Finite generation of the cohomology of some skew group algebras
aa r X i v : . [ m a t h . R T ] O c t FINITE GENERATION OF THE COHOMOLOGYOF SOME SKEW GROUP ALGEBRAS
VAN C. NGUYEN AND SARAH WITHERSPOON
Abstract.
We prove that some skew group algebras have Noetherian coho-mology rings, a property inherited from their component parts. The proof is anadaptation of Evens’ proof of finite generation of group cohomology. We applythe result to a series of examples of finite dimensional Hopf algebras in positivecharacteristic. Introduction
The cohomology ring of a Hopf algebra encodes potentially useful informationabout its structure and representations. It is always graded commutative (see,for example, [25]). For many classes of finite dimensional Hopf algebras, it is alsoknown to be finitely generated: for example, cocommutative Hopf algebras (Fried-lander and Suslin [13]), small quantum groups (Ginzburg and Kumar [14]), andsmall quantum function algebras (Gordon [16]). Etingof and Ostrik [11] conjec-tured that it is always finitely generated, as a special case of a conjecture aboutfinite tensor categories. Snashall and Solberg [23] made an analogous conjecturefor Hochschild cohomology, of finite dimensional algebras, that was seen to befalse when Xu [27] constructed a counterexample. In contrast, there is neither acounterexample nor a proof of the Hopf algebra conjecture. Each finite generationresult so far has used, in crucial ways, known structure of a particular class ofHopf algebras. Further progress will require new ideas.In this article, we present one technique for handling some types of algebrasinductively. Many (Hopf) algebras of interest are skew group algebras (that is,smash products with group algebras). Under some conditions on a skew groupalgebra, we show that its cohomology is Noetherian if the same is true of theunderlying algebra on which the group acts.
Date : October 21, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Cohomology, Hopf algebras, skew group algebras.This material is based upon work done while the first author was a Texas A&M graduatestudent. It was supported by the National Science Foundation under grant No. 0932078000while both authors were in residence at the Mathematical Sciences Research Institute (MSRI)in Berkeley, California, during the semester of Spring 2013. Both authors were also supportedby NSF grant DMS-1101399.
Specifically, if A is a finite dimensional augmented algebra over a field k , withan action of a finite group G by automorphisms, there is a spectral sequence re-lating the cohomology of the smash product A kG (definition in Section 2) asan augmented algebra to that of each of A and G . (It is essentially the Lyndon-Hochschild-Serre spectral sequence.) This allows us to use the framework of Evens’classic proof of finite generation of group cohomology [12] to prove that the coho-mology rings of some smash products are Noetherian (Theorem 3.1). In order todo this, we need a particularly nice set of permanent cycles in the cohomology of A . In the finite group case, these cycles exist due to an application of Evens’ normmap. In our setting, there may be no such norm map, and we instead hypothesizeexistence of these permanent cycles.We focus on a class of examples (in Section 5) found by Cibils, Lauve, andthe second author [9] that satisfy our hypotheses. We prove finite generation ofthe cohomology of these noncommutative, noncocommutative Hopf algebras inpositive characteristic. While our main theorem is tailored to suit these examples,we state and prove it in the abstract setting, in order to add one more tool tothe collection of techniques available for proving finite generation. Our restrictivehypotheses serve to highlight the difficulty in adapting methods designed for thefinite group setting, where serendipity reigns.We thank D. Benson and P. Symonds for very insightful conversations andsuggestions. We thank Ø. Solberg for computing the cohomology of some of theNichols algebras in Section 4; these computations led us to our general result onthis series of Nichols algebras and corresponding Hopf algebras.2. Definitions and notation
