Finite generation of cohomology for Drinfeld doubles of finite group schemes
aa r X i v : . [ m a t h . QA ] M a y FINITE GENERATION OF COHOMOLOGY FOR DRINFELDDOUBLES OF FINITE GROUP SCHEMES
CRIS NEGRON
Abstract.
We prove that the Drinfeld double of an arbitrary finite groupscheme has finitely generated cohomology. That is to say, for G any finitegroup scheme, and D ( G ) the Drinfeld double of the group ring kG , we showthat the self-extension algebra of the trivial representation for D ( G ) is a finitelygenerated algebra, and that for each D ( G )-representation V the extensionsfrom the trivial representation to V form a finitely generated module over theaforementioned algebra. As a corollary, we find that all categories rep( G ) ∗ M dual to rep( G ) are of also of finite type (i.e. have finitely generated cohomol-ogy), and we provide a uniform bound on their Krull dimensions. This papercompletes earlier work of E. M. Friedlander and the author. Introduction
Fix k an arbitrary field of finite characteristic. Let us recall some terminol-ogy [21]: A finite k -linear tensor category C is said to be of finite type (over k ) ifthe self-extensions of the unit object Ext ∗ C ( , ) are a finitely generated k -algebra,and for any object V in C the extensions Ext ∗ C ( , V ) are a finitely generated mod-ule over this algebra. In this case, the Krull dimension
Kdim C of C is the Krulldimension of the extension algebra of the unit. One is free to think of C here asthe representation category rep( A ) of a finite-dimensional Hopf algebra A , withmonoidal structure induced by the comultiplication, and unit = k provided bythe trivial representation.It has been conjectured [10, Conjecture 2.18] [14] that any finite tensor category,over an arbitrary base field, is of finite type. Here we consider the category ofrepresentations for the Drinfeld double D ( G ) of a finite group scheme G , which isidentified with the so-called Drinfeld center Z (rep( G )) of the category of finite G -representations [18, 9]. The Drinfeld double D ( G ) is the smash product O ( G ) ⋊ kG of the algebra of global functions on G with the group ring kG , under the adjointaction. So, one can think of Z (rep( G )), alternatively, as the category of coherent G -equivariant sheaves on G under the adjoint action Z (rep( G )) = rep( D ( G )) = Coh( G ) G . In the present work we prove the following.
Theorem (7.1) . For any finite group scheme G , the Drinfeld center Z (rep( G )) isof finite type and of Krull dimension Kdim Z (rep( G )) ≤ Kdim rep( G ) + embed . dim( G ) . Here embed . dim( G ) denotes the minimal dimension of a smooth (affine) alge-braic group in which G embeds as a closed subgroup. The above theorem wasproved for G = G ( r ) a Frobenius kernel in a smooth algebraic groups G in work of E. M. Friedlander and the author [11]. Thus Theorem 7.1 completes, in a sense,the project of [11].One can apply Theorem 7.1, and results of J. Plavnik and the author [21], to ob-tain an additional finite generation result for all dual tensor categories rep( G ) ∗ M (:=End rep( G ) ( M )), calculated relative to an exact rep( G )-module category M [10,Section 3.3]. Corollary 1.1.
Let G be a finite group scheme, and M be an arbitrary exact rep( G ) -module category. Then the dual category rep( G ) ∗ M is of finite type and ofuniformly bounded Krull dimension Kdim rep( G ) ∗ M ≤ Kdim rep( G ) + embed . dim( G ) . Proof.
Immediate from Theorem 7.1 and [21, Corollary 4.1]. (cid:3)
We view Theorem 7.1, and Corollary 1.1, as occurring in a continuum of now veryrich studies of cohomology for finite group schemes, e.g. [12, 14, 23, 13, 26, 7, 3].
Remark 1.2.
Exact rep( G )-module categories have been classified by Gelaki [15],and correspond to pairs ( H, ψ ) of a subgroup H ⊂ G and certain 3-cocycle ψ whichintroduces an associativity constraint for the action of rep( G ) on rep( H ). Remark 1.3.
For an analysis of support theory for Drinfeld doubles of some solv-able height 1 group schemes, one can see [20, 19]. The problem of understandingsupport for general doubles D ( G ) is, at this point, completely open.1.1. Approach via equivariant deformation theory.
In [11], where the Frobe-nius kernel G ( r ) in a smooth algebraic group G is considered, we basically use thefact that ambient group G provides a smooth, equivariant, deformation of G ( r ) parametrized by the quotient G / G ( r ) ∼ = G ( r ) in order to gain a foothold in ouranalysis of cohomology. In particular, the adjoint action of G ( r ) on G descends to atrivial action on the twist G ( r ) , so that the Frobenius map G → G ( r ) can be viewedas smoothly varying family of G ( r ) -algebras which deforms the algebra of functions O ( G ( r ) ). Such a deformation situation provides “deformation classes” in degree 2, { deformation classes } = T G ( r ) ⊂ Ext G ( r ) ) G ( r ) ( , ) = Ext D ( G ( r ) ) ( , ) . One uses these deformation classes, in conjunction with work of Friedlander andSuslin [14], to find a finite set of generators for extensions.For a general finite group scheme G , we can try to pursue a similar deformationapproach, where we embed G into a smooth algebraic group H , and consider H asa deformation of G parametrized by the quotient H /G . However, a general finitegroup scheme may not admit any normal embedding into a smooth algebraic group.(This is the case for certain non-connected finite group schemes, and should alsobe the case for restricted enveloping algebras kG = u res ( g ) of Cartan type simpleLie algebras, for example). So, in general, one accepts that G acts nontrivially onthe parametrization space H /G , and that the fibers in the family H are permutedby the action of G here. Thus we do not obtain a smoothly varying family of G -algebras deforming O ( G ) in this manner.One can, however, consider a type of equivariant deformation theory where thegroup G is allowed to act nontrivially on the parametrization space, and attemptto obtain higher deformation classes in this instance { higher deformation classes } ⊂ Ext ≥ G ) G ( , ) = Ext ≥ D ( G ) ( , ) . We show in Sections 3 and 5 that this equivariant deformation picture can indeedbe formalized, and that–when considered in conjunction with work of Touz´e andVan der Kallen [26]–it can be used to obtain the desired finite generation resultsfor cohomology (see in particular Theorems 5.4 and 6.4).
