 2 ( G ) / / F ( α ) / / F 3 ( α ) ⊕ F 4 ( α ) (0 , id ) / / F 4 ( α ) H 2 ( G )   / / F 1 ( α ) ?  O O / / F 3 ( α ) ?  O O / / / / F 4 ( α ) .  W e next giv e examples of functors satisfying (26). Examples. 1. Let G b e a group, n ≥ 1, { R ֌ F ։ G } ∈ F ext Gr ( G ), then there is a natural exact sequence o f ab elian gro ups (see [4]): (30) 0 → H 2 n ( G ) → H 0 ( F , R ⊗ n ab ) → H 1 ( F , R ⊗ n − 1 ab ) → H 2 n − 1 ( G ) → 0 . 2. Let A b e an a sso ciativ e algebra o v er Q , n ≥ 1, { I ֌ R ։ A } ∈ F ext Ass Q ( A ) . There is a natural exact se quence (see [11]): 0 → H C 2 n ( A ) → H H 0 ( R/I n +1 ) → H 1 ( R, R/I n ) → H C 2 n − 1 ( A ) → 0 . Lemma 2.1 implies t ha t, for the example 1 ab ov e, the condition (27) is satisfied for t he functor { R ֌ F ։ G } 7→ H 1 ( F , R ⊗ n − 1 ab ) for n ≥ 1 ( see [4] f or details), hence there is an isomorphism H 2 n ( G ) ≃ lim ← − H 0 ( F , R ⊗ n ab ) . F or the simplest case , namely for n = 1, functors from the exact sequenc e (30), the diagram (29) 16 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI can b e c hosen to b e the one given b elo w (with the obv ious maps): H 2 ( G )   / / R/ [ R, F ] / / F / ( R ∩ [ F , F ])     / / / / G     H 2 ( G )   / / R/ [ F , R ] / / F / [ F , F ] / / / / H 1 ( G ) Prop osition 4.3 then implies that the nat ural map H 1 ( G ) → lim ← − 2 H 2 ( G ) is the zero map. Consequen tly , t he diagram ( 2 8) implies that H 1 ( G ) is con tained in a group, whic h is an epimorphic image of lim ← − 1 ( R/ [ F , R ]). W e hav e thus pro ved the following: Theorem 4.4. If G is n ot a p erfe ct gr oup, then lim ← − 1 ( R/ [ F , R ]) i s non-trivial. 5. Concluding remarks and questions Observ e tha t giv en a n ob ject G ∈ Ob ( C ) , one can consider the category F ext 2 ( G ) of double (resp. triple etc) presen tations of G . F or simplicity , let us assume that we work in the catego r y of groups. The ob jects of F ext 2 ( G ) are triples ( F , R 1 , R 2 ) , where F is a gr o up, R 1 , R 2 normal subgroups in F , suc h that F /R 1 R 2 = G . The morphisms in F ext 2 ( G ) are t he diagrams of the form R 1 ∩ R 2 / / { { w w w w w w w w w R 2   ~ ~ } } } } } } } } R 1   / /   F   R ′ 1 ∩ R ′ 2 | | x x x x x x x x x / / R ′ 2   ~ ~ ~ ~ ~ ~ ~ ~ R ′ 1 / / F ′ whic h induce the iden tity isomorphism F /R 1 R 2 → F ′ /R ′ 1 R ′ 2 . It would b e of interes t to examine limits of functors ov er the cat ego ry F ext 2 ( G ). F or example, note that, giv en a gro up G , there is a natural homomorphism H 3 ( G ) → lim ← − ( F, R 1 , R 2 ) ∈ F ext 2 ( G ) R 1 ∩ R 2 [ R 1 , R 2 ][ F , R 1 ∩ R 2 ] ; for the construction of this map see the ho molo gy exact sequence in [1]. In the same w ay , one can make v ariations of Quillen’s description (6) o f cyclic homology . Giv en an asso ciative algebra A ov er Q , consider the category F ext 2 ( A ) whose ob jects are the triples ( R , I 1 , I 2 ), where R is a free algebra I 1 , I 