Throughout this article, let k be a field. All algebras will be associative algebrasover k , and all modules will be left modules, finite dimensional over k . Let ⊗ = ⊗ k .Let G be a finite group acting on a finite dimensional augmented k -algebra A byautomorphisms. Let A kG be the resulting smash product (or skew group algebra ),that is A ⊗ kG as a vector space, with multiplication ( a ⊗ g )( b ⊗ h ) = a ( g b ) ⊗ gh ,for all a, b ∈ A and g, h ∈ G . (For simplicity, we will drop tensor symbols in thisnotation from now on.) We assume the action of G preserves the augmentation of A , so that A kG is also augmented with augmentation map ε A kG : A kG → k defined by ε A kG ( ag ) = ε A ( a ), for all a ∈ A , g ∈ G .We use the symbol k also to denote the one-dimensional A -module (respectively, A kG -module) on which A (respectively, A kG ) acts via its augmentation. LetH ∗ ( A, k ) := Ext ∗ A ( k, k ) and H ∗ ( A kG, k ) := Ext ∗ A kG ( k, k ) . Both are algebras under Yoneda composition. The embedding of A into A kG asa subalgebra induces a restriction map res A kG,A : H ∗ ( A kG, k ) → H ∗ ( A, k ) INITE GENERATION OF COHOMOLOGY 3 on cohomology. There is an action of G on H ∗ ( A, k ) that may be defined forexample via the diagonal action of G on the components of the bar resolution for A . There is a similar action of G on H ∗ ( A kG, k ) that is trivial since it comesfrom inner automorphisms on A kG .3. Finite generation of cohomology
In this section, we prove our main theorem that under certain hypotheses, thecohomology ring H ∗ ( A kG, k ) of A kG is Noetherian: Theorem 3.1.
Let G be a finite group acting on a finite dimensional augmentedalgebra A , preserving the augmentation map. Assume that Im(res A kG,A ) containsa polynomial subalgebra over which H ∗ ( A, k ) is Noetherian and free as a mod-ule, with a free basis whose k -linear span is a kG -submodule of H ∗ ( A, k ) . Then H ∗ ( A kG, k ) is Noetherian. Remarks 3.2. (a) The hypothesis that Im(res A kG,A ) contains a polynomial subalgebra overwhich H ∗ ( A, k ) is Noetherian, together with the left module version of [15, Corol-lary 1.5], implies that H ∗ ( A, k ) is (left) Noetherian.(b) We did not specify the characteristic of the base field k in the theorem. If thecharacteristic of k does not divide the order of G , then kG is semisimple and its co-homology is trivial except in the degree 0. In this case, H ∗ ( A kG, k ) ∼ = H ∗ ( A, k ) G ,the invariant ring under the action of G . Here, one can use invariant ring theoryin the noncommutative setting to show that the conclusion of the theorem holds.(See, for example, [19, Corollary 4.3.5].) For the proof of Theorem 3.1, we assumethe characteristic of k divides the order of G . Proof.
We use the Lyndon-Hochschild-Serre spectral sequence (see, for example,[5, Chapter VI] in a very general setting): E p,q = E p,q ( k ) = H p ( G, H q ( A, k )) = ⇒ H p + q ( A kG, k ) . Let E r ( k ) denote the resulting r th page, and note that for each q , H q ( A, k ) is afinite dimensional k -vector space.Note that E , ∗∞ is a submodule of E , ∗ , since no d r : E p,qr → E p + r,q − r +1 r ends onthe vertical edge. It follows that the restriction map H ∗ ( A kG, k ) → E , ∗ ( k ) ispart of the following commuting diagram:H ∗ ( A kG, k ) (cid:15) (cid:15) res A kG,A / / H ( G, H ∗ ( A, k )) = H ∗ ( A, k ) G E , ∗∞ ( k ) ֒ → / / E , ∗ ( k )We can identify E , ∗∞ with the image of the restriction map in E , ∗ . VAN C. NGUYEN AND SARAH WITHERSPOON
Let T = k [ χ , . . . , χ m ] denote the polynomial subalgebra of Im(res A kG,A ) hy-pothesized in the statement of the theorem. The action of G on H ∗ ( A, k ) restrictsto the trivial action on T since it is a subalgebra of Im(res A kG,A ). Therefore, bythe Universal Coefficients Theorem, H ∗ ( G, T ) ∼ = H ∗ ( G, k ) ⊗ T , an isomorphism ofgraded algebras.Let S := H ∗ ( G, k ) = E ∗ , ( k ). Let R be the subring of E ( k ) generated by S and T . By the above observations, R ∼ = S [ χ , . . . , χ m ], a polynomial ring over S in m indeterminates (that we also denote by χ , . . . , χ m for convenience). Since d vanishes on the horizontal edge, R ⊆ Ker( d ). So R projects onto a subringof E ( k ) = H( E ( k ) , d ). Similarly, R projects onto a subring of E r ( k ) for every r > ∞ . Therefore, we may consider E r ( k ) to be a module over R , forevery r > ∞ . Claim 1: E ( k ) is a Noetherian module over R . Proof of Claim 1.