Remark 1.4.
From a geometric perspective, one can interpret our main theoremas a finite generation result for the cohomology of non-tame stacky local completeintersections. (Formally speaking, we only deal with the maximal codimensioncase here, but the general situation is similar.) One can compare with works ofGulliksen [16], Eisenbud [8], and many others regarding the homological algebra ofcomplete intersections.1.2.
Acknowledgements.
Thanks to Ben Briggs, Christopher Drupieski, EricFriedlander, Julia Pevtsova, Antoine Touz´e, and Sarah Witherspoon for helpfulconversations. The proofs of Lemmas 2.3 and 2.4 are due to Ben Briggs and Rag-nar Buchweitz (with any errors in their reproduction due to myself). This materialis based upon work supported by the National Science Foundation under GrantNo. DMS-1440140, while the author was in residence at the Mathematical SciencesResearch Institute in Berkeley, California, during the Spring 2020 semester.2.
Differential generalities
Throughout k is a field of finite characteristic, which is not necessarily alge-braically closed. Schemes and algebras are k -schemes and k -algebras, and ⊗ = ⊗ k .All group schemes are affine group schemes which are of finite type over k , andthroughout G denotes an (affine) group scheme.2.1. Commutative algebras and modules. A finite type commutative algebraover a field k is a finitely generated k -algebra. A coherent module over a commu-tative Noetherian algebra is a finitely generated module. We adopt this language,at times, to distinguish clearly between these two notions of finite generation.2.2. G -equivariant dg algebras. Consider G an affine group scheme. We letrep( G ) denote the category of finite-dimensional G -representations, Rep( G ) denotethe category of integrable, i.e. locally finite, representations, and Ch(Rep( G )) de-note the category of cochain complexes over Rep( G ). Each of these categories isconsidered along with its standard monoidal structure.By a G -algebra we mean an algebra object in Rep( G ), and by a dg G -algebra wemean an algebra object in Ch(Rep( G )). For T any commutative G -algebra, by a G -equivariant dg T -algebra S we mean a T -algebra in Ch(Rep( G )). Note that, forsuch a dg algebra S , the associated sheaf S ∼ on Spec( T ) is an equivariant sheaf ofdg algebras, and vice versa. Note also that a dg G -algebra is the same thing as anequivariant dg algebra over T = k .2.3. DG modules and resolutions.
For S a dg G -algebra, we let S -dgmod G and D ( S ) G denote the category of G -equivariant dg modules over S and its correspond-ing derived category D ( S ) G = ( S -dgmod G )[quis − ]. (Of course, by an equivariantdg module we mean an S -module in the category of cochains over G .) If we spec-ify some commutative Noetherian graded G -algebra T , and equivariant T -algebrastructure on cohomology T → H ∗ ( S ), then we take D coh ( S ) G := (cid:26) The full subcategory in D ( S ) G consisting of dg modules M with finitely generated cohomology over T (cid:27) . CRIS NEGRON
When T = k we take D fin ( S ) G = D coh ( S ) G .A (non-equivariant) free dg S -module is an S -module of the form ⊕ j ∈ J Σ n j S ,where J is some indexing set. A semi-projective resolution of a (non-equivariant)dg S -module M is a quasi-isomorphism F → M from a dg module F equippedwith a filtration F = ∪ i ≥ F ( i ) by dg submodules such that each subquotient F ( i ) /F ( i −
1) is a summand of a free S -module. An equivariant semi-projectiveresolution of an equivariant dg module M is a G -linear quasi-isomorphism F → M from an equivariant dg module F which is non-equivariantly semi-projective. Theusual shenanigans, e.g. [6, Lemma 13.3], shows that equivariant semi-projectiveresolutions always exist.2.4. Homotopy isomorphisms.
Consider S and A dg G -algebras, over somegiven group scheme G . By an (equivariant) homotopy isomorphism f : S → A we mean a zig-zag of G -linear dg algebra quasi-isomorphism S ∼ ← S ∼ → S · · · ∼ ← S N − ∼ → A. (1)We note that we use the term homotopy informally here, as we do not proposeany particular model structure on the category of dg G -algebras (cf. [24, 25]).Throughout the text, when we speak of homotopy isomorphisms between dg G -algebras we always mean equivariant homotopy isomorphisms.A homotopy isomorphism f : S → A as in (1) specifies a triangulated equivalencebetween the corresponding derived categories of dg modules f ∗ : D ( S ) G ∼ → D ( A ) G , (2)via successive application of base change and restriction along the maps to/from the S i . To elaborate, an equivariant quasi-isomorphism f : S → S specifies mutuallyinverse equivalences S ⊗ L S − : D ( S ) G → D ( S ) G and res f : D ( S ) G → D ( S ).So for a homotopy isomorphism f : S → A , compositions of restriction and basechange produce the equivalence (2).Note that, on cohomology, such a homotopy isomorphism f : S → A induces anactual isomorphism of algebras H ∗ ( f ) : H ∗ ( S ) → H ∗ ( A ), and one can check thatfor a dg module M over S we have H ∗ ( f ∗ M ) ∼ = H ∗ ( A ) ⊗ H ∗ ( S ) H ∗ ( M ) ∼ = res H ∗ ( f ) − H ∗ ( M ) . So, in particular, if H ∗ ( S ) and H ∗ ( A ) are T -algebras, for some commutative Noe-therian T , and H ∗ ( f ) is T -linear, then the equivalence (2) restricts to an equivalence f ∗ : D coh ( S ) G ∼ → D coh ( A ) G between the corresponding equivariant, coherent, derived categories. Definition 2.1.
