Global Dimension of Polynomial Rings in Partially Commuting Variables

GLOBAL DIMENSION OF POL YNOMIAL RINGS IN P AR TIALL Y COMMUTING V ARIABLES Ahmet A. Husa i no v No v em b er 21, 2018 Abstract F or an y free partiall y comm utativ e monoid M ( E , I ) , we c ompute the global dimension of the category of M ( E , I )-o b jects in an Ab elian catego ry with exact copro ducts. As a corollary , w e generalize Hilb ert’s Syzygy Theorem to p olynomial rings in partially co mm uting v aria bles. Keyw ords: cohomology of small categories, free partially comm utativ e monoid, t race monoid, Ho chsc hild-Mitc hell dimens ion, noncomm utative p oly- nomial ring 2000 Mathematics Sub j ect Classification: 16E05, 16 E10, 16E40, 18G10, 18G20 In tr o duction In this pap er, the global dimension o f the category of ob jects in a n Ab elian category with the action of free partially comm utativ e monoid is computed. As a corollary , a form ula f or the global dimension of p olynomial rings in partially comm uting v aria bles is obta ined. Let A b e an y Ab elian category . By [1, Chapter XI I, § 4], extension gro ups E xt n ( A, B ) are consisted of congruence classes of exact sequences 0 → B → C 1 → · · · → C n → A in A for n > 1 and E xt 0 ( A, B ) = H om ( A, B ). It allo ws us to define the g lob al dim ension of A by gl dim A = sup { n ∈ N : ( ∃ A, B ∈ Ob A ) E xt n ( A, B ) 6 = 0 } . 1 Here N is the set of nonnegative integers . (W e set sup ∅ = − 1 and sup N = ∞ .) F or a ring R with 1, gl dim R is the global dimension of the catego r y of left R - mo dules. As it is w ell kno wn [2, Theorem 4 .3.7], for any ring R with 1, gl dim R [ x 1 , . . . , x n ] = n + gl dim R . Moreo v er, by [3, Theorem 2.1], if A is any Ab elian categor y with exact copro ducts and C a bridge c ate gory , then gl dim A C = 1 + gl dim A . It follo ws that gl dim A N n = n + gl dim A for the free comm utativ e monoid N n generated by n elemen ts. W e will get one of p ossible generalizations of this form ula. L et M ( E , I ) b e a free partia lly comm utative monoid with a set of v ariables E , where I ⊆ E × E is an irreflexiv e symmetric relation assigning the pa ir s of comm uting v ariables. In this pap er, w e prov e that gl dim A M ( E , I ) = n + g l dim A for any Ab elian category with exact copro ducts where n is the sup of n umbers of m utually comm uting distinct elemen ts of E . F or example, if R [ M ( E , I )] is the p olynomial ring in v ariables E = { x 1 , x 2 , x 3 , x 4 } with the comm uting pairs ( x i , x j ) corresp onding to adjacen t ve rtices of the graph demonstrated in Figure 1, then for an y ring R with 1 w e ha ve gl dim R [ M ( E , I )] = 2 + gl dim R . x 4 x 3 x 1 x 2 Figure 1: P airs of comm uting v ariables The free par tially comm ut a tiv e monoids ha ve n umerous applications in com binatorics and computer sciences [4]. Our interes t in their homolo gy groups is concerned with the studying a top olo gy of ma t hematical mo dels for concurrency [5]. 1 Cohomolo g y of small categorie s Throughout this pap er let Ab the category of Ab e lian groups and homo- morphisms, Z the additiv e gro up of in tegers, and N the set of nonnegativ e 2 in tegers or the fr ee monoid with only one generator. F or any category A and a pair A 1 , A 2 ∈ Ob A , denote by A ( A 1 , A 2 ) the set of all morphisms A 1 → A 2 . A diagr am C → A is a functor from a small category C to a category A . Giv en a small category C w e denote by A C the catego r y o f diagrams C → A and natural transformations. F or A ∈ Ob A , let ∆ C A (shortly ∆ A ) denote a diagr a m C → A with constan t v alues A o n ob jects and 1 A on morphisms. In this section, w e recall some results from the cohomology theory of small categories. 