By hypothesis, there are (homogeneous) elements ρ , . . . , ρ t ∈ H ∗ ( A, k ) that form a free basis of H ∗ ( A, k ) as a T -module, and for which V := Span k { ρ , . . . , ρ t } is a kG -submodule of H ∗ ( A, k ). Let L := H ∗ ( G, V ) . Note that L contains a copy of S = H ∗ ( G, k ) as V must include an element indegree 0, that is in H ( A, k ) ∼ = k , which has trivial G -action. By hypothesis,H ∗ ( A, k ) = k [ χ , . . . , χ m ] · V , and so E ( k ) = H ∗ ( G, k [ χ , . . . , χ m ] · V ) . Further, k [ χ , . . . , χ m ] has trivial G -action and the module H ∗ ( A, k ) for this poly-nomial ring is free with free basis ρ , . . . , ρ t . It follows that, as a kG -module, k [ χ , . . . , χ m ] · V ∼ = M i ,...,i m ≥ χ i · · · χ i m m · V ∼ = M i ,...,i m ≥ V, a direct sum of copies of the same kG -module, V . Therefore by the UniversalCoefficients Theorem, E ( k ) is the image ofH ( G, k [ χ , . . . , χ m ]) ⊗ H ∗ ( G, V ) ∼ = k [ χ , . . . , χ m ] ⊗ L, under cup product. We thus identify E ∗ , ∗ ( k ) with S [ χ , . . . , χ m ] ⊗ S L .Since G is a finite group and V is a finite dimensional vector space over k , L =H ∗ ( G, V ) is Noetherian over S = H ∗ ( G, k ) [12]. By the Hilbert Basis Theorem forgraded commutative rings (see, for example, [15, Theorem 2.6]), S [ χ , . . . , χ m ] ⊗ S L is Noetherian over R = S [ χ , . . . , χ m ]. Therefore, E ∗ , ∗ ( k ) is Noetherian over R .We have proven Claim 1. Claim 2: The spectral sequence stops, i.e., E r = E ∞ for some r < ∞ . INITE GENERATION OF COHOMOLOGY 5
Proof of Claim 2.
Let Z i be the space of i -cocycles and B i be the space of i -coboundaries in E i = E i ( k ). Recall that E = Z and E = Z /B . Considerthe “pull back” B r in E of d r ( E r ) as follows:Each element of E on which d vanishes determines an element of E . Suppose d vanishes on that element, so that it in turn determines an element of E . Con-tinue placing such restrictions until we determine an element of E r , and supposethat element is in the image of d r . We define: B r := { τ ∈ E : τ ∈ Ker( d i ), for 2 ≤ i ≤ r −
1, and τ ∈ Im( d r ) } . Note that B r is an R -submodule of E since d j is a derivation for all j , 2 ≤ j ≤ r ,and the image in each E j of R consists of universal cycles. Moreover, B r ⊆ B r +1 so we obtain an ascending chain of R -submodules of E :0 = B ⊆ B ⊆ · · · Since E is Noetherian over R by Claim 1, this chain must stabilize by the as-cending chain condition. Thus there exists some r finite such that B r = B r +1 = B r +2 = . . . , and so d r = 0 for all r > r . This implies E r = E ∞ for all r > r .We have proven Claim 2.We can put this together to finish the proof of the theorem: Each Z r , B r is asubmodule of E over R = S [ χ , . . . , χ m ]. Thus, each E r , which is a submoduleof a quotient module of E r − , is Noetherian over R by Claim 1 and induction on r . By Claim 2, E ∞ is Noetherian over R , and so by [15, Corollary 1.5] it is aNoetherian ring.Now, H ∗ ( A kG, k ) has a filtration whose filtered quotients are E p,q ∞ ( k ) ∼ = F p H p + q ( A kG, k ) F p +1 H p + q ( A kG, k ) . Suppose that H ∗ ( A kG, k ) is not Noetherian and let T ⊆ T ⊆ · · · ⊆ H ∗ ( A kG, k )be an infinite ascending chain of ideals of H ∗ ( A kG, k ). Let F p T i := T i ∩ F p H ∗ ( A kG, k )and U i := M p ≥ F p T i /F p +1 T i ⊆ E ∞ ( k ) . If x ∈ T i +1 \ T i , then for some p , x ∈ F p T i +1 but x / ∈ F p T i and x / ∈ F p +1 T i +1 , so x + F p +1 T i +1 is not in the image of the inclusion F p T i /F p +1 T i ֒ → F p T i +1 /F p +1 T i +1 that is, x ∈ U i +1 \ U i . So U i +1 properly contains U i , for all i . Therefore, we havean infinite ascending chain of ideals of E ∞ ( k ): U $ U $ · · · VAN C. NGUYEN AND SARAH WITHERSPOON