We say a dg G -algebra S is equivariantly formal if S is equivari-antly homotopy isomorphic to its cohomology H ∗ ( S ).2.5. Derived maps and derived endomorphisms.
Fix S a dg G -algebra, overa group scheme G . For such S , the dg Hom functor Hom S on S -dgmod G naturallytakes values in Ch(Rep( G )). Namely, for x in the group ring kG = O ( G ) ∗ , we acton functions f ∈ Hom S ( M, N ) according to the formula( x · f )( m ) := x f ( S ( x ) m ) . With these actions each Hom S ( M, N ) is a dg G -representation, and compositionHom S ( N, L ) ⊗ Hom S ( M, N ) → Hom S ( M, L ) is a map of dg G -representations. In particular, End S ( M ) is a dg G -algebra for anyequivariant dg module M over S . Remark 2.2.
One needs to use cocommutativity of kG here to see that x · f is infact S -linear for S -linear f .We derive the functor Hom S to Ch(Rep( G )) by takingRHom S ( M, N ) := Hom S ( M ′ , N ) , where M ′ → M is any equivariant semi-projective resolution of M . One can applytheir favorite arguments to see that RHom S ( M, N ) is well-defined as an object in D (Rep( G )), or refer to the following lemma. Lemma 2.3.
For any two equivariant resolutions M → M and M → M there isan equivariant semi-projective dg module F which admits two surjective, equivari-ant, quasi-isomorphisms F → M and F → M .Proof. By adding on acyclic semi-projective summands we may assume that thegiven maps f i : M i → M are surjective. For example, one can take a surjectiveresolution N → M , consider the mapping cone cone( id N ), then replaces the M i with (Σ − cone( id N )) ⊕ M i . So, let us assume that the f i here are surjective.We consider now the fiber product F of the maps f and f to M . Note thatthe structure maps F → M i are surjective, since the f i are surjective. We havethe exact sequence 0 → F → M ⊕ M f − f ] T → M → → H ∗ ( F ) → H ∗ ( M ) ⊕ H ∗ ( M ) → H ∗ ( M ) → , with the map from H ∗ ( M ) ⊕ H ∗ ( M ) the sum of isomorphisms ± H ∗ ( f i ). It followsthat the composites H ∗ ( F ) → H ∗ ( M ) ⊕ H ∗ ( M ) → H ∗ ( M i ) are both isomor-phisms, and hence that the maps F → M and F → M are quasi-isomorphisms.One considers F → F any surjective, equivariant, semi-projective resolution toobtain the claimed result. (cid:3) For M in D ( S ) G we take REnd S ( M ) = End S ( M ′ ), for M ′ → M any equivariantsemi-projective resolution. The following result should be known to experts. Theproof we offer is due to Benjamin Briggs and Ragnar Buchweitz. I thank Briggs forcommunicating the proof to me, and allowing me to repeat it here. Lemma 2.4.
REnd S ( M ) is well-defined, as a dg G -algebra, up to homotopy iso-morphism. Furthermore, if M and N are isomorphic in D ( S ) G , then REnd S ( M ) and REnd S ( N ) are homotopy isomorphic as well. Given an explicit isomorphism ξ : M → N in D ( S ) G , the homotopy isomor-phism RHom S ( M ) → RHom S ( N ) can in particular be chosen to lift the canonicalisomorphism Ad ξ : Ext ∗ S ( M, M ) → Ext ∗ S ( N, N ) on cohomology.
Proof.
Consider two equivariant semi-projective resolutions M → M and M → M . By Lemma 2.3 we may assume that the map M → M lifts to a surjective, CRIS NEGRON equivariant, quasi-isomorphism f : M → M . In this case we have the two quasi-isomorphisms f ∗ and f ∗ of Hom complexes, and consider the fiber product B s s ❤ ❤ ❤ ❤ ❤ ❤ + + ❲❲❲❲❲❲ End S ( M ) f ∗ * * ❱❱❱❱❱❱❱ End S ( M ) f ∗ t t ❤❤❤❤❤❤❤ Hom S ( M , M ) (3)As f ∗ and f ∗ are maps of dg G -representations, B is a dg G -representation. Further-more, one checks directly that B is a dg algebra, or more precisely a dg subalgebrain the product End( M ) × End( M ). So the top portion of (3) is a diagram of mapsof dg G -algebras.As M is projective, as a non-dg module, the map f ∗ is a surjective quasi-isomorphism. One can therefore argue as in the proof of Lemma 2.3 to see that thestructure maps from B to the End S ( M i ) are quasi-isomorphisms. So we have theexplicit homotopy isomorphismEnd S ( M ) ∼ ← B ∼ → End S ( M ) . Now, if M is isomorphic to N in D ( S ) G , then there is a third equivariant dgmodule Ω with quasi-isomorphisms M ∼ ← Ω ∼ → N . Any resolution F ∼ → Ω thereforeprovides a simultaneous resolution of M and N , and we may take REnd S ( M ) =End S ( F ) = REnd S ( N ). (cid:3) Equivariant deformations and Koszul resolutions
In Sections 3 and 5 we develop the basic homological algebra associated withequivariant deformations. Our main aim here is to provide equivariant versions ofresults of Bezrukavnikov and Ginzburg [4], and Pevtsova and the author [20, § Equivariant deformations.
