aa r X i v : . [ m a t h . R T ] F e b - SUPPORT THEORY FOR EXTENDED DRINFELD DOUBLES

ERIC M. FRIEDLANDER

Abstract.

Following earlier work with Cris Negron on the cohomology ofDrinfeld doubles D ( G ( r ) ), we develop a “geometric theory” of support varietiesfor “extended Drinfeld doubles” ˜ D ( G ( r ) ) of Frobenius kernels G ( r ) of smoothlinear algebraic groups G over a ﬁeld k of characteristic p >

0. To a ˜ D ( G ( r ) )-module M we associate the space Π( ˜ D ( G ( r ) )) M of equivalence classes of “pairsof π -points” and prove most of the desired properties of M Π( ˜ D ( G ( r ) )) M .Namely, this association satisﬁes the “tensor product property” and admitsa natural continuous map Ψ ˜ D to cohomological support theory. Moreover,for M ﬁnite dimensional and with suitable conditions on G ( r ) , this associa-tion provides a “projectivity test”, Ψ ˜ D is a homeomorphism, and identiﬁesΠ( ˜ D ( G ( r ) )) M with the cohomological support variety of M for various classesof ˜ D ( G ( r ) )-modules M . Introduction

Cohomological support varieties for kG -modules for a ﬁnite group G constitutenatural extensions of D. Quillen’s geometric description of Spec H • ( G, k ) [19], [20].As introduced by J. Carlson in [3], the cohomological support variety | G | M of a ﬁnitedimensional kG -module M is the closed subvariety of Spec H • ( G, k ) associated tothe (radical of) the kernel of the natural map H • ( G, k ) → Ext ∗ G ( M, M ) of k -algebras. To make this more geometric and more computable in the special casethat G is an elementary abelian p -group, J. Carlson conjectured and G. Avrunin andL. Scott proved (in [1]) that | E | M can be identiﬁed with the “rank variety” V ( E ) M of cyclic shifted subgroups along which the pull-backs of M are not projective.This “geometric interpretation” of cohomological support varieties for modules foran elementary abelian p -group was successively extended by B. Parshall and theauthor to modules for restricted Lie algebras [6], by A. Suslin, C. Bendel, andthe author in [23] to modules for inﬁnitesimal group schemes, and to modules forarbitrary ﬁnite group schemes by J. Pevstova and the author [8]. The interweavingof the two approaches (one, primarily cohomological; the second, more geometric)has led to various constructions and invariants.In this paper, we adapt the theory of π -points for ﬁnite group schemes to atheory of π -point pairs for a special class of ﬁnite dimensional Hopf algebras over k which are neither commutative nor cocommutative. Namely, beginning with asmooth linear algebraic group G over k , we consider the Frobenius kernel G ( r ) (i.e.,the kernel of the r -th iterate of the Frobenius map, F r : G → G ( r ) for some r > D ( G ( r ) ) ≡ k [ G ( r +1) ] k G ( r ) which mapssurjectively to the Drinfeld double D ( G ( r ) ) of G ( r ) . Date : February 5, 2021.2010

Mathematics Subject Classiﬁcation.

Key words and phrases.

Drinfeld doubles, Frobenius kernels, Support varieties.Partially supported by the Simons Foundation.

What enables our approach for ˜ D ( G ( r ) )-modules (and does not work for modulesfor the Drinfeld double D ( G ( r ) )) is the existence of a sub Hopf algebra O ( G ( r ) ) ⊂ ˜ D ( G ( r ) ) which essentially detects the cohomology of ˜ D ( G ( r ) ) and which is iso-morphic as an algebra to the “group algebra” for a ﬁnite group scheme. Indeed,almost by deﬁnition, the π -point pair space Π( ˜ D ( G ( r ) )) maps bijectively to the π -point pair space Π( O ( G ( r ) )) which is homeomorphic to the π -point space of theassociated ﬁnite group scheme. To establish the desired tensor product propertyΠ( ˜ D ( G ( r ) )) M ⊗ M ′ = Π( ˜ D ( G ( r ) )) M ∩ Π( ˜ D ( G ( r ) )) N in Theorem 3.4, we utilize thistensor product property for ﬁnite group schemes. The key step requires the veriﬁ-cation that this property remains valid upon changing coproduct structure on thisgroup algebra to the coproduct structure on O ( G ( r ) ). This invariance of coprod-uct of supports of a tensor product is the content of Proposition 3.2. The tensorproduct property enables a natural topology on Π( ˜ D ( G ( r ) )).In Section 4, we extend the arguments of C. Negron and the author in [5] for thecohomology of Drinfeld doubles to the cohomology of the extended doubles ˜ D ( G ( r ) )which allows a comparison in Corollary 4.3 of the cohomology of ˜ D ( G ( r ) ) with thatof the simpler subalgebra O ( G ( r ) ). In Proposition 4.6, we establish a commutativesquare of continuous maps relating π -point pair spaces for O ( G ( r ) ) and ˜ D ( G ( r ) )with (projectivized) prime ideal spectra of their cohomology.In order to proceed further, we require the determination of the cohomology alge-bra H • ( ˜ D ( G ( r ) ) , k ), the commutative subalgebra of the Ext-algebra Ext ∗ ˜ D ( G ( r ) ) ( k, k )of classes of even degree for p > p = 2). This isachieved in Theorem 5.4 provided that G admits a quasilogarithm (see Example5.2) and that p r +1 > dim ( G ), again using techniques of [5]. This computationallows us to show that Ψ ˜ D : Π( ˜ D ( G ( r ) )) → P ( H • ( ˜ D ( G ( r ) ) , k )) is a homeomor-phism. As a consequence, we conclude in Theorem 5.6 that a ﬁnite dimensional˜ D ( G ( r ) )-module is projective if and only if Π( ˜ D ( G ( r ) )) M is empty.The fact that Ψ ˜ D is a homeomorphism mapping Π( ˜ D ( G ( r ) )) M to P ( H • ( ˜ D ( G ( r ) ) , k )) M does not a priori identify Π( ˜ D ( G ( r ) )) M with P ( H • ( ˜ D ( G ( r ) ) , k )) M . In Section 6,we achieve this identiﬁcation for various special classes of ˜ D ( G ( r ) )-modules, in-cluding Carlson’s L ζ -modules, modules whose action factors through the quotient˜ D ( G ( r ) ) ։ k G ( r ) and modules whose restriction to k [ G ( r +1) ] are projective.In Section 7, we observe that the same techniques we use to investigate ˜ D ( G ( r ) )-modules apply to modules for closely related Hopf algebras. We also mention acouple of questions worthy of further investigation.We thank both Cris Negron and Julia Pevtsova for their contributions to thefoundations of this work, and the Institute for Advanced Study for its (virtual)hospitality. 1. D ( G ( r ) ) , ˜ D ( G ( r ) ) , O ( G ( r ) ) , G ( r ) × ( g ( r ) ) (1) Throughout this work, G will denote a connected linear algebraic group smoothover k . If A is a k -algebra and K/k is a ﬁeld extension, then we denote by A K thebase change K ⊗ A ; for an A -module M , we use the notation M K for the A K -module K ⊗ M . For a ﬁnite dimensional k -algebra A , we denote by H • ( A, k ) ⊂ Ext ∗ A ( k, k )the commutative subalgebra of even dimensional classes if p > Ext ∗ A ( k, k ) if p = 2. UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 3

For any r >

0, we denote by F r : G → G ( r ) the r -th iterate of the Frobeniusmap F : G → G (1) given by the k -linear map F ∗ : k [ G (1) ] ≡ k ⊗ φ k [ G ] → k [ G ] , λ ⊗ f λf p of coordinate algebras (where k ⊗ φ − denotes the base change along the p -th powermap φ : k → k ). The r -th Frobenius kernel of G is the ﬁnite group scheme(1.0.1) G ( r ) ≡ ker { F r : G → G ( r ) } ֒ → G , whose coordinate algebra k [ G ( r ) ] is the local commutative algebra deﬁned as thequotient of the localization ( k [ G ]) ( e ) of k [ G ] at the identity e ∈ G modulo the idealgenerated by elements x p r , x ∈ m e . For example, k [ GL n ( r ) ] = k [ x i,j ] / ( x p r i,j − δ i,j ).The coordinate algebra k [ G ] of G is a Hopf algebra with comultiplication deter-mined by the multiplication of G . For a ﬁnite group scheme G such as G ( r ) , weconsider the k -linear dual kG of k [ G ] which is a ﬁnite dimensional, co-commutativeHopf algebra (over k ); we refer to kG as the group algebra of G . For such a ﬁnitegroup scheme G , a (rational) G -module is a k -vector space equipped with a k -linearmodule structure, kG × M → M . We shall frequently refer to this structure as the“action of G on M ”.We recall that the Drinfeld double of a Hopf algebra H is the double crossedproduct of H ∗ cop and H , where H ∗ cop has the same algebra structure as H ∗ andcoproduct given by the opposite of the coproduct of H ∗ . If H is ﬁnite dimensionaland cocommutative, then D ( H ) is isomorphic to the smash product H ∗ op H (see[18, 10.3.10]).In Hopf algebra standard notation, the coadjoint action of an element h ∈ H onan element f ∈ H ∗ is given by P f ( h ⇀ f ↼ s − ( h )) where s is the antipode of H , ∆ is the coproduct of H ∗ and ∆( h ) is written as P h ⊗ h . By a result of D.Radford [21, 10.3.5], the product structure of D ( H ) is given for ﬁnite dimensional H by ( f h )( f ′ h ′ ) = X f ( h ⇀ f ′ ↼ s − ( h )) h h ′ . If H is both ﬁnite dimensional and cocommutative, this multiplication is that ofthe smash product algebra H ∗ H :( f h )( f ′ h ′ ) = X f h ( f ) h h ′ ;the coproduct of D ( H ) is the tensor product of the opposite coproduct of H ∗ andthe coproduct of H . Deﬁnition 1.1.

As discussed above, the Drinfeld double of the ﬁnite dimensionalcocommutative Hopf algebra k G ( r ) has algebra structure given by the smash prod-uct(1.1.1) D ( G ( r ) ) ≡ k [ G ( r ) ] k G ( r ) . with respect to the right coadjoint action of G ( r ) on k [ G ( r ) ] and coalgebra structuregiven by the tensor product of the co-opposite cooproduct on k [ G ( r ) ] (i.e., theopposite product structure of k G ( r ) ) and the coproduct of k G ( r ) .The extended Drinfeld double ˜ D ( G ( r ) ) is the analogous smash product withrespect to the restriction to G ( r ) of the coadjoint action of G ( r +1) on k [ G ( r +1) ]:(1.1.2) ˜ D ( G ( r ) ) ≡ k [ G ( r +1) ] k G ( r ) ֒ → D ( G ( r +1) ) . ERIC M. FRIEDLANDER

To verify that (1.1.2) is a well deﬁned inclusion of Hopf algebras, we observe thatthe restriction of the coadjoint action of G ( r +1) on k [ G ( r +1) ] along G ( r ) ֒ → G ( r +1) is the action used to deﬁne ˜ D ( G ( r ) ), then check directly that the multiplication andcomultiplication of D ( G ( r +1) ) restricts via the evident inclusion of k -vector spaces k [ G ( r +1) ] ⊗ k G ( r ) ⊂ k [ G ( r +1) ] ⊗ k G ( r +1) to the multiplication and comultiplicationof ˜ D ( G ( r ) ).We shall be interested in (left) modules for the extended Drinfeld double ˜ D ( G ( r ) )which can be viewed as k -vector spaces N with an action ˜ D ( G ( r ) ) × N → N . Thecondition on such a pairing to be a left module structure can be expressed as therestriction to elements in ˜ D ( G ( r ) ) ⊂ D ( G ( r +1) ) of the condition that a pairing D ( G ( r +1) ) × N → N be a D ( G ( r +1) ) -module structure. This latter condition ismade explicit in the following proposition (with H = k G ( r +1) ).We ﬁrst observe that for any Hopf algebra H there are natural algebra embed-dings of H, H ∗ in D ( H ): h ∈ H h ; f ∈ H ∗ f Proposition 1.2. [18, 10.6.16] , [16] Let H be a ﬁnite dimensional Hopf Algebra.The data of a D ( H ) -module structure on a vector space N consists of module struc-tures on N for both H ∗ and H such that h · ( f · n ) = X ( h ⇀ f ) · (( h ↼ f ) · n ) ∈ N. This is equivalent to a left/right Yetter-Drinfeld module structure on N whichconsists of the data of a k G ( r ) -module structure and a right k G ( r ) -comodule struc-ture on N satisfying the Yetter-Drinfeld condition: X h · n o ⊗ h n = X ( h n ) ⊗ ( h · n ) h ∈ N ⊗ k G ( r ) . Here, we have written the coaction N → N ⊗ H as n P n ⊗ h . We relate the extended Drinfeld double ˜ D ( G ( r ) ) to various other structures. Deﬁnition 1.3.

We deﬁne(1.3.1) O ( G ( r ) ) ≡ k [( G ( r ) ) (1) ] ⊗ k G ( r ) . Using the natural identiﬁcation of ( G ( r ) ) (1) with G ( r +1) / G ( r ) , we obtain the naturalembedding(1.3.2) i O : O ( G ( r ) ) ֒ → ˜ D ( G ( r ) ) . Proposition 1.4.

The embedding G ( r ) ⊂ G ( r +1) determines a G ( r ) -equivariantquotient map k [ G ( r +1) ] ։ k [ G ( r ) ] of Hopf algebras and thus a quotient map of Hopfalgebras (1.4.1) q : ˜ D ( G ( r ) ) ։ D ( G ( r ) ) . Similarly, the quotient map G ( r +1) ։ ( G ( r ) ) (1) determining i O of (1.3.2) is anembedding of Hopf algebrasThe coproduct structure inherited by O ( G ( r ) ) is the tensor product of the co-opposite of the coproduct on k [ G ( r ) ) (1) ] (in other words the dual of the oppositemultiplication of k ( G ( r ) ) (1) ) and the coproduct of k G ( r ) (the dual of the multipli-cation of k [ G ( r ) ] ). Thus, O ( G ( r ) ) is co-commutative if and only if g = Lie ( G ) iscommutative. UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 5

Proof.

Since each G ( r ) is normal in G , the embedding G ( r ) ⊂ G ( r +1) is G -equivariant so that k [ G ( r +1) ] ։ k [ G ( r +1) / G ( r ) ] is a G -equivariant quotient mapof Hopf algebras; thus, this G ( r ) -equivariant quotient determines the map of Hopfalgebras q : ˜ D ( G ( r ) ) ։ D ( G ( r ) ).Similarly, the quotient G ( r +1) ։ G ( r ) ) (1) is also G -equivariant, determining the G -equivariant embedding k [ G ( r ) ) (1) ] ֒ → k [ G ( r +1) ] of Hopf algebras. Observe that G ( r ) ⊂ G acts trivially on k [ G ( r ) ) (1) ], so that we obtain the embedding (as algebras)of the tensor product k [ G ( r ) ) (1) ] ⊗ k G ( r ) ֒ → ˜ D ( G ( r ) ). (cid:3) Proposition 1.5.

The subalgebra k [ G ( r ) ) (1) ] ≃ k [ G ( r +1) / G ( r ) ] of the commutativealgebra k [ G ( r +1) ] gives k [ G ( r +1) ] the structure of a free k [ G ( r ) ) (1) ] -module with basis { ˜ f i } given by lifting along k [ G ( r +1) ] → k [ G ( r ) ] a k -basis { f i } for k [ G ( r ) ] .This structure determines a free (right) O ( G ( r ) ) -module structure on ˜ D ( G ( r ) ) with basis { ˜ f i } .Proof. We choose a k -linear section σ : k [ G ( r ) ] → k [ G ( r +1) ] of the quotient map k [ G ( r +1) ] ։ k [ G ( r ) ] and consider the k [ G ( r +1) / G ( r ) ]-bilinear map(1.5.1) k [ G ( r ) ] ⊗ k [ G ( r +1) / G ( r ) ] → k [ G ( r +1) ] , f i ⊗ h σ ( f i ) · h. We readily check that this induces a k -linear isomorphism, for example by showingthat tensors of elements of k -bases for k [ G ( r ) ] and k [ G ( r +1) / G ( r ) ] determine a k -linearly independent set in k [ G ( r +1) ].Using the fact that k [ G ( r +1) / G ( r ) ] ⊗ ⊂ ˜ D ( G ( r ) ) is central, we readily checkthat k [ G ( r ) ] ⊗ O ( G ( r ) ) → ˜ D ( G ( r ) ) , f ⊗ ( f ′ ⊗ h ) ( σ ( f ) · ( f ′ h )is an O ( G ( r ) )-linear isomorphism (cid:3) The following observation about O ( G ( r ) ) will prove central in our investigationof ˜ D ( G ( r ) )-modules. The reader should not confuse the ﬁnite group scheme ( g ( r ) ) (1) occurring in the proposition below with ( G ( r ) ) (1) . Proposition 1.6.

