Semigroup cohomology as a derived functor

Semigroup cohomology as a deriv ed functor A. A. Kostin, B. V. No vik o v Abstract In this work we construct an extension for the catego ry o f 0-mo dules by analogy with [5]. The 0-cohomo logy functor b e c omes a der ived functor in the extended ca tegory . A s an a pplication of this constr uctio n we calculate the cohomolog ic a l dimension of so-called 0 -free monoids. 1. 0-cohomolo gy of semigroups app eared in researc h of pr o j ective rep- resen tations of semigroups [1]. Besides, it w as useful in s tu dying of matrix algebras [ 3] and Brauer mo noids [4 ] (see also su r v ey [2] a nd references there). Ho wev er the further study of its prop erties is complicated. O n e of the reasons is that the semigroup 0-cohomolo gy is not a deriv ed functor in th e catego ry w here it is built (so-called category of 0-mo d ules). The pur p ose of this pap er is to describ e the extension of 0-cohomology on a larger category where it b ecomes a derive d fun ctor. Our construction is sim ilar to Ba ues theory for cohomolo gy of small categ ories [5]. Therefore w e omit some pro ofs replacing them b y references to [5]. As an example of app lication of our construction w e prov e that a co- homologica l d imension of a so-called 0-free semigroup equals one. In p ar- ticular, it follo ws that all pro jectiv e represent ations of a free semigroup are linearizable. 2. W e b egin with definitions. Let S b e a monoid. W e ma y assume that S has a zero elemen t (if not, let us join it to S ). By analogy with [5] the c ate gory of factorizations in S is giv en as follo ws. The ob jects are all n onzero elemen ts of S and the set of morp hisms Mor ( a, b ) consists of all triples ( α, a, β ) ( α, β ∈ S ) su c h that αaβ = b . W e will denote ( α, a, β ) by ( α, β ) if this cannot lead to confu sion. The comp osition is d efined b y the ru le: ( α ′ , β ′ )( α, β ) = ( α ′ α, β β ′ ); hence w e hav e ( α, β ) = ( α, 1)(1 , β ) = (1 , β )( α, 1) . Denote this category by F ac S . A natur al system on S is a fun ctor D : F ac S − → A b . The category N at S = A b F ac S is an Ab elian category with enough p ro jectiv es and injec- tiv es [6]. Denote the v alue of D at the ob ject a ∈ Ob F ac S by D a . By α ∗ 1 and β ∗ denote v alues of D at morphisms ( α, 1) and (1 , β ) resp ectiv ely . W e ha v e D ( α, β ) = α ∗ β ∗ for all morphism s ( α, β ). F or giv en natural num b er n denote by N er n S the set of all n -tuples ( a 1 , . . . , a n ) , a i ∈ S , su c h that a 1 · · · a n 6 = 0. By definition N er 0 S = { 1 } . A n-c o chain assigns to ea c h p oin t a = ( a 1 , . . . , a n ) of N er n S an elemen t on D a 1 ··· a n . The set of all n -c o c hains is an Ab elian group C n ( S, D ) with resp ect to the p oin t w ise addition. S et C 0 ( S, D ) = D 1 . The c ob oundary δ = δ n : C n ( S, D ) − → C n +1 ( S, D ) is given by the form ula ( n ≥ 1) ( δ f )( a 1 , . . . , a n +1 ) = a 1 ∗ f ( a 2 , . . . , a n +1 ) + n X i =1 ( − 1) i f ( a 1 , . . . , a i a i +1 , . . . , a n +1 ) + ( − 1) n +1 a ∗ n +1 f ( a 1 , . . . , a n ) . F or n = 0 let δ 0 : C 0 ( S, D ) − → C 1 ( S, D ) b e d efined by δ f ( x ) = x ∗ f − x ∗ f ( f ∈ D 1 , x ∈ S \ 0) . One can c hec k directly that δ n δ n − 1 = 0 . By H n ( S, D ) denote the cohomol- ogy groups of the co mplex { C n ( S, D ) , δ n } n ≥ 0 . 