We recall that a deformation of an algebra R ,parametrized by an augmented commutative algebra Z , is a choice of flat Z -algebra Q along with an algebra map Q → R which reduces to an isomorphism k ⊗ Z Q ∼ = R .We call such a deformation Q → R an equivariant deformation if all of the algebraspresent are G -algebras, and all of the structure maps Z → Q , Z → k , and Q → R are maps of G -algebras.The interesting point here, and the point of deviation with other interpretationsof equivariant deformation theory, is that we allow G to act nontrivially on theparametrization space Spec( Z ) (or Spf( Z ) in the formal setting).3.2. An equivariant Koszul resolution.
We fix a group scheme G , and equivari-ant deformation Q → R of a G -algebra R with formally smooth parametrizationspace space Spf( Z ). We require specifically that Z is isomorphic to a power series k [[ x , . . . , x n ]] in finitely many variables. As the distinguished point 1 ∈ Spf( Z ) isa fixed point for the G -action, the cotangent space T Spf( Z ) = m Z /m Z admits anatural G -action, and so does the graded algebraSym(Σ m Z /m Z ) = ∧ ∗ ( m Z /m Z ) , which we view as a dg G -algebra with vanishing differential. Lemma 3.1 (cf. [1, Lemma 5.1.4]) . One can associate to the parametrization al-gebra Z a commutative equivariant dg Z -algebra K Z such that (1) K Z is finite and flat over Z , and (2) K Z admits quasi-isomorphisms K Z ∼ → k and k ⊗ Z K Z ∼ → Sym(Σ m Z /m Z ) of equivariant dg algebras.Construction. We first construct an unbounded dg resolution K ′ of k , as in [5,Section 2.6], then truncate to obtain K . We construct K ′ as a union K ′ = lim −→ i ≥ K ( i )of dg subalgebras K ( i ) over Z . We define the K ( i ) inductively as follows: Take K (0) = Z and, for V a finite-dimensional G -subspace generating the maximal ideal m Z in Z , we take K (1) = Sym Z ( Z ⊗ Σ V ) with differential d (Σ v ) = v , v ∈ V .Suppose now that we have K ( i ) an equivariant dg algebra which is finite andflat over Z in each degree, and has (unique) augmentation K ( i ) → k which is aquasi-isomorphism in degrees > − i . Let V i be an equivariant subspace of cocyclesin K ( i ) − i which generates H − i ( K ( i )), as a Z -module. Define K ( i +1) = Sym Z ( Z ⊗ Σ V i ) ⊗ Z K ( i ) , with extended differential d (Σ v ) = v for v ∈ V i . We then have the directed system of dg algebras K (0) → K (1) → . . . with colimit K ′ = lim −→ i K ( i ). By construction K ′ is finite and flat over Z in each degree, and hascohomology H ∗ ( K ′ ) = k .Since Z is of finite flat dimension, say n , the quotient( K Z :=) K = K ′ / (( K ′ ) < − n + B − n ( K ′ ))is finite and flat over Z in all degrees. Furthermore, K inherits a G -action sothat the quotient map K ′ → K is an equivariant quasi-isomorphism. So we haveproduced a finite flat dg Z -algebra K with equivariant quasi-isomorphism K ∼ → k .We consider a section m Z /m Z → V of the projection V → m Z /m Z , and let¯ S ⊂ V denote the image of this section. Take S = Sym Z ( Z ⊗ Σ ¯ S ) with differentialspecified by d (Σ v ) = v for v ∈ ¯ S . Then S the the standard Koszul resolution for k , and the inclusion S → K is a (non-equivariant) dg algebra quasi-isomorphism.Since K and S are bounded above and flat over Z in each degree, the reduction k ⊗ Z S → k ⊗ Z K remains a quasi-isomorphism and we have an isomorphism ofalgebras Sym(Σ m Z /m Z ) ∼ = H ∗ ( k ⊗ Z S ) ∼ = → H ∗ ( k ⊗ Z K ) . Note that the dg subalgebra Sym(Σ V ) ⊂ k ⊗ Z K consists entirely of cocycles,and furthermore Z − ( k ⊗ Z K ) = Σ V . We see also that the intersection V ∩ m Z consists entirely of coboundaries, as such vectors v lift to cocycles in the acycliccomplex K which are of the form v + m Z ⊗ V . A dimension count now implies thatthe projection V = Z − ( k ⊗ Z K ) → H − ( k ⊗ Z K )reduces to an isomorphism V / ( m Z ∩ V ) = H ( k ⊗ Z K ). Hence, for the degree − k ⊗ Z K , we have B − = V ∩ m Z . One now consults the diagramSym(Σ m Z /m Z ) ∼ = (cid:15) (cid:15) incl / / Sym(Σ V ) (cid:15) (cid:15) proj / / Sym(Σ V ) / ( B − ) ∼ = Sym(Σ m Z /m Z ) t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ H ∗ ( k ⊗ Z S ) ∼ = / / H ∗ ( k ⊗ Z K ) , CRIS NEGRON to see that the intersection B ∗ ( k ⊗ Z K ) ∩ Sym(Σ V ) is necessarily the ideal ( B )generated by the degree − f : k ⊗ Z K →
Sym(Σ V ) / ( B ) ∼ = Sym(Σ m Z /m Z )which annihilates (the images of) all cells Σ V i with i > (cid:3) In the following Z a commutative G -algebra which is isomorphic to a powerseries in finitely many variables, as above. Definition 3.2. An equivariant Koszul resolution of k over Z is a G -equivariant dg Z -algebra K Z which is finite and flat over Z , comes equipped with an equivariantdg algebra quasi-isomorphism ǫ : K Z ∼ → k , and also comes equipped with an equi-variant dg map π : K Z → Sym(Σ m Z /m Z ) which reduces to a quasi-isomorphism k ⊗ Z K Z ∼ → Sym(Σ m Z /m Z ) along the augmentation Z → k .Lemma 3.1 says that equivariant Koszul resolutions of k , over such Z , alwaysexists.3.3. The Koszul resolution associated to an equivariant deformation.