Let g denote the vector group scheme Spec S • ( g ∗ ) (ignoring theLie algebra structure on g ). Then the group algebra of the ﬁnite group scheme ( g ( r ) ) (1) × G ( r ) is isomorphic as an algebra to O ( G ( r ) ) .On the other hand, the dual of the Hopf algebra O ( G ( r ) ) is not commutative(unlike the coordinate algebra of ( g ( r ) ) (1) × G ( r ) ).Proof. Essentially by deﬁnition, elements X ( r ) ∈ Lie ( G ( r ) ) are distributions at theidentity of G ( r ) of k [ G ( r ) ] of order 1 (see [13, I.7.10]), This identiﬁcation determinesan embedding of g ( r ) ∗ into the maximal ideal of k [ G ( r +1) / G ( r ) ] which necessarilysends each ( X ( r ) ) p ∈ S p ( g ( r ) ∗ ) to 0. The resulting map ν : S • ( g ( r ) ∗ ) / (( X ( r ) ) p , X ∈ g ∗ ) → k [ G ( r +1) / G ( r ) ]is injective, and thus surjective by dimension reasons. The proof is completed byobserving that the group algebra of ( g ( r ) ) (1) equals S • ( g ( r ) ∗ ) / (( X ( r ) ) p , X ∈ g ∗ ). (cid:3) We conclude this section with some remarks about π -point spaces Π( G × G ) ofa product of ﬁnite group schemes which we shall use for G × G = ( g ( r ) ) (1) × G ( r ) .Recall that a π -point α K : K [ t ] /t p → kG of a ﬁnite group scheme G over k is a ﬂat ERIC M. FRIEDLANDER map of algebras which factors through the group algebra of some unipotent abeliansubgroup scheme C K ⊂ G K , where K is some ﬁeld extension of k . Thus, a π -pointof G × G is a pair of π -points α K, : K [ t ] /t p → KG , α K, : K [ t ] /t p → KG corresponding to a ﬂat map α K, + α : K [ t ] /t p → KC ⊗ KC , t α K, ( t ) ⊗ ⊗ α K, ( t )with C K,i ⊂ G K,i unipotent abelian subgroup schemes for i = 1 , α K, ( t ) , α K, ( t ) to commute in order that α K, + α K, is well deﬁned.By [8, Thm 3.6], two π -points α K : K [ t ] /t p → KG, β L : L [ t ] /t p → LG areequivalent if and only if the two ideals of H • ( G, k ) ker { α ∗ K : H • ( G, K ) → H ∗ ( K [ t ] /t p , K ) } ∩ H • ( G, k ) ,ker { β ∗ L : H • ( G, L ) → H ∗ ( L [ t ] /t p , L ) } ∩ H • ( G, k )are equal. In other words, this gives a natural bijection of sets (which is sharpenedto an isomorphism of schemes in [8])(1.6.1) Π G : Π( G ) ∼ → P H • ( G, k ) . Observe that the ﬂat map ( α K, , α K, ) : K [ t ] /t p → KG × KG induces the map( α ∗ K, ⊗

1) + (1 ⊗ α K, ) ∗ : H ∗ ( C , K ) ⊗ K H • ( C , K ) → H ∗ ( K [ t ] /t p , K ) . We denote the composition of this map with the restriction map( α K, + α K, ) ∗ : H ∗ ( G × G , K ) → H ∗ ( K [ t ] /t p , K ) . Thus, we have proved the following proposition.

Proposition 1.7.

Two π -points ( α K, , α K, ) and ( β L, , β L, ) of the ﬁnite groupscheme G × G are equivalent if and only if the two ideals of H • ( G × G , k ) ker { ( α K, + α K, ) ∗ : H • ( G × G , K ) → H ∗ ( K [ t ] /t p , K ) } ∩ H • ( G × G , k ) ,ker { ( β L, + β L, ) ∗ : H • ( G × G , L ) → H ∗ ( L [ t ] /t p , L ) } ∩ H • ( G × G , k ) are equal.In particular, if α K i is equivalent to β L,i for i = 1 , , then ( α K, , α K, ) and ( β L, , β L, ) are equivalent. The space

Π( ˜ D ( G ( r ) )) of π -point pairs In this section we introduce an extension of “rank varieties” which applies to˜ D ( G ( r ) )-modules. Our construction is a modiﬁcation of the “ π -point construction”of [8]. The strategy we follow is relatively straight-forward. In view of the homo-logical computations of [5], it seems natural to use pairs of π -points ( α K , β K ) with α K : K [ t ] /t p → K [ G r +1) ] and β K : K [ T ] /t p → K G ( r ) . Yet the most natural wayto do this (which would also apply to D ( G ( r ) )) appears to fail because α K ( t ) , β K ( t )need not commute. We work around this diﬃculty by restricting ˜ D ( G ( r ) )-modulesto O ( G ( r ) ). Deﬁnition 2.1. A π -point pair ( α K , β K ) of ˜ D ( G ( r ) ) is a pair of maps of K -algebrasfor some ﬁeld extension K/k , α K : K [ t ] /t p → K [ G ( r +1) / G ( r ) ] ֒ → K [ G ( r +1) ] , β K : K [ t ] /t p → K G ( r ) , UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 7 such that β K either sends t to 0 or is a π -point of G ( r ) , α K either sends t to 0 oris ﬂat, and at least one of α K , β K is ﬂat.The following lemma provides an essential property of a π -point pair ( α K , β K ). Lemma 2.2. A π -point pair ( α K , β K ) of ˜ D ( G ( r ) ) determines a ﬂat map α K + β K : K [ t ] /t p → O ( G ( r ) ) K , t α K ( t ) β K ( t ) whose composition with i O : O ( G ( r ) ,K ) ֒ → ˜ D ( G ( r ) ,K ) is also ﬂat.Proof. We consider the composition K [ t ] ∆ → K [ t ] /p ⊗ α K ⊠ β K → K [ G ( r +1) / G ( r ) ] ⊗ K G ( r ) = O ( G ( r ) ) K , t α K ( t ) ⊗ β K ( t ) . Since α K ( t ) , β K ( t ) commute in O ( G ( r ) ) K , this composition factors through α K + β K : K [ t ] /t p → O ( G ( r ) ) K . Since ∆ : K [ t ] /t p ∆ → K [ t ] /p ⊗ K [ t ] /t p is ﬂat, sinceboth α K and β K are either ﬂat or “trivial”, since O ( G ( r ) ) is ﬂat over each ofits tensor factors, and since α K ⊠ β K is ﬂat if α K ( t ) = 0 = β K ( t ), we concludethat α K + β K : K [ t ] /t p → O ( G ( r ) ) is ﬂat. By Proposition 1.5, this implies thecomposition i O ◦ ( α K + β K ) : K [ t ] /t p → ˜ D ( G ( r ) ) is also ﬂat. (cid:3) The following deﬁnition of equivalence for π -point pairs ( α K , β K ) , ( α ′ L , β ′ L ) ismotivated by Proposition 1.7. Deﬁnition 2.3.

Two π -point pairs ( α K , β K ) , ( α ′ L , β ′ L ) of ˜ D ( G ( r ) ) are said to beequivalent (respectively, O ( G ( r ) )-equivalent) if there exists some common ﬁeld ex-tension Ω of both K and L such that for all ﬁnite dimensional ˜ D ( G ( r ) )-modules M (resp., all ﬁnite dimensional O ( G ( r ) )-modules M ) ( α Ω + β Ω ) ∗ ( M Ω ) is projective asan Ω[ t ] /t p -module if and only if ( α ′ Ω + β ′ Ω ) ∗ ( M Ω ) is projective.We denote by Π( ˜ D ( G ( r ) )) the set of equivalence classes of π -point pairs of˜ D ( G ( r ) ), and we denote by Π( O ( G ( r ) )) the set of O ( G ( r ) )-equivalence classes of π -point pairs of ˜ D ( G ( r ) ). For any ˜ D ( G ( r ) )-module M , we deﬁne the Π -pair set Π( ˜ D ( G ( r ) )) M ⊂ Π( ˜ D ( G ( r ) ))to be the subset of equivalence classes of π -point pairs of ˜ D ( G ( r ) ) such that ( α K + β K ) ∗ ( M ) is not projective. Similarly, for any O ( G ( r ) )-module M ′ , we deﬁneΠ( O ( G ( r ) )) M ′ ⊂ Π( O ( G ( r ) )) to be the subset of equivalence classes of π -point pairsof ˜ D ( G ( r ) ) such that ( α K + β K ) ∗ ( M ′ ) is not projective. Proposition 2.4.