3. No w w e define a trivial natur al system Z . T o eac h ob ject a ∈ S \ 0 it assigns the in finite cyclic group Z a generated by a symb ol [ a ]; and to eac h morphism ( α, β ) : a − → b it assigns a homomorphism of the groups Z ( α, β ) : Z a − → Z b whic h ta k es [ a ] to [ b ]. Since N at S has enough p ro jectiv e and injectiv e, hence there exists th e deriv ed functor Ext n N a t S ( Z , − ). This fun ctor is isomorphic to the cohomolo- gy fu nctor H n ( S, − ) whic h is defined in Section 2. T o pro v e this statement w e co nstruct a suitable p ro jectiv e r esolution of Z . F or every n ≥ 0 w e denote by B n : F ac S − → A b th e follo w ing natural system. F or an ob ject a ∈ S \ 0 the group B n ( a ) is a fr ee Ab elian group generated b y the set of sym b ols [ a 0 , . . . , a n +1 ] suc h that a 0 · · · a n +1 = a. T o eac h morphism ( α, β ) we assign a homomorphism of groups by the formula B n ( α, β ) : [ a 0 , . . . , a n +1 ] 7− → [ αa 0 , . . . , a n +1 β ] . The functors B n ( n ≥ 0) constitute a chain complex { B n , ∂ n } n ≥ 0 , where ∂ n : B n . − → B n − 1 ( n ≥ 1) is a natur al transformation with th e set of its comp onen ts ( ∂ n ) a : B n ( a ) − → B n − 1 ( a ) , 2 ( ∂ n ) a [ a 0 , . . . , a n +1 ] = n X i =0 ( − 1) i [ a 0 , . . . , a i a i +1 , . . . , a n +1 ] . 4. Lemma. The natur al system B n is a pr oje ctive obje ct in N at S . Pr oof. Consider the follo wing diagram with the exact ro w B n     y ν D µ − − − − − → E − − − − − → 0 and construct a natural transformation τ : B n . − → D whic h turn s this diagram in to comm utativ e. Let s = s 0 · · · s n +1 , ˆ s = s 1 · · · s n . Cho ose a ( s 1 ,...,s n ) ∈ D ( ˆ s ) su c h that µ ˆ s a ( s 1 ,...,s n ) = ν ˆ s [1 , s 1 , . . . , s n , 1], and put τ s [ s 0 , . . . , s n +1 ] = D ( s 0 , s n +1 ) a ( s 1 ,...,s n ) . The natural tr an s formation is we ll defined. Indeed, τ αsβ B n ( α, β )[ s 0 , . . . , s n +1 ] = D ( αs 0 , s n +1 β ) a ( s 1 ,...,s n ) = D ( α, β ) D ( s 0 , s n +1 ) a ( s 1 ,...,s n ) = D ( α, β ) τ s [ s 0 , . . . , s n +1 ] . ✷ 5. Lemma. Th e chain c omplex { B n , ∂ n } n ≥ 0 is a pr oje ctive r esolution of the natur al system Z . The pro of is similar to [5]. 6. No w w e are ready to prov e the main result of th is p ap er. Theorem. F or any mo noid S with a zer o element ther e is an isomor phism of the fu nctors: H n ( S, − ) ∼ = Ext n N a t S ( Z , − ) . Pr oof. Define an isomorph ism of complexes Ψ ∗ D : { Hom N a t S ( B n , D ) , ∂ n } n ≥ 0 − → { C n ( S, D ) , δ n } n ≥ 0 (here we denote ∂ n = Hom N a t S ( ∂ n − 1 , D )) as follo w s . Let the h omomorphism of Ab elian group Ψ n D : Hom N a t S ( B n , D ) − → C n ( S, D ) 3 b e giv en by (Ψ n D τ )( a 1 , . . . , a n ) = τ a 1 ··· a n [1 , a 1 , . . . , a n , 1] ∈ D a 1 ··· a n for a 1 · · · a n 6 = 0 . Let a = a 0 · · · a n +1 , i.e. [ a 0 , . . . , a n +1 ] ∈ B n ( a ) . Since the diagram B n ( a 1 · · · a n ) τ a 1 ··· a n − − − − − → D n ( a 1 · · · a n ) B n ( a 0 , a n +1 )     y     y D n ( a 0 , a n +1 ) B n ( a ) τ a − − − − − − − − − − → D n ( a ) is comm utative w e ha v e τ a [ a 0 , . . . , a n +1 ] = D ( a 0 , a n +1 ) τ a 1 ··· a n [1 , a 1 , . . . , a n , 1] . Therefore Ψ n D τ = 0 implies that τ a v anishes on all generators of the group B n ( a ) . Hence Ψ n D is in jectiv e. F urther, for an y f ∈ C n ( S, D ) d efine a natural transformation ϕ : B n . − → D : ϕ a [ a 0 , . . . , a n +1 ] = D ( a 0 , a n +1 ) f ( a 1 , . . . , a n ) It is clear that Ψ n D ϕ = f and hence Ψ n is su rjectiv e. The comm u tativit y of the diagram Hom N a t S ( B n , D ) ∂ n − − − − − → Hom N a t S ( B n +1 , D ) Ψ n D     y     y Ψ n +1 D C n ( S, D ) δ n − − − − − − − − − − → C n +1 ( S, D ) is established immediately . It can easily b e chec k ed that the family Ψ n = { Ψ n D | D ∈ N at S } is a natural transformation. F rom ab o v e w e see that Ψ n induces an isomorphism of functors H n and Ext n . ✷ 7. Let us discuss the relation b et w een cohomology which is d efined ab o v e and cohomology groups of other kind s. In Section 1 w e note that th e 0-cohomolo gy is a particular case of our construction. This can b e sho w n in the follo wing w a y . Let A b e an Ab elian group and A b e a natur al system giv en b y A ( s ) = A and α ∗ β ∗ a = αa 4 for all s ∈ F ac S , ( α, β ) ∈ Mor F ac S . In other words, A is so-called 0- mo dule ov er S [1]: an action ( S \ { 0 } ) × A − → A is giv en , whic h satisfies the follo wing conditions: s ( a + b ) = sa + s b, st 6 = 0 ⇒ s ( ta ) = ( st ) a, where s , t ∈ S \ 0 and a, b ∈ A. 0-Cohomolo gy groups are denoted b y H n 0 ( S, A ). Note that Eilen b erg-MacLane cohomology of semigroups [8] can b e con- sidered as a particular case o f the 0-co homology . Namely , if S is a semigroup (p ossibly w ithout a zero), then H n ( S, − ) ∼ = H n 0 ( S 0 , − ), wher e S 0 is th e semi- group S with an adj oin t zero. The cat egory of 0-mo dules arises naturally in applications of 0-cohomo- logy theory [4]. Ho w ev er it is easily sho w n that the second 0-cohomology group of the comm u tativ e semigroup S = { u, v , w , 0 } with u 2 = v 2 = uv = w, uw = v w = 0 (see [1]) is not trivial for all nonzero 0-mo du le o ver S . Therefore the 0-cohomology is n ot a derived functor on the category of 0-mo d ules. This is th e reason for in tr o ducing the category N at S . Our construction d iffers from Baues’ cohomology theory f or monoids [5] in the fi rst step only . Actually in [5] a monoid S is regarded as a category with a sin gle ob ject. At the same time the Baues’ category of factorizations in S is equal to F ac S 0 out of Section 2 . Ther efore th e Baues’ cohomology groups of monoid S and cohomology grops of S 0 in our sense are the same. Ho wev er if S p ossesses a zero elemen t then the category F ac S and Baues’ one are n ot equiv alen t and we obtain the differen t cohomology groups. 8. Let us consider an application of the obtained results. Cohomolo gi- c al dimension c.d. S of monoid S is the greatest natural num b er suc h that H n ( S, D ) 6 = 0 for some D ∈ N at S . Th e Theorem from Section 6 allo ws us to use a pro jectiv e resolution for calculation of the dimension. It is well-kno wn that in man y cohomolo gical theories c.d. of free ob jects equals 1. F ree ob jects in the class of monoids with zero are free monoids with adjoint zero elemen t. Nev ertheless in our case the family of m onoids ha ving c.d.1 is larger. A mon oid is called a 0-fr e e mono id if it is isomorphic to a R ees factor monoid of a fr ee monoid. F ree monoids w ith adjoint zero will b e regarded as 0-free monoids to o. 9. W e shall need the follo w ing Lemma. L et A , B b e c ate g ories, F : A − → B , G : B − → A b e adjoint 5 functors ( F ⊣ G ) , functor G pr eserves epimorphisms and the c ounit ε : F G . − → Id B is identic al. If an obje ct a ∈ A is pr oje ctive then F ( a ) is pr oje ctive to o. Pr oof. Let a ∈ A b e a pro jectiv