Consider Q → R an equivariant deformation, parameterized by a formally smoothspace Spf( Z ), as in Section 3.2. For any equivariant Koszul resolution K Z ∼ → k over Z , the product K Q := Q ⊗ Z K Z (4)is naturally a dg G -algebra which is a finite and flat extension of Q . Since finite flatmodules over Z are in fact free, K Q is more specifically free over Q in each degree.Flatness of Q over Z implies that the projection id Q ⊗ Z ǫ : K Q ∼ → Q ⊗ Z k = R is a quasi-isomorphism of dg G -algebras (cf. [1, Section 5.2], [4, Section 3], [2,Section 2]). We call the dg algebra (4), deduced from a particular choice of equi-variant Koszul resolution for Z , the (or a ) Koszul resolution of R associated to theequivariant deformation Q → R .4. Deformations associated to group embeddings
Consider now G a finite group scheme, and a closed embedding of G into asmooth affine algebraic group H . (We mean specifically a map of group schemes G → H which is, in addition, a closed embedding.) We explain in this sectionhow such an embedding G → H determines an equivariant deformation O → O ( G )which fits into the general framework of Section 3.Note that such closed embeddings G → H always exists for finite G . For example,if we choose a faithful G -representation V then the corresponding action map G → GL( V ) is a closed embedding of G into the associated general linear group.4.1. The quotient space.
For any embedding G → H of G into smooth H weconsider the quotient space H /G . The associated quotient map H → H /G is G -equivariant, where we act on H via the adjoint action and on H /G via translation.This is all clear geometrically, but let us consider this situation algebraically.Functions on the quotient O ( H /G ) are the right G -invariants O ( H ) G in O ( H ), or rather the left O ( G )-coinvariants. Then O ( H /G ) is a right O ( H )-coideal subalgebrain O ( H ), in the sense that the comultiplication on O ( H ) restricts to a coaction ρ : O ( H /G ) → O ( H /G ) ⊗ O ( H ) . We project along O ( H ) → O ( G ) to obtain the translation coaction of O ( G ) on O ( H /G ). The left translation coaction of O ( G ) on O ( H ) restricts to a trivialcoaction on O ( H /G ). So, O ( H /G ) is a sub O ( G )-bicomodule in O ( H ).We consider the dual action of the group ring kG = O ( G ) ∗ on O ( H ), and findthat the inclusion O ( H /G ) → O ( H ) is an inclusion of G -algebras, where we act on O ( H ) via the adjoint action and on O ( H /G ) by translation. We have the followingclassical result, which can be found in [17, Proposition 5.25 and Corollary 5.26]. Theorem 4.1.
Consider a closed embedding G → H of a finite group scheme intoa smooth algebraic group H . The algebra of functions O ( H ) is finite and flat over O ( H /G ) , and O ( H /G ) is a smooth k -algebra. The associated equivariant deformation sequence.
Consider G → H asabove and let 1 ∈ H /G denote the image of the identity in H , by abuse of notation.We complete the inclusion O ( H /G ) → O ( H ) at the ideal of definition for G to geta finite flat extension b O H /G → b O H . Take Z = b O H /G and O = b O H . So we have the deformation O → O ( G ), with formally smooth parametrizing alge-bra Z . A proof of the following Lemma can be found at [20, Lemma 2.10]. Lemma 4.2.
The completion O = b O H is Noetherian and of finite global dimension. Note that the ideal of definition for G is the ideal m O ( G ), where m ⊂ O ( H /G )is associated to the closed point 1 ∈ H /G . Proposition 4.3.
Consider a closed embedding G → H of a finite group schemeinto a smooth algebraic group H . Take O = b O H and Z = b O H /G , where we completeat the augmentation ideal m in O ( H /G ) . Then (a) the quotients O ( H /G ) / m n and O ( H ) / m n O ( H ) inherit G -algebra structuresfrom O ( H /G ) and O ( H ) respectively. (b) The completions Z and O inherit unique continuous G -actions so that theinclusions O ( H /G ) → Z and O ( H ) → O are G -linear. (c) Under the actions of (b) , the projection O → O ( G ) is an equivariant de-formation of O ( G ) parametrized by Spf( Z ) = ( H /G ) ∧ .Proof. All of (a)–(c) will follow if we can simply show that m ⊂ O ( H /G ) is stableunder the translation action of kG . This is clear geometrically, and certainly well-known, but let us provide an argument for completeness. If we let ker( ǫ ) ⊂ O ( H )denote the augmentation ideal, we have m = ker( ǫ ) ∩ O ( H /G ).For the adjoint coaction ρ ad : f f ⊗ S ( f ) f of O ( H ) on itself, and f ∈ ker( ǫ ),we have ( ǫ ⊗ ◦ ρ ad ( f ) = ǫ ( f ) S ( f ) f = S ( f )( ǫ ( f ) f ) = S ( f ) f = ǫ ( f ) = 0 . So we see that under the adjoint coaction ρ ad (ker( ǫ )) ⊂ ker( ǫ ) ⊗ O ( H ). It followsthat ker( ǫ ) is preserved under the adjoint coaction of O ( G ), and hence the adjointaction of kG , as well. So, the intersection m = O ( H /G ) ∩ ker( ǫ ) is an intersection of G -subrepresentations in O ( H ), and hence m is stable under the action of kG . (cid:3) Equivariant formality results and deformation classes
We observe cohomological implications of the existence of a (smooth) equivariantdeformation, for a given finite-dimensional G -algebra R . The main results of thissection can been seen as particular equivariantizations of [4, Theorem 1.2.3] and [20,Corollary 4.7], as well as of classical results of Gulliksen [16, Theorem 3.1].5.1. We fix an equivariant deformation.