This contradicts the result that E ∞ ( k ) is Noetherian. Hence, H ∗ ( A kG, k ) isNoetherian. (cid:3) Remark 3.3.
Theorem 3.1 parallels the main step in Evens’ proof of finite gener-ation of group cohomology: Let H be a finite p -group (where k has characteristic p ), A = kZ is the group algebra of a central subgroup Z of H of order p , and G = H/Z . (In case Z is complemented in H , we obtain kH ∼ = A kG , whereasmore generally, kH is a crossed product of A with G .) In this case, Evens’ normmap is applied to show that Im(res kH,kZ ) contains a polynomial subalgebra k [ ζ ](in one indeterminate). One observes that H ∗ ( kZ, k ) is a free module over k [ ζ ],and that the k -linear span of any free basis is a kG -submodule. This special caseis somewhat simpler than our more general context as it uses a polynomial ringin one indeterminate.We are particularly interested in those actions of finite groups G on algebras A for which A kG is a Hopf algebra. We turn to a class of such examples in theremainder of the paper.4. Examples: Nichols algebras in positive characteristic
In this section, we first recall the Nichols algebras A from [9, Corollary 3.14]and the corresponding Hopf algebras A kG from the same paper. We will provethat these Hopf algebras have finitely generated cohomology. This will follow fromTheorem 3.1 and explicit calculation using Anick’s resolution [4]. In this sectionwe explain these calculations for A , and in the next we complete the proof of finitegeneration of cohomology of A kG . The results of this section were anticipatedby Solberg [24] as a consequence of computer calculations (for small p ) that gavethe graded vector space structure and generators of cohomology.In the remainder of the paper, k will be a field of characteristic p >
2. (Thecase p = 2 is included in [9], but is different, and we will not consider that casehere.) Let A be the augmented k -algebra generated by a , b , with relations a p = 0 , b p = 0 , ba = ab + 12 a , and augmentation ε : A → k given by ε ( a ) = ε ( b ) = 0. Let G be a cyclic group oforder p with generator g , acting on A on the right by g ( a ) = a, g ( b ) = a + b. Then A kG is a Hopf algebra with comultiplication given by∆( g ) = g ⊗ g, ∆( a ) = a ⊗ g ⊗ a, ∆( b ) = b ⊗ g ⊗ b. To apply results in [20], we changed from the left G -module structure: g ( a ) = a, g ( b ) = b − a to this right G -module structure. INITE GENERATION OF COHOMOLOGY 7
It is useful to consider A as a quotient of a larger algebra. Let(4.1) B := k h a, b i / ( ba − ab − a ) , so that A ∼ = B/ ( a p , b p ). We will show that B is a PBW algebra in the senseof [8] or [22, Section 2], although we will not need this fact for our cohomologycalculations.Choose the lexicographic order on N for which (0 , < (1 , a ) = (0 , , deg( b ) = (1 , ba − ab − a is a Gr¨obner basis for theideal of the free algebra k h a, b i that it generates. It follows that { a i b j | i, j ≥ } is a vector space basis of B . The relation ab = ba − a satisfies the requiredcondition in the definition of a PBW algebra since deg( a ) < deg( ab ), so B isa PBW algebra. Moreover, B is a Koszul algebra by a theorem of Priddy [21,Theorem 5.3].Applying [9, (3.9)], one finds that the elements a p , b p are in the center of B . Wemay thus apply a theorem of Shroff, [22, Theorem 4.3], to the Nichols algebra A to conclude that