1.1 Homology groups of a nerv e Recall a definition of a nerve of the category and prop erties of homology groups of simplicial sets. W e refer the reader to [1] and [6] f or the pro of s. 1.1.1 A nerve of the category Let C b e a small category . Its nerve N ∗ C is the simplicial set in whic h N n C consists o f all sequences o f comp osable mor phisms c 0 α 1 → c 1 α 2 → · · · α n → c n in C for n > 0 and N 0 C = Ob C . F or n > 0 and 0 6 i 6 n , b oundary op erators d n i : N n C → N n − 1 C acts as d n i ( c 0 α 1 → · · · α n → c n ) = c 0 α 1 → · · · α i → ˆ c i α i +1 → · · · α n → c n . Here c 0 α 1 → c 1 α 2 → · · · α i → ˆ c i α i +1 → · · · α n → c n ∈ N n − 1 C is the ( n − 1)-fold sequence obtained from c 0 α 1 → c 1 α 2 → · · · α n → c n for 0 < i < n by substitution the mor phisms c i − 1 α i → c i α i +1 → c i +1 b y t heir comp osition c i − 1 α i +1 ◦ α i − → c i +1 . The map d n 0 remo v es α 1 with c 0 and d n n remo v es α n with c n . Degeneracy op erators s n i : N n C → N n +1 C insert in c 0 → · · · → c n the iden tity morphism c i → c i for eve ry 0 6 i 6 n . 1.1.2 Homology gr oups of simplicial sets Let X b e a simplicial set giv en b y b oundary op erat o rs d n i and degener- acy op erators s n i for 0 6 i 6 n . Consider a c hain complex C ∗ ( X ) of free Ab elian groups C n ( X ) generated by the sets X n for n > 0. Differ- en tia ls d n : C n ( X ) → C n − 1 ( X ) are defined on the basis elemen ts x ∈ X n b y d n ( x ) = P n i =0 ( − 1) i d n i ( x ). Let C n ( X ) = 0 for n < 0. The groups H n ( X ) = Ker d n / Im d n +1 are called n -th homo l o gy gr oups of the sim p l i c ial 3 set X . The g roups H n ( X ) a r e isomorphic to n -th singular homology groups of the geometric realization of X b y the Eilen b erg theorem [6, Appl. 2]. 1.1.3 Cohomology of a category with coefficients in an Abelian group F or a small category C , let H n ( C ) denote the n -th homology group of the nerve N ∗ C . F or a simplic ial set X and a n Ab elian group A , c oho- molo gy gr oups H n ( X , A ) are defined as cohomology gr o ups of the complex H om ( C ∗ ( X ) , A ). L et C b e a small category . W e introduce its c ohomolo gy gr oups H n ( C , A ) with coefficien ts in A as H n ( N ∗ C , A ). It follows from [1, Chapter I I I, Theorem 4.1] that there is the follo wing exact se qunce (Univ ersal Co efficien t Theorem) 0 → E xt ( H n − 1 ( C ) , A ) → H n ( C , A ) → H om ( H n ( C ) , A ) → 0 1.2 Cohomology of categories with co efficien ts in dia- grams Recall the definition and prop erties of righ t derive d functors lim ← − n C : Ab C → Ab o f the limit functor. 1.2.1 Definition of c ohomology of categories with co efficien ts in diagrams Let C b e a small category . F or ev ery family { A i } i ∈ I of Ab elian g roups w e consider the direct pro duct Q i ∈ I A i as the Ab elian group of maps ϕ : I − → S i ∈ I A i suc h those ϕ ( i ) ∈ A i for a ll i ∈ I . F or an y functor F : C → Ab, consider the sequnce of Ab elian groups C 0 ( C , F ) = Y c 0 ∈ Ob C F ( c 0 ) , . . . , C n ( C , F ) = Y c 0 α 1 → ··· α n → c n F ( c n ) , . . . 4 and homomorphisms δ n : C n ( C , F ) → C n +1 ( C , F ) defined by ( δ n ϕ )( c 0 α 1 → · · · α n +1 → c n +1 ) = n X i =0 ( − 1) i ϕ ( c 0 α 1 → · · · α i → ˆ c i α i +1 → · · · α n +1 → c n +1 )+ ( − 1) n +1 F ( c n α n +1 → c n +1 )( ϕ ( c 0 α 1 → · · · α n → c n )) . Let C n ( C , F ) = 0 for n < 0. The equalities δ n +1 δ n = 0 ho ld for all in teger n . The obtained co c hain complex will b e denoted by C ∗ ( C , F ). Ab elian groups H n ( C ∗ ( C , F )) = Ker δ n +1 / Im δ n are called c ohomolo gy gr o ups of the smal l c ate gory C with c o efficients in a diagr am F and denoted by lim ← − n C F . It follows from [6 , Appl. 2, Prop. 3.3] by the substitution A = Ab op that the f unctors lim ← − n C are n -th righ t satellites of lim ← − C : Ab C → Ab. Since the category Ab C has enough injectiv es, the functors lim ← − n C are isomorphic to righ t derived of the limit functor. 