We fix a G -equivariant deformation Z → Q → R , with Z isomorphic to a power series in finitely many variables. Fixalso a choice of equivariant Koszul resolution K := K Z , with ǫ : K ∼ → k and π : K →
Sym(Σ m Z /m Z ) . Recall the associated dg resolution K Q ∼ → R , with K Q = Q ⊗ Z K . Via generalphenomena (Section 2.4) we observe Lemma 5.1.
Restriction provides a derived equivalence D fin ( R ) G ∼ → D coh ( K Q ) G . Following the notation of [20], we fix A Z := Sym(Σ − T Spf( Z )) = Sym(Σ − ( m Z /m Z ) ∗ ) . (5)5.2. Equivariant formality and deformation classes.Lemma 5.2.
Consider K the regular dg K -bimodule. There is a ( G -)equivarianthomotopy isomorphism REnd K⊗ Z K ( K ) ∼ → A Z . In particular,
REnd K⊗ Z K ( K ) is equivariantly formal.Proof. Consider our algebra A = A Z from (5) and take B = Sym(Σ m Z /m Z ). Let F → k be the standard resolution of the trivial module over B . The resolution F isof the form B ⊗ A ∗ , as a graded space, with differential given by right multiplicationby the identity element P i x i ⊗ x i in B − ⊗ A , and so F admits a natural dg( B, A )-bimodule structure. The action map for A now provides an equivariantquasi-isomorphism A ∼ → End B ( F ) = REnd B ( k ).For the Koszul resolution K over Z , we have the equivariant quasi-isomorphism π ⊗ Z ǫ : K ⊗ Z K ∼ → B and corresponding restriction and base change equivalences D ( K ⊗ Z K ) G ⇆ D ( B ) G , which are mutually inverse. Restriction sends the trivialrepresentation k over B to the regular K -bimodule k ∼ = K . Hence the base change B ⊗ L K⊗ Z K K is isomorphic to k . We then get then an equivariant quasi-isomorphism B ⊗ L K⊗ Z K − : REnd K⊗ Z K ( K ) ∼ → REnd B ( B ⊗ L K⊗ Z K K ) , with the latter algebra homotopy isomorphic to REnd B ( k ) ∼ = A by Lemma 2.4. (cid:3) Remark 5.3.
In odd characteristic, one can replace the quasi-isomorphism π ⊗ Z ǫ : K ⊗ Z K → B with the more symmetric map mult ( 12 π ⊗ Z − π ) : K ⊗ Z K → B. The point is to provide an equivariant quasi-isomorphism which is a retract of thenon-equivariant quasi-isomorphism B → K ⊗ Z K implicit in [4, Lemma 2.4.2]. Recall that we are considering an equivariant deformation Q → R , with associ-ated dg resolution K Q ∼ → R , as in Section 3.3. We have the natural action of A Z on D coh ( K Q ) [20, § A Z = End ∗ D ( K⊗ Z K ) ( K ) → Z ( D coh ( K Q )) (6)to the center of the derived category Z ( D coh ( K Q )) = ⊕ i Hom
Fun ( id, Σ i ). Specif-ically, for any endomorphism f : K → Σ n K in the derived category of Z -centralbimodules, and M in D coh ( K Q ), we have the induced endomorphism f ⊗ L K M : M → Σ n M. Suppose, for convenience, that Q is of finite global dimension. We lift the maps − ⊗ L K M : End ∗ D ( K⊗ Z K ) ( K ) → End ∗ D ( K Q ) ( M ) (7)to a dg level, for equivariant M , as follows [4]: Fix an equivariant semi-projectiveresolution F → K over K ⊗ Z K and, at each M , chose an equivariant quasi-isomorphism M ′ → M from a dg K Q -module which is bounded and projectiveover Q in each degree. (Such a resolution exists since Q is of finite global dimen-sion.) Then F ⊗ K M ′ → M is an equivariant semi-projective resolution of M over K Q [20, Lemma 4.4]. We now have the lift − ⊗ K M ′ : End K⊗ Z K ( F ) → End K Q ( F ⊗ K M ′ )of (7), and we write this lift simply as def GM : REnd K⊗ Z K ( K ) → REnd K Q ( M ) . Direct calculation verifies that def GM , constructed in this manner, is in fact G -linear.The following result is an equivariantization of [20, Corollary 4.7]. Theorem 5.4.
Consider a G -equivariant deformation Q → R , with R finite-dimensional, Q of finite global dimension, and parametrization algebra Z isomor-phic to a power series in finitely many variables. Let R denote the formal dg algebra REnd K⊗ Z K ( K ) (Lemma 5.2).For any M in D coh ( K Q ) G , the equivariant dg algebra map def GM : R → REnd K Q ( M ) defined above has the following properties: (1) The induced map on cohomology H ∗ ( def GM ) : A Z → End ∗ D ( K Q ) ( M ) is afinite morphism of graded G -algebras. (2) For any N in D coh ( K Q ) G , the induced action of R on RHom K Q ( M, N ) issuch that RHom K Q ( M, N ) ∈ D coh ( R ) G . By D coh ( R ) G we mean the category of G -equivariant dg modules over R withfinitely generated cohomology over A Z = H ∗ ( R ). Proof.