The natural surjection Π ˜ D : Π( O ( G ( r ) )) ։ Π( ˜ D ( G ( r ) )) is abijection.Moreover, for any ﬁnite dimensional ˜ D ( G ( r ) ) -module M , Π ˜ D restricts to a bi-jection Π ˜ D,M : Π( O ( G ( r ) )) M ∼ → Π( ˜ D ( G ( r ) )) M . Proof.

If ( α K , β K ) , ( α ′ L , β ′ L ) are O ( G ( r ) )-equivalent, then they are necessarily equiv-alent since equivalence involves the same condition as O ( G ( r ) )-equivalence exceptthat it only requires consideration of ﬁnite dimensional ˜ D ( G ( r ) )-modules. Thus, wehave well deﬁned, surjective map Π( O ( G ( r ) )) ։ Π( ˜ D ( G ( r ) )). ERIC M. FRIEDLANDER

To prove injectivity, let ( α K , β K ) , ( α ′ L , β ′ L ) represent distinct points in Π( ˜ D ( G ( r ) ))and choose some ﬁnite dimensional O ( G ( r ) )-module N such that ( α Ω + β Ω ) ∗ ( N Ω ) isprojective whereas ( α ′ Ω + β ′ Ω ) ∗ ( N Ω ) is not projective for some some common ﬁeld ex-tension Ω of both K and L . Consider the ˜ D ( G ( r ) )-module M ≡ ˜ D ( G ( r ) ) ⊗ O ( G ( r ) ) N and set N ′ ≡ i ∗ O ( M ). By Proposition 1.5, N ′ is isomorphic to a direct sum of copiesof N (indexed by a k -basis of k [ G ( r ) ]). Thus, ( α Ω + β Ω ) ∗ ( N ′ Ω ) is projective whereas( α ′ Ω + β ′ Ω ) ∗ ( N ′ Ω ) is not projective. Since N ′ is the restriction of the D ( G ( r ) )-module M , we conclude that the images of ( α K , β K ) , ( α ′ L , β ′ L ) in Π( ˜ D ( G ( r ) )) are distinct.Finally, the bijectivity of Π ˜ D,M follows immediately from the deﬁnitions and thebijectivity of Π ˜ D . (cid:3) We next observe that the condition of equivalence of π -point pairs is equiva-lent to the seemingly stronger equivalence relation obtained by dropping the ﬁnitedimensionality condition on M in Deﬁnition 2.3. Proposition 2.5.

Two π -point pairs ( α K , β K ) , ( α ′ L , β ′ L ) of ˜ D ( G ( r ) ) (respectively O ( G ( r ) ) ) are equivalent if and only if there exists some common ﬁeld extension Ω ofboth K and L such that for all ˜ D ( G ( r ) ) -modules M (resp., all O ( G ( r ) ) -modules M ) ( α Ω + β Ω ) ∗ ( M Ω ) is projective as an Ω[ t ] /t p -module if and only if ( α ′ Ω + β ′ Ω ) ∗ ( M Ω ) is projective.Proof. Denote by ˜Π( ˜ D ( G ( r ) )) (respectively, ˜Π( O ( G ( r ) ))) the set of equivalenceclasses of π -point pairs of ˜ D ( G ( r ) ) (resp., O ( G ( r ) )) using the equivalence relationsof the statement of this proposition. We readily verify that we have a commutativesquare of sets(2.5.1) ˜Π( O ( G ( r ) )) Π ˜ D / / ˜Π O (cid:15) (cid:15) ˜Π( ˜ D ( G ( r ) )) ˜Π D (cid:15) (cid:15) Π( O ( G ( r ) )) Π D / / Π( ˜ D ( G ( r ) )) . The map Π D is a bijection as seen in Proposition 2.4; the proof of that propositionalso proves that Π ˜ D is a bijection. The fact that ˜Π O is a bijection follows from [8,Thm 4.6] applied to the ﬁnite group scheme ( g ( r ) ) (1) × G ( r ) (see Proposition 2.7below). Thus, the fact that ˜Π D is a bijection follows from the surjectivity of Π ˜ D and a simple diagram chase. (cid:3) The following properties of M Π( ˜ D ( G ( r ) )) M are each veriﬁed by restrictingalong ﬂat maps ( α Ω + β Ω ) ∗ associated to π -point pairs ( α K , β K ), thereby reducingeach assertion to the corresponding assertion about K [ t ] /t p -modules. Proposition 2.6.

Let M , M , M be ˜ D ( G ( r ) ) -modules. (1) Π( ˜ D ( G ( r ) )) k = Π( ˜ D ( G ( r ) )) . (2) If M is a projective ˜ D ( G ( r ) ) -module, then Π( ˜ D ( G ( r ) )) M = ∅ . (3) If → M → M → M → is exact and if σ is a permutation of { , , } ,then Π( ˜ D ( G ( r ) )) M σ (1) ⊂ Π( ˜ D ( G ( r ) )) M σ (2) ∪ Π( ˜ D ( G ( r ) )) M σ (3) . (4) Π( ˜ D ( G ( r ) )) M ⊕ M = Π( ˜ D ( G ( r ) )) M ∪ Π( ˜ D ( G ( r ) )) M . UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 9

We proceed to show that Π( O ( G ( r ) )) can be “identiﬁed” with the set of Zariskipoints of the π -point scheme of the ﬁnite group scheme ( g ( r ) ) (1) × G ( r ) introducedbelow. Proposition 2.7.

Let g ≃ ( G a ) × dim ( g ) denote the vector group scheme associatedto the underlying vector space of g = Lie ( G ) . As an algebra O ( G ( r ) ) is isomorphicto the group algebra of the inﬁnitesimal group scheme ( g ( r ) ) (1) × G ( r ) .Choose an isomorphism of k -algebras O ( G ( r ) ) ∼ → k (( g ( r ) ) (1) × G ( r ) ) . Then send-ing a π -point pair ( α K , β K ) of ˜ D ( G ( r ) ) to α K + β K : K [ t ] /t p → O ( G ( r ) ,K ) ∼ → K (( g ( r ) ) (1) × G ( r ) ) determines a bijection (2.7.1) Π O : Π( O ( G ( r ) )) ∼ → Π(( g ( r ) ) (1) × G ( r ) ) , where we abuse notation by interpreting the right hand side as the set of (Zariski)points of the indicated Π -point scheme.Moreover, for every O ( G ( r ) ) -module M , Π O restricts to a bijection (2.7.2) Π O,M : Π( O ( G ( r ) )) M ≃ Π(( g ( r ) ) (1) × G ( r ) ) M . Proof.

Since ( g ( r ) ) (1) is an abelian unipotent ﬁnite group scheme, since β K is a π -point of G ( r ) and thus factors through an abelian unipotent subgroup scheme, andsince α K + β K : K [ t ] /t p → KG is ﬂat by Lemma 2.2, we conclude that α K + β K is a π -point of the ﬁnite group scheme ( g ( r ) ) (1) × G ( r ) . Moreover, the discussionpreceding shows that every point of Π(( g ( r ) ) (1) × G ( r ) ) (i.e., every equivalence classof π -points of g ( r ) ) (1) × G ( r ) ) is represented by such a ﬂat map α K + β K associatedto a π -point pair of ˜ D ( G ( r ) ).Thus, to prove that Π O of (2.7.1) is well deﬁned and injective (as well as sur-jective), it suﬃces to show that two π -point pairs ( α K , β K ) , ( α ′ K , β ′ K ) of ˜ D ( G ( r ) )are O ( G ( r ) )-equivalent if and only if α K + β K , α ′ K + β ′ K are equivalent π -points of( g ( r ) ) (1) × G ( r ) . This follows immediately from Deﬁnition 2.3, Proposition 1.7, andthe bijection Ψ g ( r ) ) (1) × G ( r ) of (1.6.1).The fact that (2.7.1) restricts to a bijection Π( O ( G ( r ) )) M ∼ → Π(( g ( r ) ) (1) × G ( r ) ) M follows immediately from the deﬁnitions (i.e., from Deﬁnition 2.3 plusProposition 2.4 and [7, Defn 3.2]). (cid:3) The tensor product property for ˜ D ( G ( r ) ) -modules Propositions 2.4 and 2.7 identify M Π( ˜ D ( G ( r ) )) M with M Π(( g ( r ) ) (1) × G ( r ) ) M . This immediately tells us that M Π( ˜ D ( G ( r ) ) M satisﬁes most of theproperties required for a good theory of support varieties of ˜ D ( G ( r ) )-modules. How-ever, this identiﬁcation does not imply that M Π( ˜ D ( G ( r ) ) M satisﬁes the “tensorproduct property” (namely, that Π( ˜ D ( G ( r ) ) M ⊗ M ′ = Π( ˜ D ( G ( r ) ) M ∩ Π( ˜ D ( G ( r ) ) M ′ )because O ( G ( r ) ,K ) ∼ → K (( g ( r ) ) (1) × G ( r ) ) is not a map of Hopf algebras. In thissection, we prove this tensor product property.The following proposition of [7] is the key to our proof of Theorem 3.4 establishingthe tensor product property for M Π( ˜ D ( G ( r ) )) M . Proposition 3.1. [7, Prop 2.2]

Let V be a k -vector space and α, β, γ be pairwisecommuting V -endomorphisms. Assume further α, β are p -nilpotent and γ is p r -nilpotent in End k ( V ) for some ≥ . Then V is projective as a k [ u ] /u p -modulewhere the action of u is given by α if and only if it is projective as a k [ v ] /v p modulewhere the action of v is given by α + βγ . We apply Proposition 3.1 to prove the following comparison of pull-backs alonga given π -point of tensor products of A -modules with two diﬀerent A -module struc-tures determined by diﬀerent coproducts on A . Proposition 3.2.