1.2.2 Cohomology of categories without ret ractions A mor phism α : a → b C is a r etr action if there exists a morphism β : b → a suc h that αβ = 1 b . Prop osition 1.1. [7, Prop. 2.2] If a sma l l c ate gory C do es not c ontain non- identity r etr actions, then for any diagr am F : C → Ab , the gr oups lim ← − n C F ar e isomorphic to the h omolo gy gr oups of the sub c omplex C ∗ + ( C , F ) ⊆ C ∗ ( C , F ) c omp ose d of the pr o ducts C n + ( C , F ) = Y c 0 6 = →··· 6 = → c n F ( c n ) , n > 0 , wher e indic es run the se quenc es c 0 α 1 − → c 1 α 2 − → · · · α n − → c n such those α i 6 = id c i for al l 1 6 i 6 n . Corollary 1.2. If a smal l c ate gory C do es no t c ontain nonidentity r etr ac- tions an d the length m of ev e ry se quenc e o f nonidentity morphisms c 0 α 1 − → · · · α m − → c m is not gr e ater than n , then lim ← − k C = 0 for k > n . Example 1.1 . L et Θ b e the c ate gory with Ob Θ = { a, b } and Mor Θ = { 1 a , 1 b , a α 1 → b, a α 2 → b } . It fol lows fr om Pr op osition 1.1 that for any di a gr am F : Θ → Ab an d n > 1 , the gr oups lim ← − n Θ F e qual 0 . 5 F or an y Ab elian gro up A , lim ← − n C ∆ A ∼ = H n ( C , A ). Lemma 1.3. L e t Θ n b e the n -th p ower of the c ate gory Θ for n > 1 . The functors lim ← − k Θ n e qual 0 for al l k > n . F or any Ab e l i a n gr oup A , ther e is an isomorphism lim ← − n Θ n ∆ A ∼ = A . Pr o of. T he first assertion follo ws from Corollary 1.2. Since the geometric realization of the nerv e o f Θ n is the n -dimensional torus, H k (Θ n ) ∼ = Z ( n k ) for 0 6 k 6 n . Here  n k  is the binomial co efficien ts. Univ ersal Co efficien t Theorem for the cohomolo g y groups of the nerve of Θ n giv es H n (Θ n , A ) ∼ = A . 1.2.3 Strongly c oinitial functors A small category C is acyclic if H n ( C ) = 0 for all n > 0 and H 0 ( C ) = Z . Let S : C → D b e a functor from a small category to a n a rbitrary category . F or an y d ∈ Ob D , a fibr e (or c omma-c ate gory ) S/d is the category whic h ob jects are giv en by pairs ( c, α ) where c ∈ O b ( C ) and α ∈ D ( S ( c ) , d ). Morphisms ( c 1 , α 1 ) → ( c 2 , α 2 ) in S/ d are triples ( f , α 1 , α 2 ) with f ∈ C ( c 1 , c 2 ) satisfying α 2 ◦ S ( f ) = α 1 . If S is a f ull em b ed ding C ⊆ D , then S/d is denoted b y C /d . Definition 1.2 . A functor S : C → D b etwe en smal l c ate gories is c al le d strongly coinitia l if S/d is acycli c for e ach d ∈ D . Lemma 1.4 (Ob erst) . L et C and D b e smal l c ate gories. If S : C → D b e a str o n gly c oinitial functor, then the c anon i c al h omomorphisms lim ← − n D F → lim ← − n C F S ar e isomorphisms f o r al l n > 0 . Pr o of. I t follo ws from the opp osite assertion [8, 2.3 ] for the functors S op : C op → D op and F op : D op → Ab op . 1.3 Cohomological dimension of a small category Let N b e the set o f nonnegative in teger num b ers. W e will b e consider it as the subset of {− 1 } ∪ N ∪ { ∞} ordered b y − 1 < 0 < 1 < 2 < · · · < ∞ . Definition 1.3 . Cohomolo gic al dim ension cd C of a smal l c ate gory C is the sup in {− 1 } ∪ N ∪ { ∞} of the set n ∈ N for which the functors lim ← − n C : Ab C → Ab ar e not e qual 0 . 