2 are ideals in R and R/ ( I 1 + I 2 ) = A . The LIMITS OVER CA TEGORIES OF EXTENSIO NS 17 description (6) implies, for example, that for n ≥ 2 , n > k ≥ 1 , there is a natural mo r phism lim ← − ( R, I 1 , I 2 ) ∈ F ext 2 ( A ) R I n +1 − k 1 I k 2 + I k 1 I n +1 − k 2 + [ R, R ] → H C 2 n ( A ) and its inv estigation ma y b e of in terest for cyclic homolgy . Finally , one now kno ws how to define ev en-dimens iona l homology of g r oups, Lie algebras, cyclic homolog y of asso ciativ e algebras as limits of certain functors ov er the categories o f extensions. What can one sa y ab out higher limits of the functors yielding this relationship? 6. Ackno wledgement The authors w ould lik e to thank Ioannis Emmanouil and Gosha Sharygin for useful discus- sions and imp o r t an t suggestions. Reference s [1] W. Bogley and M. Gutierrez: Ma yer-Vietoris sequences homotopy of 2-complexes a nd in homology of groups, J. Pur e Appl. Algeb r a 7 7 (1992 ) 39-6 5. [2] A. Bousfield a nd D. Kan: Homotopy limits, completions and lo caliz a tions. Lecture Notes in Mathematics, 304, Springer-V er lag, 197 2. [3] A. Dold and D. Puppe: Homologie nich t-additiver F unktoren; Anw endugen. Ann. Inst. F ourier 11 (1961) 201-3 12. [4] I. Emmano uil and R. Mikhailov: A limit approach to g roup homology , J. Algeb r a 3 19 (2008), 1450 -1461 . [5] I. E mmanouil and I. B. S. Passi: Group homolo gy and extensions of gro ups, Homolo gy, Homotopy and Applic ations 10 (2008), 23 7 -257. [6] K.W. Grue nber g and Kla us W. Roggenk amp: Extension categor ies of gro ups and mo dules I: E ssential cov ers, J. Algebr a 49 (1 977), 564 -594. [7] K.W. Gr uenberg and Kla us W. Roggenk amp: Extens ion categor ies o f g roups and mo dules I I: Stem extensions, J. A lgebr a 67 (1 9 80), 3 42-36 8. [8] C. U. Jensen: Les functeurs d´ eriv´ es de lim ← − et leurs applications en th ´ e orie des modules, Le c ture Notes in Mathematics, Springer -V erlag , 25 4 (1 972). [9] B. K¨ oc k: Co mputing the homo lo gy of Ko szul c o mplexes, T r ans. Amer. Math. So c. 353 (2001), 311 5 -3147 . [10] Ro man Mik ha ilov and Inder Bir Singh Passi: L ower c entr al and dimension series of gr oups , Lecture Notes in Mathematics 1952 (2009 ). Ber lin: Springer. [11] D. Quillen: Cyclic co homology a nd algebra extensio ns . K - Theory 3 , 2 05-24 6 (1989) [12] C . A. W eib el, A n intr o duction t o homolo gic al algebr a , C a mbridge Stud. Adv. Ma th. 38 , Ca m br idge Univ er sity Press, Ca m br idg e, 1 994. 18 ROMAN MIKHA ILOV AND IND ER BIR S. P A SSI Roman Mikhailov Steklo v Mathematical Institute Departmen t of Algebra Gubkina 8 Mosco w 11999 1 Russia email: roma nvm@mi.ras.ru Inder Bir S. Pas si INSA Senior Scien tist Cen tre for Adv a nced Study in Mathematics P anjab Univ ersit y Chandigarh 160014 India and Indian Institute of Science Education and Researc h Mohali MGSIP A Complex, Sector 19 Chandigarh 160019 India email: ibspassi@y aho o.co.in