Let A be a ﬁnite dimensional, local, commutative k -algebra withmaximal ideal m satisfying the property that a p = 0 for all a ∈ m . Assume that A is equipped with two A -linear coproducts ∆ , ∆ ′ : A → A ⊗ A with the property that (3.2.1) (∆( a ) − a ⊗ − ⊗ a ) , (∆ ′ ( a ) − a ⊗ − ⊗ a ) ∈ m ⊗ m , ∀ a ∈ m . Let

M, M ′ be A -modules and let M ⊠ M ′ be the A ⊗ A module given as the externaltensor product. Then for any ﬂat map α : k [ t ] /t p → A , the pull-back of M ⊠ M ′ via ∆ ◦ α is a projective k [ t ] /t p -module if and only if the pull-back of M ⊠ M ′ via ∆ ′ ◦ α is a projective k [ t ] /t p -module.More generally, let H be ﬁnite dimensional, local, commutative Hopf algebrawith maximal ideal m H such that c p = 0 for all c ∈ m H and equip A ⊗ H with thecoproducts ∆ ⊗ ∆ H , ∆ ′ ⊗ ∆ H . Consider ( A ⊗ H ) -modules M, M ′ , and let M ⊠ M ′ denote the ( A ⊗ H ) ⊗ ( A ⊗ H ) module given as the external tensor product. Thenfor any ﬂat map α : k [ t ] /t p → A ⊗ H , the pull-back of M ⊠ M ′ via (∆ ⊗ ∆ H ) ◦ α isa projective k [ t ] /t p module if and only if the pull-back of M ⊠ M ′ via (∆ ′ ⊗ ∆ H ) ◦ α is a projective k [ t ] /t p -module.Proof. We apply Proposition 3.1 to the k -vector space V = M ⊗ M ′ and considertriples of p -nilpotent elements in the image of A ⊗ A → End k ( V ) given by viewing V as the external tensor product M ⊠ M ′ (and thus an A ⊗ A module for thecommutative k -algebra A ⊗ A ).Consider an element a ∈ m ⊂ A and write ∆( a ) − ∆ ′ ( a ) = P ti =1 b i ⊗ b ′ i ∈ m ⊗ m . Observe that ∆( a ) − ∆ ′ ( a ) ∈ A ⊗ A acts on V = M ⊠ M ′ as the sum of the actionscommuting elements ( b i ⊗ · (1 ⊗ b ′ i ) , ≤ i ≤ , t each of which is p -nilpotent.. Weapply Proposition 3.1 with V = M ⊠ M ′ successively to the triples α = ((∆ ′ ( a ) + i X s =1 b s ⊗ b ′ s ) , β = ( b i +1 ⊗ , γ = (1 ⊗ b ′ i +1 )with i + 1 ≤ t to conclude that V is projective for k [ u ] /u p with u acting as ∆ ′ ( a ) + P is =1 b s ⊗ b ′ s if and only if V is projective for k [ v ] /t p with v acting as ∆ ′ ( a ) + P i +1 s =1 b s ⊗ b ′ s . This establishes the ﬁrst assertion.We extend this argument to A ⊗ H , a commutative algebra with coproducts∆ ⊗ ∆ H , ∆ ′ ⊗ ∆ H . We denote by n ⊂ A ⊗ H the maximal ideal of A ⊗ H generatedby m ⊗ , ⊗ m H , and observe that every element in n has p -th power 0. Moreover,(∆ ⊗ ∆ H ) − (∆ ′ ⊗ ∆ H ) = (∆ − ∆ ′ ) ⊗ ∆ H applied to a ⊗ ∈ n lies in n ⊗ n whereas(∆ − ∆ ′ ) ⊗ ∆ H applied to 1 ⊗ h ∈ ⊗ m H is 0. Consequently,((∆ ⊗ ∆ H ) − (∆ ′ ⊗ ∆ H ))( b ) ∈ n ⊗ n , ∀ b ∈ n . Thus we may repeat the preceding argument with ( A, m ) replaced by ( A ⊗ H, n ) tocomplete the proof. (cid:3) UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 11

Deﬁnition 3.3.

Consider O ( G ( r ) )-modules M, M ′ .We denote by M ⊗ M ′ the O ( G ( r ) )-module determined by the coproduct of O ( G ( r ) ) deﬁned as by restriction of the coproduct of ˜ D ( G ( r ) ) (see Proposition 1.4).We denote by M ⊗ G M ′ the k -vector space M ⊗ M ′ equipped with the O ( G ( r ) ) ≃ ( k (( g ( r ) ) (1) ) × G ( r ) )-module structure determined by the coproduct for the groupalgebra k (( g ( r ) ) (1) ) × G ( r ) ) of the ﬁnite group scheme ( g ( r ) ) (1) ) × G ( r ) .In the following theorem, we prove the “tensor product property” for ˜ D ( G ( r ) )-modules; namely, M Π( ˜ D ( G ( r ) )) M sends tensor product of modules to inter-section of the supports of its tensor factors. Theorem 3.4.

Let ( α K , β K ) be a π -point pair of ˜ D ( G ( r ) ) as in Deﬁnition 2.1 andlet M, M ′ be O ( G ( r ) ) -modules. Then ( α K + β K ) ∗ ( M K ⊗ M ′ K ) is a free K [ t ] /t p -module if and only if ( α K + β K ) ∗ ( M K ⊗ G M ′ K ) is a free K [ t ] /t p -module. Thus, (3.4.1) Π( ˜ D ( G ( r ) )) M ⊗ M ′ = Π( ˜ D ( G ( r ) )) M ∩ Π( ˜ D ( G ( r ) )) M ′ . Proof.

We recall that any β K : K [ t ] /t p → K G ( r ) is equivalent to a map whichfactors through a map of group algebras induced by a map of ﬁnite group schemes G a ( r ) ,L → G ( r ) ,L [7, Prop 4.2]. By replacing K if necessary by a ﬁeld extension K ′ /K , we may assume K = L . Thus when comparing restrictions along a given π -point pair, we may restrict to K [ G ( r +1) / G ( r ) ] ⊗ K G a ( r ) ֒ → O ( G ( r ) ) ⊗ K andreplace M, M ′ by their restrictions to K [ G ( r +1) / G ( r ) ] ⊗ K G a ( r ) .We apply Proposition 3.2 with A equal to K [ G ( r +1) / G ( r ) ] and H = K G a ( r ) ,and equip A ⊗ K G a ( r ) with the two coproduct structures obtained by restrictionof the two coproduct structures on O ( G ( r ) ) discussed in Deﬁnition 3.3. As shownin [13, I.2.4], ∆ , ∆ ′ satisfy condition (3.2.1) of Proposition 3.2. In our applicationof Proposition 3.2, we take α : K [ t ] /t p → K [ G ( r +1) / G ( r ) ] ⊗ K G a ( r ) to be thecomposition of the coproduct K [ t ] /t p → K [ t ] /t p ⊗ K [ t ] /t p , t t ⊗ ⊗ t and theexternal tensor product α K ⊗ β K : K [ t ] /t p → K [ t ] /t p ⊗ K [ t ] /t p → K [ G ( r +1) / G ( r ) ] ⊗ K G a ( r ) .By applying Proposition 3.2 with these values for A, H, α , we conclude that itsuﬃces to prove the equality (3.4.1) after replacing the tensor product for K [ G ( r +1) / G ( r ) ] ⊗ K G a ( r ) -modules (given by the restriction of the coproduct of O ( G ( r ) )) by the tensor product given by the restriction of the tensor product of( g ( r ) ) (1) × G ( r ) -modules. To conclude the proof of equality (3.4.1), we apply thetensor product property for modules for the (inﬁnitesimal) group scheme G =( g ( r ) ) (1) × G ( r ) [8, Prop 3.2]; in other words, for M ⊗ G M ′ in the notation ofDeﬁnition 3.3. (cid:3) Theorem 3.4 enables topological structures on the sets Π( ˜ D ( G ( r ) )) and Π( O ( G ( r ) )). Corollary 3.5.