6 It follo ws from Lemma 1.3 that cd Θ n = n . L emma 1.4 giv es the follo wing Corollary 1.5. If ther e exists a str ongly c oinitial functor S : C → D b etwe en smal l c ate gories, then cd C > cd D . A sub category D ⊆ C is said to b e close d if D is a full sub category con taining the domain fo r any morphism whose co domain is in D . Corollary 1.6. L et D j ⊆ D b e a fa m ily of close d sub c ate gories for al l j ∈ J . If the inclusion S j ∈ J D j ⊆ D is str o n gly c oinitial, then cd D = sup j ∈ J { cd D j } . Pr o of. F or c ∈ Ob C , let C c ⊆ C b e denote a full sub category whic h consists of c ′ ∈ O b C hav ing morphisms c ′ → c . It follo ws from [9, Corollary 7] t hat the equalit y cd C = sup c ∈ Ob C cd C c holds. Consequen tly sup j ∈ J { cd D j } = cd S j ∈ J D j 6 cd D . Since the inclusion S j ∈ J D j ⊆ D is strongly coinitial, the equality follows from Corollary 1.5. 2 Dimension o f a free partially comm ut ati v e monoids W e will prov e the main results. W e compute the Baues-Wirsc hing dimension of a free partially comm utativ e monoids and show a formula for the global dimension of the category of ob jects with actions of a free partially com- m utativ e monoid. W e prov e that for any graded R [ M ( E , I )]-mo dule , there exists a free resolution. 2.1 Cohomological d imension of the factorization c at- egory W e consider the category of fa ctorization of a small category , although we applicate it for the case o f the small category is a monoid. 2.1.1 The category of factorizations Let C b e a small category . Ob jects of the c ate gory of factorizations F C [10] are all morphisms of C . F or an y α, β ∈ Ob F C = Mor C , the set of morphisms α → β consists fro m all pairs ( f , g ) of morphisms in C satisfying 7 g ◦ α ◦ f = β . The comp osition of α ( f 1 ,g 1 ) − → β and β ( f 2 ,g 2 ) − → γ is defined b y α ( f 1 ◦ f 2 ,g 2 ◦ g 1 ) − → γ . The iden tit y of an ob ject a α → b of F C equals α (1 a , 1 b ) − → α . 2.1.2 Baues-Wirsc hing dimension A na tur al system of Ab elian gr oups on C is an y functor F : F C → Ab. Baues and Wirsc hing intro duce cohomology gro ups H n ( C , F ) of C with co efficien ts in a natural system F and ha v e prov ed that these groups are isomor phic to lim ← − n F C F . The Baues-Wirsching dimensi on Dim C is the cohomolo gical dimension of F C . Example 2.1 . L e t N = { 1 , a, a 2 , . . . } b e the fr e e monoid gener ate d by one element. I t e asy to se e that the inclusion Θ a ⊆ F N of the ful l sub c ate gory with the obje cts Ob Θ a = { 1 , a } is str ong l y c oinitial. The sub c ate gory T a is close d in F N . I t is iso morphic to Θ fr o m Example 1.1. Conse quently Dim N = 1 . Prop osition 2.1. F or a n y inte ge r n > 1 , Dim N n = n . Pr o of. C onsider the f ull sub category Θ n a ⊆ F N n with ob jects ( a ε 1 , . . . , a ε n ) where ε i ∈ { 0 , 1 } for all 1 6 i 6 n . It not hard to see that it is isomorphic to Θ n and the fibre of the inclusion o ver ( a k 1 , . . . , a k n ) ∈ N n is isomorphic to the pro duct Θ a /a k 1 ×· · · Θ a /a k n . Since H i (Θ a /a k ) = 0 for i > 0 and H 0 (Θ a /a k ) ∼ = Z , it follows t ha t the catego ry F N n con tains the srongly coinitial sub category Θ n a , whic h is isomorphic to Θ n . It is clear that Θ n a is closed in F N n . Hence, Dim N n = cd Θ n = n . 2.2 The d imension of a free partially comm u tativ e monoid This subsection is dev oted to computing the Baues-Wirsc hing dimension o f free par tially commutativ e monoids. 2.2.1 The indep endence graph Let E be a set and I ⊆ E × E an irreflexiv e symmetric binary relation o n E . Monoid given by a g enerating set E and relations ab = ba for all ( a, b ) ∈ I is called fr e e p artial ly c ommutative and denoted b y M ( E , I ). The pair ( E , I ) ma y b e considered as a simple indep end e nc e gr aph of M ( E , I ) with the set of v ertices E and edges { a, b } for a ll pairs ( a, b ) ∈ I . 