e ob ject. Consider a diagram F ( a )     y α c β − − − − − → b with the exact r o w ( c, b ∈ B ) . Since fu nctor G p reserv es epim orp hisms w e obtai n the diagram: a     y G ( α ) η a G ( c ) G ( β ) − − − − − → G ( b ) (1) where η : Id A . − → GF is the unit of the adj unction F ⊣ G . Since a is pro jectiv e, th er e is a homomorphism γ : a − → G ( c ) w h ic h makes d iagram (1) comm u tativ e. This means that G ( β ) γ = G ( α ) η a and β F γ = α F ( η a ) . Using the equalities F ( η a ) = Id F ( a ) and F G = Id B w e ge t β F γ = α. ✷ 10. Theorem. c.d.M ≤ 1 for al l 0-fr e e monoids M . Pr oof. F or a giv en monoid M consider the exact sequence 0 − → P M . − → B M . − → Z M − → 0 where Z M , B M are natural systems defined in Section 3 , P M = Ker( B M . − → Z M ). W e need to prov e that P M is a pro jectiv e functor. It follo ws from Section 7 that P M is a free functor whenever M is a free monoid with adj oin t zero (see [5], Lemma 6.7). No w let M b e a 0-free monoid, M ∼ = W /I wh ere W is a free mon oid and I is an ideal in W . Consider the catego ry of factorizatio ns F W whic h was defined in [5], i.e. F W = F ac ( W 0 ). Define the fu nctor K : F ac M − → F W whic h tak es eac h nonzero elemen t from M to its preimage under the canonic homomorphism W − → W /I . F unctor K is well d efined and in duces the functor K ∗ : Nat W − → N a t M , where N at W = A b F W . Consider the exa ct sequence wh ic h is defined in [5 ], Sec.5: 0 − → ˜ P W ˜ δ W − → ˜ B W ˜ ε W − → ˜ Z W − → 0 , 6 where ˜ P W , ˜ B W , ˜ Z W : F W − → A b are natural s ystems on W . W e ha ve K ∗ ( ˜ Z W ) = Z M , K ∗ ( ˜ B W ) = B M , K ∗ ( ˜ ε W ) = ε M hence K ∗ ( ˜ P W ) = P M . Consider the functor L : N at M − → Nat W whic h is given by L ( G ) a = ( G a , if a 6∈ I 0 , if a ∈ I where G ∈ N a t M , and L ( G )( x, a, y ) = ( G ( x, a, y ) , if xay 6∈ I 0 , if xay ∈ I Eviden tly K ∗ L = Id N a t M and there is a natural transformation Id Nat W . − → LK ∗ . It implies that L is righ t adjoin t to K ∗ . Besides, L preserv es epi- morphisms and b y [5] ˜ P W is a fr ee ob ject. Using Lemma 9 we get P M is a pro jectiv e ob j ect. ✷ 11. The semigroup is called 0-c anc el lative if ax = bx 6 = 0 ⇒ a = b and xa = xb 6 = 0 ⇒ a = b for all elemen ts a, b, x . I n view of Th eorem 10 the follo wing question arises: is a 0-cancellativ e monoid of cohomological dimension one a 0-free monoid? References [1] B. V. No viko v. On 0-c ohomolo gies of semigr oups. T eor.Appl.Quest. Diff.Eq.and Algebra, Kiev, 1978, 18 5-188 (Russian). [2] B. V. No viko v. Semigr oup c ohomolo gy and applic ations. Algebra — Rep- resen tation Th eory (ed. K. W. Roggenk amp and M. S ¸ tefˇ anescu), K lu w er, 2001, 311-318. [3] W. E. Clark. Cohomolo gy of semigr oups via top olo gy with an applic ation to semigr oup algebr as. C omm un. Algebra, 4 (1 976), 979-9 97. [4] B. V. No viko v. O n the Br auer monoid. Matem. zametki, 57 (199 5), No. 4, 633-63 6 (Russian). 7 [5] H.-J. Baues, G. Wirsh ing. Cohom olo gy of smal l c ate gories. J. Pur e Appl. Algebra. 1985. V.38, N 2/3, 187–21 1. [6] A. Grothendiec k. Sur quelque p oints d’alg ` ebr e homolo gique. T ohoku Math.J. 9 (1957) 119- 221. [7] A. H. Clifford, G. B. Preston. A lgebr aic The ory of Semigr oups. Amer. Math. Soc., Pro vidence, 196 4. [8] H. Cartan, S. Eilenb er g. Homolo gic al Algebr a. Princeton, 1956. E-mails: andr eykostin@mail.c om b oris.v.novikov@univer.kharkov.ua 8