Deﬁning a subset of

Π( ˜ D ( G ( r ) )) to be closed if and only if it is ofthe form Π( ˜ D ( G ( r ) )) M ⊂ Π( ˜ D ( G ( r ) )) for some ﬁnite dimensional ˜ D ( G ( r ) ) -module M deﬁnes a topology on Π( ˜ D ( G ( r ) )) . Similarly, deﬁning a subset of Π( O ( G ( r ) )) tobe closed if and only if it is of the form Π( O ( G ( r ) )) M ⊂ Π( O ( G ( r ) )) for some ﬁnitedimensional O ( G ( r ) ) -module M deﬁnes a topology on Π( O ( G ( r ) )) .With these topologies, the bijection Π ˜ D : Π( ˜ D ( G ( r ) )) ։ Π( O ( G ( r ) )) of Propo-sition 2.4 is a homeomorphism. Proof.

The asserted formulation of a topology on Π( ˜ D ( G ( r ) )) is justiﬁed by Propo-sition 2.6(4) and Theorem 3.4. This applies as well to justiﬁy the topology onΠ( O ( G ( r ) )), since i O : O ( G ( r ) ) → ˜ D ( G ( r ) ) is a map of Hopf algebras. (cid:3)

4. Ψ ˜ D : Π( ˜ D ( G ( r ) )) → P ( H • ( ˜ D ( G ( r ) ) , k ))Cohomological support varieties can be readily deﬁned for any ﬁnite dimensionHopf algebra H whose cohomology is ﬁnitely generated (i.e., such that H ∗ ( H, k )is a ﬁnitely generated k -algebra and that H ∗ ( H, M ) is a ﬁnite H ∗ ( H, k )-modulefor any ﬁnite dimensional H -module). In this section, we exhibit in Proposition4.6 a continuous map Ψ ˜ D from the π -point pair space Π( ˜ D ( G ( r ) )) introduced inSection 2 to the (projectivized) cohomology support variety P ( H • ( ˜ D ( G ( r ) ) , k )).To do this, we utilize the subalgebra O ( G ( r ) ) ⊂ ˜ D ( G ( r ) ) and the related ﬁnitegroup scheme G ( r ) × ( g ( r ) ) (1) , relying upon the isomorphism Π( G ( r ) × ( g ( r ) ) (1) ) ∼ → P ( H • ( G ( r ) × ( g ( r ) ) (1) , k ).Associated to a ﬁnite dimensional ˜ D ( G ( r ) )-module M , we consider the annihi-lator ideal I M ⊂ H • ( ˜ D ( G ( r ) ) , k ) of the H • ( ˜ D ( G ( r ) ) , k )-module Ext ∗ ˜ D ( G ( r ) ) ( M, M )which equals the annihilator of 1 ∈ Ext ∗ ˜ D ( G ( r ) ) ( M, M ) which in turn equals thekernel of the graded map of k -algebras H • ( ˜ D ( G ( r ) ) , k ) → Ext ∗ ˜ D ( G ( r ) ) ( M, M ). Weshall consider the projective scheme P ( H • ( ˜ D ( G ( r ) ) , k )) whose (Zariski) points are(non-trivial) homogenous prime ideals of H • ( ˜ D ( G ( r ) ) , k ) and the reduced projec-tive scheme P ( H • ( ˜ D ( G ( r ) ) , k )) M whose points are (non-trivial) homogenous primeideals of H • ( ˜ D ( G ( r ) ) , k ) containing I M .Consider the quotient map G ( r +2) ։ ( G ( r +1) ) (1) , the image of F r +1 : G ( r +2) → ( G ( r +1) ) ( r +2) whose kernel is G ( r +1) . This quotient map determines the ﬂat mapof k -algebras k [( G ( r +1) ) (1) ] ≃ k [ G ( r +2 ) / G ( r +1) ] → k [ G ( r +2) ]with the property that k [ G ( r +1) ] ≃ k [ G ( r +2) ] ⊗ k [ G ( r +2 ) / G ( r +1) ] k .The following proposition is a summary of a central technique of [5]. Proposition 4.1.

The deformation k [ G ( r +2 ) / G ( r +1) ] → k [ G ( r +2) ] of k [ G ( r +1) ] (parametrized by k [ G ( r +2 ) / G ( r +1) ] ≃ k [( G ( r +1) ) (1) ] ) determines a G -equivariantmap (4.1.1) g ( r +1) → H ( k [ G ( r +1) ] , k ) . The deformation k [ G ( r +2 ) / G ( r +1) ] k G ( r ) → k [ G ( r +2) ] k G ( r ) (also parametrizedby k [ G ( r +2 ) / G ( r +1) ] ) provides a G ( r ) -equivariant lifting of (4.1.1) (4.1.2) σ ˜ D : g ( r +1) → H ( ˜ D ( G ( r ) ) , k ) . . Consequently, we obtain a map of k -algebras (4.1.3) ˜ θ r : H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) deﬁned as the product of the inﬂation map on H ∗ ( G ( r ) , k ) and the algebra extensionof σ ˜ D on S • ( g ( r +1) [2]) . UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 13

Proof.

Each element of the tangent space at the identity of ( G ( r +1) ) (1) (natu-rally identiﬁed with g ( r +1) ) corresponds to a map k [( G ( r +1) ) (1) ] → k [ ǫ ] /ǫ of k -algebras. Extending the deformation k [ G ( r +2 ) / G ( r +1) ] → k [ G ( r +2) ] along sucha map determines a deformation of k [ G ( r +1) ] parametrized by k [ ǫ ] /ǫ . Thus, themap (4.1.1) is a consequence of the naturality of Gerstenhaber’s bijection [12] be-tween deformations of k [ G ( r +1) ] parametrized by k [ ǫ ] /ǫ and the second Hochschildcohomology group HH ( k [ G ( r +1) ]) of k [ G ( r +1) ] together with the natural map HH ( k [ G ( r +1) ]) → H ( k [ G ( r +1) ] , k )) (see [5, Prop 3.4]). The map σ ˜ D of (4.1.2)follows exactly the same way. Moreover, the naturality of Gerstenhaber’s bijectionimplies that σ ˜ D lifts (4.1.1).The construction of ˜ θ r : H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) is implicitin the last statement of this proposition. (cid:3) Finite generation of the cohomology algebra H ∗ ( D ( G ( r ) ) , k ) and the ﬁnitenessof H ∗ ( D ( G ( r ) ) , M ) as an H ∗ ( D ( G ( r ) ) , k )-module (for a ﬁnite dimensional D ( G ( r ) )-module M ) was established in [5, Thm 5.3]. Using Proposition 4.1, we extend thisﬁnite generation property to ˜ D ( G ( r ) ) and O ( G ( r ) ) as stated below; the proof is astraight-forward adaptation of that of [5, Thm 5.3] which in turn is an adaptationof [11, Thm 1.1]. Theorem 4.2.