8 b a = = = = = = = = c d e Figure 2: The indep endence gr a ph It is show n in Fig ure 2 the indep endence graph of the monoid giv en by the generators E = { a, b, c, d, e } and relations ab = ba , bc = cb , cd = dc , ad = d a , ae = ea , de = ed . The clique numb er ω ( E , I ) of a simple g raph with veric es E and edges I is the sup o f cardinalities of its finite complete subgraphs. If ( E , I ) contains complete graphs K n for a ll n ∈ N , then ω ( E , I ) = ∞ . F or example, the clique num b er of the graph in Figure 2 is equal to 3. 2.2.2 Computing the dimensions of free partially comm utativ e monoids Let V b e the set of maximal cliques o f the indep endence graph of M ( E , I ). (These cliques may be infinite.) F o r example, the set V for the gr a ph in Figure 2 , consists o f the sets { a, b } , { b, c } , { c, d } , { a, d, e } . Let E v b e the set of v erices b elonging to a clique v and M ( E v ) the submonoid of M ( E , I ) generated b y E v . It is clear that M ( E v ) are commu tativ e monoids. The category of fa ctorization F M ( E v ) is a closed sub category o f F M ( E , I ). Lemma 2.2. [1 1] The inclusion S v ∈ V F M ( E v ) ⊆ F M ( E , I ) is str ongly c oini- tial. Theorem 2.3. D im M ( E , I ) = ω ( E , I ) . Pr o of. F or ev ery subset of m utually comm uting elemen ts { e 1 , . . . , e n } ⊆ E , the full sub category F M ( { e 1 , . . . , e n } ) is closed in F M ( E , I ). Hence, the equalit y is true in the case of ω ( E , I ) = ∞ . The sub categories F M ( E v ) are closed in F M ( E , I ). It follows from Lemma 2.2 that we can use Corollary 1.6. W e get cd F M ( E , I ) = sup v ∈ V { cd F M ( E v ) } . If E v are finite, then w e get the assertion Dim M ( E , I ) = ω ( E , I ) b y Prop osition 2.1. 9 2.3 The generalized syzygy theorem In this subsection, we pro v e t he main theorem. 2.3.1 The global dimension of the category of M ( E , I ) -ob jects By [1 2, Corollary 13.4’] for an y small catego ry C and Ab elian category with exact copro ducts, there exists the inequalit y gl dim A C 6 dim C + gl dim A . Here dim is the Ho chschild-Mitchel l dimen sion . W e will sho w that if C = M ( E , I ), t hen the equalit y holds. Theorem 2.4. L et A b e an Ab elian c ate gory with ex act c opr o ducts. Then gl dim A M ( E , I ) = ω ( E , I ) + gl dim A . Pr o of. F or any finite subset E ′ ⊆ E , the s ubmonoid generated b y E ′ is cancellativ e [4]. It follows that M ( E , I ) is cancellative and D im M ( E , I ) is equal to Ho c hsc hild-Mitche ll dimension dim M ( E , I ) [7, Theorem 3.1 ]. Con- sequen tly gl dim A M ( E , I ) 6 ω ( E , I ) + gl dim A . F or eac h fin ite subset of m utually comm uting elemen ts S ⊆ E there exists a retraction M ( E , I ) → M ( S ). It follows by [3 , Corolla r y 1.4] that gl dim A M ( E , I ) > gl dim A M ( S ) = | S | + gl dim A . Since dim M ( E , I ) is equal to sup of cardinalities | S | of finite subsets S ⊆ E of m utually comm uting elemen ts, w e get gl dim A M ( E , I ) > ω ( E , I ) + gl dim A . 2.3.2 Graded syzygies Let R be a ring with 1. Th e monoid ring has the natural g raduation R [ M ( E , I )] = L n ∈ Z R [ M ( E , I )] n b y R -mo dules R [ M ( E , I )] n = { r µ : r ∈ R, µ ∈ M ( E , I ) , | µ | = n } . In particular R [ M ( E , I )] 0 = R . Let R [ M ( E , I )] n = 0 for all n < 0 . The ring R with 1 is called pr oje ctive fr e e if any pro jective R -mo dule is free. By [1 3, § 8.7, Corollary 2] and Theorem 2.4 w e get: Corollary 2.5. L et M ( E , I ) b e a fr e e p artial ly c ommutative monoid and R pr oje ctive fr e e ring with gl dim R = n < ∞ . 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