The map def GM was already constructed above. We just need to verify theimplications for cohomology, which actually have nothing to do with the G -action.We note that the cohomology H ∗ ( def GM ) is, by construction, obtained by evaluatingthe functor − ⊗ L K M : D ( K ⊗ Z K ) → D ( K Q ) at the object K . (Again, we forget about equivariance here.) We can factor thisfunctor through the category of K Q -bimodules D ( K ⊗ Z K ) −⊗ L Z Q −→ D ( K Q ⊗ K Q ) −⊗ L K Q M −→ D ( K Q )to see that the corresponding map to the center (6) agrees with that of [4, (3.1.5)][20, Section 3.4]. So the finiteness claims of (1) and (2) follow from [20, Corollary4.7]. (cid:3) Via Lemma 5.2 we may replace D ( R ) G with D ( A Z ) G , and view RHom K Q , orequivalently RHom R , as a functor to D ( A Z ) G . Alternatively, we could work withthe dg scheme (shifted affine space) T ∗ = T ∗ Spf( Z ) = Spec( A Z ), and view RHom R as a functor taking values in the derived category of equivariant dg sheaves on T ∗ .From this perspective, Theorem 5.4 tells us that RHom R has image in the sub-category of dg sheaves on T ∗ with coherent cohomology,RHom R : ( D fin ( R ) G ) op × D fin ( R ) G → D coh ( A Z ) G ∼ = D coh ( T ∗ ) G . Remark 5.5.
We only use the finiteness claims of Theorem 5.4 in the case in whichall of Z , Q , and R are commutative. In this case in particular, claims (1) and (2)of Theorem 5.4 should be obtainable directly from Gulliksen [16, Theorem 3.1]. Remark 5.6.
One may compare the above analyses with the formality argumentsof [1, Sections 5.4–5.8].6.
Touz´e-Van der Kallen and derived invariants
We recall some results of Touz´e and Van der Kallen [26]. Our aim is to takederived invariants of Theorem 5.4 to obtain a finite generation result for equivariantextensions Hom ∗ D ( R ) G . We successfully realize this aim via an invocation of [26].Throughout this section G is a finite group scheme.6.1. Basics and notations.
For V any G -representation we have the standardgroup cohomology H ∗ ( G, V ) = Ext ∗ G ( , V ). For more general objects in D (Rep( G ))we adopt a hypercohomological notation. Notation 6.1.
We let ( − ) R G : D (Rep( G )) → D ( V ect ) denote the derived invari-ants functor, ( − ) R G = RHom G ( , − ). For M in D (Rep( G )) we take H ∗ ( G, M ) := H ∗ ( M R G ) . We note that the hypercohomology H ∗ ( G, M ) is still identified with morphismsHom ∗ D (Rep( G )) ( , M ) in the derived category. Since G is assumed to be finite, weare free to employ an explicit identification( − ) R G = Hom G ( Bar G , − ) , where Bar G is the standard Bar resolution. For any dg G -algebra S the derivedinvariants S R G are naturally a dg algebra in V ect , and for any equivariant dg S -module M , M R G is a dg module over S R G . (Under our explicit expression ofderived invariants in terms of the bar resolution, these multiplicative structures areinduced by a dg coalgebra structure on Bar G , see e.g. [22, § G -algebra a functor( − ) R G : D ( S ) G → D ( S R G ) . (8) The following well-known fact can be proved by considering the hypercohomology H ∗ ( G, S ) as maps → Σ n S in the derived category. Lemma 6.2. If A is a commutative dg G -algebra, then the hypercohomology H ∗ ( G, A ) is a also commutative. Derived invariants and coherence of dg modules.
We have the followingresult of Touz´e and Van der Kalen.
Theorem 6.3 ([26, Theorems 1.4 & 1.5]) . Consider G a finite group scheme, and A a commutative G -algebra which is of finite type over k . Then the cohomology H ∗ ( G, A ) is also of finite type, and for any finitely generated equivariant A -module M , the cohomology H ∗ ( G, M ) is a finite module over H ∗ ( G, A ) . One can derive this results to obtain
Theorem 6.4.