Consider the map ˜ θ r : H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) of (4.1.3). (1) The composition (4.2.1) i ∗ O ◦ ˜ θ r : H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) → H ∗ ( O ( G ( r ) ) , k ) ≃ H • ( G ( r ) , k ) ⊗ H ∗ ( k [( G ( r ) ) (1) ] , k ) is a split injection with H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) -module complement H ∗ ( G ( r ) , k ) ⊗ Λ > ( g [1]) ⊗ S • ( g ( r +1) [2]) which consists entirely of nilpotent elements (2) For any ﬁnite dimensional ˜ D ( G ( r ) ) -module M , H ∗ ( ˜ D ( G ( r ) ) , M ) is a ﬁnitelygenerated H • ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) -module (with action given by the re-striction along ˜ θ r of the natural action of . H ∗ ( ˜ D ( G ( r ) ) , k ) on H ∗ ( ˜ D ( G ( r ) ) , M ) ). (3) For any ﬁnite dimensional O ( G ( r ) ) -module M , H ∗ ( O ( G ( r ) ) , M ) is also aﬁnitely generated H • ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) -module.Proof. We utilize the spectral sequence(4.2.2) E s,t = H s ( G ( r ) , H t ( k [ G ( r +1) ] , k )) ⇒ H s + t ( ˜ D ( G ( r ) ) , k ) , which can be derived as a special case of Grothendieck’s spectral sequence for thederived functors of a composition of left exact functors exactly as in [5, Prop 5.2].The composition i ∗ O ◦ ˜ θ r restricts to the identity on the tensor factor H ∗ ( G ( r ) , k )by the deﬁnition of ˜ θ r and the fact that the composition k G ( r ) → O ( G ( r ) ) → ˜ D ( G ( r ) ) is the evident inclusion. The deformation construction leading to the deﬁni-tion of ˜ θ r is designed to insure that the composition S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) → H ∗ ( k [ G ( r +1) / G ( r ) ] , k ) sends elements of g ( r +1) [2] to elements of H ( k [ G ( r +1) / G ( r ) ] , k )which generate H ∗ ( k [ G ( r +1) / G ( r ) ] , k ) modulo nilpotent elements. The identiﬁca-tion of the restriction of the map i ∗ O ◦ ˜ θ r to 1 ⊗ S • ( g ( r +1) [2]) and the computation of H ∗ ( k [( G ( r ) ) (1) ] , k ) are given in [5, § H ∗ ( ˜ D ( G ( r ) ) , M ) is a ﬁnitely generated H • ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2])-module for any ﬁnite dimensional ˜ D ( G ( r ) )-module M arises from the natural pairingof (4.2.2) with the corresponding spectral sequence(4.2.3) E s,t ( M ) = H s ( G ( r ) , H t ( k [ G ( r +1) ] , M )) ⇒ H s + t ( ˜ D ( G ( r ) ) , M )with M coeﬃcients (as in the proof of [5, Prop 5.2]). The same argument with˜ D ( G ( r ) ) replaced by O ( G ( r ) ) implies that H ∗ ( O ( G ( r ) ) , M ) is a ﬁnitely generated H • ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2])-module. (cid:3) Corollary 4.3.

The map of commutative k -algebras ˜ θ r induces a ﬁnite, surjectivemap ˜Θ r : Spec H • ( ˜ D ( G ( r ) ) , k ) → Spec H • ( G ( r ) , k ) × g ∗ ( r +1) . Consequently, dim (Spec H • ( ˜ D ( G ( r ) ) , k )) = dim (Spec H • ( G ( r ) , k ) + dim ( g ) . Proof.

By Theorem 4.2, the composition i ∗ ◦ ˜ θ r is injective, so that ˜ θ r is injective;this theorem also tells us that ˜ θ r is a ﬁnite map. Consequently, ˜ θ r is an integralextension which implies that Θ r is ﬁnite and surjective (see [17, Thm 9.3]).The determination of dim (Spec H • ( ˜ D ( G ( r ) ) , k )) follows from the observationthat dim (Spec H • ( G ( r ) , k ) × g ∗ ( r +1) ) = dim (Spec H • ( G ( r ) , k )) + dim ( g ) . (cid:3) We establish the following notation.

Notation 4.4.

We denote by P ( H • ( ˜ D ( G ( r ) ) , k )) (respectively, P ( H • ( O ( G ( r ) ) , k ));respectively, P ( H • ( G ( r ) , k ))) the projective scheme associated to the commutativegraded k -algebra H • ( ˜ D ( G ( r ) ) , k ) (resp., H • ( O ( G ( r ) ) , k ); resp., H • ( G ( r ) , k ))). Thus,the points of P ( H • ( ˜ D ( G ( r ) ) , k )) (resp., P ( H • ( O ( G ( r ) ) , k ); resp., P ( H • ( G ( r ) , k ))) arethe non-trivial homogeneous prime ideals of H • ( ˜ D ( G ( r ) ) , k ) (resp., H • ( O ( G ( r ) ) , k );resp., H • ( G ( r ) , k ))). For any ﬁnite dimensional ˜ D ( G ( r ) )-module M , we denote by P ( H • ( ˜ D ( G ( r ) ) , k )) M ⊂ P ( H • ( ˜ D ( G ( r ) ) , k )) , P ( H • ( O ( G ( r ) )) , k )) M ⊂ P ( H • ( O ( G ( r ) )) , k )) , P ( H • ( G ( r ) , k )) M ⊂ P ( H • ( G ( r ) , k ))the reduced closed subscheme given by the radicals of the ideals ker { H • ( ˜ D ( G ( r ) ) , k ) → Ext ∗ ˜ D ( G ( r ) ) ( M, M ) } ,ker { H • ( O ( G ( r ) ) , k ) → Ext ∗ O ( G ( r ) )) ( M, M ) } ,ker { H • ( G ( r ) , k ) → Ext ∗ G ( r ) ( M, M ) } respectively.The equivalence relation on π -point pairs ( α K , β K ) of ˜ D ( G ( r ) ) (and thus on O ( G ( r ) )) has been designed to enable the validity of the following proposition. UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 15

Proposition 4.5.

Deﬁne Ψ O : Π( O ( G ( r ) )) → P ( H ∗ ( O ( G ( r ) ) , k )) to be the map ofsets sending the equivalence class of ( α K , β K ) ∈ Π( O ( G ( r ) )) to ker { ( α K + β K ) ∗ : H • ( O ( G ( r ) ) K , K ) → H • ( K [ t ] /t p , K ) } ∩ H ∗ ( O ( G ( r ) ) , k ) in P ( H ∗ ( O ( G ( r ) ) , k )) . A choice of isomorphism of k algebras j : O ( G ( r ) ) ∼ → k ( g ( r ) ) (1) × G ( r ) ) as in Proposition 1.6 determines the vertical maps in the followingcommutative square of homeomorphisms (4.5.1) Π( O ( G ( r ) )) Ψ O / / Π O (cid:15) (cid:15) P ( H • ( O ( G ( r ) ) , k )) P O (cid:15) (cid:15) Π( g ( r ) ) (1) × G ( r ) ) Ψ G / / P ( H • ( g ( r ) ) (1) × G ( r ) , k )) whose lower horizontal map is the natural homeomorphism of C for G the ﬁnitegroup scheme ( g ( r ) ) (1) × G ( r ) .Moreover, for every ﬁnite dimensional O ( G ( r ) ) -module M , Ψ O restricts to abijection Ψ O,M : Π( O ( G ( r ) )) M ∼ → P ( H • ( O ( G ( r ) ) , k )) M .Proof. The deﬁnition of Ψ O is compatible with the deﬁnition of Ψ G : Π( g ( r ) ) (1) × G ( r ) ) → P ( H • ( g ( r ) ) (1) × G ( r ) , k )) (see ( g ( r ) ) (1) × G ( r ) and [7, Prop 3.5]), so that(4.5.1) commutes. The fact that Ψ O is well deﬁned follows from the fact thatΨ G is well deﬁned and that the maps Π O , P O are bijections. The topology onΠ( ˜ D ( G ( r ) )) = Π( O ( G ( r ) )) introduced in Theorem 3.4 is equal to that induced bythe bijection Π O , so that this bijection is a homeomorphism. The bijection P O is ahomeomorphism since the isomorphism j induces an isomorphism between the cat-egory of ﬁnite dimensional O ( G ( r ) )-modules and the category of ﬁnite dimensional k ( g ( r ) ) (1) × G ( r ) )-modules.The fact that Ψ O,M is a bijection for any ﬁnite dimensional O ( G ( r ) )-modulefollows from the fact that Ψ O restricts to a bijection Ψ O,M : Π( g ( r ) ) (1) × G ( r ) ) ∼ → P ( H • ( g ( r ) ) (1) × G ( r ) , k )) M . (cid:3) The following proposition complements Proposition 4.5 while introducing Ψ ˜ D . Proposition 4.6.

The composition Ψ ˜ D ≡ P ˜ D ◦ Ψ O ◦ Π − D : Π( ˜ D ( G ( r ) )) ∼ → Π( O ( G ( r ) )) ∼ → P ( H • ( O ( G ( r ) ) , k )) → P ( H • ( ˜ D ( G ( r ) ) , k )) is continuous, ﬁtting in the commutative square (4.6.1) Π( ˜ D ( G ( r ) )) Ψ ˜ D / / P ( H • ( ˜ D ( G ( r ) ) , k ))Π( O ( G ( r ) )) Π ˜ D ≃ O O Ψ O ∼ / / P ( H • ( O ( G ( r ) ) , k )) P ˜ D O O with the properties that Π ˜ D and Ψ O are homeomorphisms, and that the map P ˜ D (induced by i O : O ( G ( r ) ) → ˜ D ( G ( r ) ) ) is a closed immersion.For any ﬁnite dimensional ˜ D ( G ( r ) ) -module M , Ψ ˜ D restricts to (4.6.2) Ψ ˜ D,M : Π( ˜ D ) M ֒ → P ( H • ( ˜ D ( G ( r ) ) , k )) M . Proof.