the cohomology ring H ∗ ( A, k ) of A is Noetherian.We will need some details about this cohomology of A for the next section. Forthis, we will construct Anick’s resolution [4] for A , and show that it is minimal.We use the combinatorial description of the resolution given by Cojocaru andUfnarovski [10], however we index differently, and use left modules instead ofright. This is a free resolution of the trivial A -module k , of the form · · · −→ A ⊗ kC −→ A ⊗ kC −→ A ⊗ kC −→ k → , for (finite) sets C n , where kC n denotes the vector space with basis C n . Let C := { } and C := { a, b } . Then C := { a p , b p , ba } is the set of “tips” or“obstructions.” To define C n in general, consider the graph1 } } ④④④④④④④④④ ! ! ❈❈❈❈❈❈❈❈❈ a (cid:10) (cid:10) b o o (cid:10) (cid:10) a p − I I b p − h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ I I The elements of C n correspond to paths of length n that start at 1. We label suchpaths with the product of all elements through which the path passes (includingthe endpoint). In this way we obtain C = { a p +1 , b p +1 , b p a, ba p } ,C = { a p , b p , b p +1 a, b p a p , ba p +1 } , VAN C. NGUYEN AND SARAH WITHERSPOON and in general C m − = { b kp a ( m − − k ) p +1 , b kp +1 a ( m − − k ) p | k = 0 , , . . . , m − } ,C m = { b mp , b kp a ( m − k ) p , b kp +1 a ( m − − k ) p +1 | k = 0 , , . . . , m − } . For qualitative understanding of the differentials, give each of the generators a, b of A the degree 1. We claim that the differentials preserve degree, where thegraded module structure of a tensor product A ⊗ kC i is given by deg( a ⊗ x ) =deg( a ) + deg( x ) if a, x are homogeneous. This claim results from the recursivedefinition of the differential d in each homological degree: By construction, d applied to elements of A ⊗ kC is multiplication, and to A ⊗ kC takes eachtip to the Gr¨obner basis element to which it corresponds, suitably expressed asan element of A ⊗ kC . The remaining differentials are defined iteratively, viasplitting maps in each homological degree that are also defined iteratively. Sincethe relations are homogeneous and differentials in low homological degrees preservedegrees of elements, the splitting maps and differentials in higher degrees may bechosen to have the same property.Now note that C m − consists of elements of degree ( m − p +1, and C m consistsof elements of degrees mp and ( m − p + 2. Therefore elements of C n and of C n − never have the same degree. As a consequence the differential takes elements of C n to elements of A + ⊗ C n − where A + denotes all elements of A of positive degree(and these are in the kernel of the augmentation map ε ). When applying thefunctor Hom A ( − , k ) then, the induced differentials all become 0. Therefore in thiscase, Anick’s resolution is minimal, and for each n , the dimension of H n ( A, k ) is n + 1.5. Examples: Pointed Hopf algebras in positive characteristic
We wish to apply Theorem 3.1 to the Hopf algebras A kG introduced in the pre-vious section. In order to do this, we next give some of the details from Shroff [22,Section 4] as they apply to these examples in particular. Recall the PBW algebra B defined in (4.1). Let ξ a , ξ b : B ⊗ B → k be the k -linear functions given by ξ a ( r ⊗ s ) = γ a , ξ b ( r ⊗ s ) = γ b , where γ a (respectively, γ b ) is the scalar coefficient of a p (respectively, b p ) inthe product rs in B . (These functions are denoted e ζ , e ζ in [22].) Extend-ing to left B -module homomorphisms in Hom B ( B ⊗ , k ) under the isomorphismHom B ( B ⊗ , k ) ∼ = Hom k ( B ⊗ , k ), the functions ξ a , ξ b are coboundaries on the barresolution of B , as shown in [22], and they factor through A ∼ = B/ ( a p , b p ). Theresulting functions (which we will also denote ξ a , ξ b by abuse of notation) are nolonger coboundaries. They represent nonzero elements in the cohomology of A ,corresponding to permanent cycles in the May spectral sequence for A as a filteredalgebra (see [18, Theorem 3] or [26, 5.4.1]). On page E of this spectral sequence, INITE GENERATION OF COHOMOLOGY 9 their counterparts generate a polynomial ring over which E is finitely generated(by the elements 1 , η a , η b , η a η b , where η a , η b have cohomological degree 1, functionsdual to a and b in Hom k (gr A, k ) ∼ = Hom gr A (gr A ⊗ gr A, k )). The cohomologyH ∗ ( A, k ) is finitely generated over its subalgebra generated by ξ a , ξ b , as a conse-quence of the proof of [22, Theorem 4.3]. We will see below that the subalgebragenerated by ξ a , ξ b is in fact a polynomial ring in ξ a , ξ b , which is Noetherian, soapplying the left module version of [15, Corollary 1.5], H ∗ ( A, k ) is itself (left)Noetherian.To verify the hypothesis of Theorem 3.1, we will want 2-cocycles representingelements in H ∗ ( A kG, k ): We use results in [20], where the notation is slightlydifferent, with x in place of a and y in place of b . There it is shown directly thatthere are 2-cocycles ξ a , ξ b in H ∗ ( A, k ) generating a polynomial subring k [ ξ a , ξ b ].Results in [20, Section 5.1] also imply that ξ a , ξ b are in Im(res A kG,A ); the neededelements in H ∗ ( A kG, k ) are constructed explicitly using a twisted tensor productresolution in [20, Section 3.3]. We next claim that H ∗ ( A, k ) is free with free basis { , η a , η b , η a η b } over the polynomial subalgebra k [ ξ a , ξ b ]. This will follow oncewe see that the set { ξ ia ξ jb η la η mb | i, j ≥ , l, m = 0 , } represents a basis of H ∗ ( A, k ), since ξ a , ξ b commute with each other. Note thatthe cohomology of S = gr A is well-known, and has a basis precisely of this form.Recall that Anick’s resolution for A is minimal, and a comparison shows thatin each degree, the dimensions of H ∗ ( A, k ) and of H ∗ ( S, k ) are the same. Thisforces the May spectral sequence [18] for A to collapse at E = H ∗ ( S, k ), and sogr H ∗ ( A, k ) ∼ = H ∗ ( S, k ), and H ∗ ( A, k ) has basis as claimed. Therefore H ∗ ( A, k ) isindeed free as a k [ ξ a , ξ b ]-module. Further, the k -linear span of { , η a , η b , η a η b } isa kG -submodule of H ∗ ( A, k ): We compute g η a = η a + η b , g η b = η b , g ( η a η b ) = η a η b . We have shown that the hypotheses of Theorem 3.1 are satisfied. Therefore,H ∗ ( A kG, k ) is Noetherian. Question 5.1.
Are there more examples of Nichols algebras in positive charac-teristic to which Theorem 3.1 applies?Nichols algebras and their bosonizations, which are Hopf algebras, have onlyjust begun to be explored in positive characteristic. There is a vast (and recent)literature on Nichols algebras in characteristic 0. See, for example, [1, 2, 3, 17]. Since B is a Koszul algebra, H ∗ ( B, k ) ∼ = B ! , the Koszul dual of B , which is generated by η a , η b (by abuse of notation) with relations dual to those of B , that is, η a = η a η b , η b = 0 , η b η a = − η a η b . These relations also hold in H ∗ ( A, k ), however we do not need this fact.
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Department of Mathematics, Hood College, Frederick, MD 21701
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