Consider G a finite group scheme, and S a dg G -algebra which isequivariantly formal and has commutative, finite type, cohomology. Suppose addi-tionally that the cohomology of S is bounded below. Then the derived invariantsfunctor (8) restricts to a functor ( − ) R G : D coh ( S ) G → D coh ( S R G ) . Equivalently, for any equivariant dg S -module M with finitely generated cohomologyover H ∗ ( S ) , the hypercohomology H ∗ ( G, M ) is finite over H ∗ ( G, S ) .Proof. Take A = H ∗ ( S ). We are free to view, momentarily, A as a non-dg object.We have that A is finite over its even subalgebra A ev , which is a commutative alge-bra in the classical sense, so that Theorem 6.3 implies that cohomology H ∗ ( G, − )sends A to a finite extension of H ∗ ( G, A ev ), and any finitely generated A -moduleto a finitely generated H ∗ ( G, A ev )-module. Hence H ∗ ( G, A ) is of finite type over k , and H ∗ ( G, N ) is finite over H ∗ ( G, A ) for any finitely generated, equivariant,non-dg, A -module N .Since G is a finite group scheme, A is also a finite module over its (usual)invariant subalgebra A G , and any A -module is finitely generated over A if andonly if it is finitely generated over A G . Theorem 6.3 then tells us that, for anyfinitely generated A -module N , the cohomology H ∗ ( G, N ) is finitely generated over H ∗ ( G, A G ) = H ∗ ( G, ) ⊗ A G , where H ∗ ( G, A G ) acts through the algebra map H ∗ ( G, incl) : H ∗ ( G, A G ) → H ∗ ( G, A ) . Consider now any dg module M in D coh ( S ) G . Formality implies an algebraisomorphism S ∼ = A in D (Rep( G )) and so identifies H ∗ ( G, S ) with H ∗ ( G, A ) = H ∗ ( G, A ). We want to show that, for such a dg module M , the hypercohomology H ∗ ( G, M ) is a finitely generated module over H ∗ ( G, S ) ∼ = H ∗ ( G, A ). It suffices toshow that H ∗ ( G, M ) is finite over H ∗ ( G, A G ) = H ∗ ( G, ) ⊗ A G . We have the firstquadrant spectral sequence (via our bounded below assumption) E ∗ , ∗ = H ∗ ( G, H ∗ ( M )) ⇒ H ∗ ( G, M ) , and the E -page is finite over H ∗ ( G, A G ) by the arguments given above. Since H ∗ ( G, A G ) is Noetherian, it follows that the associated graded module E ∗ , ∗∞ =gr H ∗ ( G, M ) is finite over H ∗ ( G, A G ), and since the filtration on H ∗ ( G, M ) isbounded in each cohomological degree it follows that the hypercohomology H ∗ ( G, M )is indeed finite over H ∗ ( G, A G ) ⊂ H ∗ ( G, S ) [14, Lemma 1.6]. (cid:3) Finite generation of cohomology for Drinfeld doubles
Consider G a finite group scheme. Fix a closed embedding G → H into a smoothalgebraic group H , and fix also the associated G -equivariant deformation Z → O → O ( G ) , Z = b O H /G , O = b O H , as in Section 4.2. Here kG acts on O ( G ) and O via the adjoint action, and thisadjoint action restricts to a translation action on Z . We recall that the embeddingdimension of G is the minimal dimension of such smooth H admitting a closedembedding G → H .We consider the tensor category Z (rep( G )) ∼ = rep( D ( G )) ∼ = Coh( G ) G of representations over the Drinfeld double of G , aka the Drinfeld center of rep( G ).We prove the following below. Theorem 7.1.
For any finite group scheme G , the Drinfeld center Z (rep( G )) isof finite type and of bounded Krull dimension Kdim Z (rep( G )) ≤ Kdim rep( G ) + embed . dim( G ) . One can recall our definition of a finite type tensor category, and of the Krulldimension of such a category, from the introduction. For T ∗ the cotangent space T ∗ Spf( Z ), considered as a variety with a linear G -action, we show in particular thatthere is a finite map of schemes Spec Ext ∗ Z (rep( G )) ( , ) → ( G \ T ∗ ) × Spec H ∗ ( G, ).7.1. Preliminaries for Theorem 7.1: Derived maps in Z (rep( G )) . We let G act on itself via the adjoint action, and have Coh( G ) G = rep( O ( G )) G . The unitobject ∈ Coh( G ) G is the residue field of the fixed point 1 : Spec( k ) → G . Wehave REnd Coh( G ) G ( ) = REnd Coh( G ) ( ) R G , as an algebra, and for any V in Coh( G ) G we haveRHom Coh( G ) G ( , V ) = RHom Coh( G ) ( , V ) R G , as a dg REnd Coh( G ) G ( )-module.One can observe these identifications essentially directly, by noting that for theprojective generator O ( G ) ⋊ kG we have an identification of G -representationsHom Coh( G ) ( O ( G ) ⋊ kG, V ) = Hom k ( kG, V ) = O ( G ) ⊗ V, and O ( G ) ⊗ V is an injective over kG for any V . Hence the functor Hom Coh( G ) ( − , V )sends projectives objects in Coh( G ) G to injectives in Rep( G ), and for a projectiveresolution F → we have identifications in the derived category of vector spacesRHom Coh( G ) G ( , V ) = Hom Coh( G ) G ( F, V )= Hom
Coh( G ) ( F, V ) G ∼ = Hom Coh( G ) ( F, V ) R G = RHom Coh( G ) ( , V ) R G andREnd Coh( G ) G ( , ) = End Coh( G ) ( F ) G ∼ = End Coh( G ) ( F ) R G = REnd Coh( G ) ( ) R G . The middle identification for derived endomorphisms comes from the diagramEnd( F ) G / / ∼ (cid:15) (cid:15) End( F ) R G ∼ (cid:15) (cid:15) Hom( F, ) G ∼ / / Hom( F, ) R G . Proof of Theorem 7.1.
Proof.
Fix an embedding G → H and associated equivariant deformation O → O ( G ) as above, and take A = A Z = Sym(Σ − ( m Z /m Z ) ∗ ), as in (5). Take also R the dg G -algebra REnd K Z ⊗ Z K Z ( K Z ). We recall from Lemma 5.2 that R isequivariantly formal, and so homotopy isomorphic to A . We adopt the abbreviatednotations RHom = RHom Coh( G ) and REnd = REnd Coh( G ) when convenient.We consider the equivariant dg algebra map def G : R → REnd
Coh( G ) ( )of Theorem 5.4, and the action of R on each REnd Coh( G ) ( , V ) through def G . ByTheorems 5.4 and 6.4, the hypercohomology H ∗ ( G, REnd( )) is a finite algebraextension of H ∗ ( G, R ), and H ∗ ( G, RHom( , V )) is a finitely generated module over H ∗ ( G, R ) for any V in Coh( G ) G . In particular, H ∗ ( G, RHom( , V )) is finite over H ∗ ( G, REnd( )).Since H ∗ ( G, R ) ∼ = H ∗ ( G, A ) is of finite type over k , by Touz´e-Van der Kallen(Theorem 6.4), the above arguments imply that H ∗ ( G, REnd
Coh( G ) ( )) = Ext ∗ Coh( G ) G ( , )is a finite type k -algebra, and that each H ∗ ( G, RHom
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Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599
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