By Proposition 2.4, Π ˜ D is a homeomorphism, whereas Ψ O is a homeomor-phism by Proposition 4.5. Theorem 4.2(1) implies that i ∗ O is a ﬁnite map and issurjective modulo nilpotents which implies that P O is a closed embedding. Thus,Ψ ˜ D is also a closed embedding and thus is continuous.To prove that Ψ ˜ D (Π( ˜ D ( G ( r ) )) M ) is contained in P ( H • ( ˜ D ( G ( r ) ) , k )) M , we con-sider some ( α K , β K ) representing a point of Π( ˜ D ( G ( r ) )) M ; in other words, assumethat ( α K + β K ) ∗ ( M K ) is not projective. Consider the following commutative square(4.6.3) H • ( ˜ D ( G ( r ) ) , k ) ( α K + β K ) ∗ / / (cid:15) (cid:15) H • ( K [ t ] /t p , K ) (cid:15) (cid:15) Ext ∗ ˜ D ( G ( r ) )) ( M, M ) / / Ext ∗ K [ t ] /t p (( α K + β K ) ∗ ( M K ) , ( α K + β K ) ∗ ( M K ))whose horizontal maps are induced by restriction along α K + β K . Since ( α K + β K ) ∗ ( M K ) is not projective, the right vertical map is injective. A simple diagramchase then implies that ker { ( α K + β K ) ∗ : H • ( ˜ D ( G ( r ) )) , k ) → H • ( K [ t ] /t p , K ) } contains ker { H • ( ˜ D ( G ( r ) )) , k ) → Ext ∗ K [ t ] /t p (( α K , β K ) ∗ ( M K ) , ( α K , β K ) ∗ ( M K )) } , sothat Ψ ˜ D applied to the class of ( α K , β K ) lies in P ( H • ( ˜ D ( G ( r ) ) , k )) M . (cid:3)

5. Π( ˜ D ( G ( r ) )) for G admitting a quasilogarithm One missing aspect of our understanding of Ψ ˜ D in (4.6.1) is whether or not itis a homeomorphism. By Proposition 4.6, this is equivalent to whether or not P ˜ D is surjective. With the added hypothesis that G admits a quasilogarithm and that p r +1 > dim ( G ), we prove this surjectivity of P ˜ D in Theorem 5.4. This enables usto conclude in Theorem 5.6 that M Π( ˜ D ( G ( r ) ) M detects projectivity: a ﬁnitedimensional ˜ D ( G ( r ) )-module M is projective if and only if Π( ˜ D ( G ( r ) )) M is empty.We ﬁrst recall the deﬁnition of a quasilogarithm for G . Deﬁnition 5.1. [14], [5, Defn 6.2] Let G be a linear algebraic group over k . Aquasilogarithm for G is a G -equivariant map of k -schemes(5.1.1) L : G → Spec S • ( g ∗ ) ≡ g such that L ( e G ) = 0 and such that the diﬀerential d e G L : T e G G → T g = g is theidentity.We denote by I r ⊂ S • ( g ∗ ) the G -stable ideal generated by { f p r , f ∈ g ∗ } . Example 5.2. [2, Lem C3], [5, 6.4] If G is a simple algebraic group for which p is very good, then G admits a quasilogarithm. Furthermore any Borel subgroupof G also admits a quasilogarithm. If U is a unipotent subgroup of a semi-simplealgebraic group G which is normalized by a maximal torus and if p is greater thanthe nilpotent class of U , then U admits a quasilogarithm.For us, a critical consequence of the condition that G admits a quasilogarithm L : G → g is the following result, a summary of the discussion prior to and includingLemma 6.6 of [5]. UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 17

Proposition 5.3.

Assume that G is a linear algebraic group smooth over k whichadmits a quasilogarithm. The G -equivariant map on coordinate algebras associatedto the composition L ◦ i : G ( r ) ⊂ G → g has the form S • ( g ∗ ) → S • ( g ∗ ) /I r ∼ → k [ G ( r ) ] . Moreover, the isomorphism S • ( g ∗ ) /I r ∼ → k [ G ( r ) ] induces an isomorphism of k -algebras S • ( g ∗ ) /I r k G ( r ) ∼ → k [ G ( r ) ] k G ( r ) ≃ D ( G ( r ) ) . Similarly, L determines the isomorphism of k -algebras S • ( g ∗ ) /I r +1 k G ( r ) ∼ → k [ G ( r +1) ] k G ( r ) ≃ ˜ D ( G ( r ) ) . These smash products algebras S • ( g ∗ ) /I r k G ( r ) , S • ( g ∗ ) /I r +1 k G ( r ) are asso-ciated to the coadjoint action of G ( r ) on g ∗ . They both admit a G ( r ) -equivariantgrading, with g ∗ homogenous of degree 1 and k G ( r ) homogeneous of degree 0. Using the structure of a quasilogarithm on G , we obtain the following strength-ening of Corollary 4.3. The proof of this theorem is essentially that of [5, Thm 6.9]which gives the analogous result for D ( G ( r ) ). Theorem 5.4.

Assume that G is a linear algebraic group smooth over k whichadmits a quasilogarithm. Further assume that p r +1 > dim ( G ) . Then the map ˜ θ r : H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) of Theorem 4.2 is a split injection with H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) -module comple-ment consisting entirely of nilpotent elements.Similarly, the map i ∗ O : H ∗ ( ˜ D ( G ( r ) ) , k ) → H ∗ ( O ( G ( r ) ) , k ) has nilpotent kernelin H ∗ ( ˜ D ( G ( r ) ) , k ) ; and the image in H ∗ ( O ( G ( r ) ) , k ) has H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) -module complement consisting entirely of nilpotent elements.Thus, P O : P ( H ∗ ( O ( G ( r ) ) , k )) → P ( H ∗ ( ˜ D ( G ( r ) ) , k )) is a homeomorphism. Con-sequently, Ψ ˜ D : Π( ˜ D ( G ( r ) )) → P ( H • ( ˜ D ( G ( r ) ) , k )) is also a homeomorphismProof. The statement that ˜ θ r is a split injection is that of Theorem 4.2(1).The gradings on S • ( g ∗ ) /I r k G ( r ) , S • ( g ∗ ) /I r +1 k G ( r ) (and thus also on O ( G ( r ) ))determine internal gradings on the spectral sequences of Theorem 4.2; we viewthese gradings as Z /p r +1 -gradings. As in [5, Lem 6.7], the internal grading on g ( r +1) [2] ⊂ H ( k [ G ( r +1) ] , k ) is p r +1 , hence ≡ mod p r +1 and the internal gradingof g [1] ⊂ H ( k [ G ( r +1) ] , k ) is 1.We conclude that the map ˜ θ r of Theorem 4.2 is an isomorphism onto the de-gree ≡ D ( G ( r ) ). As argued in the proof of [5,Thm 6.9], the hypothesis that p r +1 > dim ( G ) implies that this image has an H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2])-module complement in H ∗ ( ˜ D ( G ( r ) ) , k ) which is con-tained in the nilradical.We may repeat this argument with the spectral sequence (4.2.2) for ˜ D ( G ( r ) )mapping to the spectral sequence(5.4.1) E s,t = H s ( G ( r ) , H t ( k [ G ( r +1) / G ( r ) ] , k )) ⇒ H s + t ( O ( G ( r ) ) , k ) . Thus, the composition H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2]) → H ∗ ( ˜ D ( G ( r ) ) , k ) → H ∗ ( O ( G ( r ) ) , k )is also a split injection with H ∗ ( G ( r ) , k ) ⊗ S • ( g ( r +1) [2])-module complement con-sisting entirely of nilpotent elements. This readily implies the second statement concerning i ∗ O : H ∗ ( ˜ D ( G ( r ) ) , k ) → H ∗ ( O ( G ( r ) ) , k ). Consequently, P O is a bijec-tion as well as a closed embedding, and thus is a homeomorphism. This, togetherwith Proposition 4.6 implies that Ψ ˜ D is also a homeomorphism. (cid:3) We “recall” the following projectivity criterion for a ﬁnite dimensional ˜ D ( G ( r ) )-module in terms of cohomological support varieties. Proposition 5.5. (see [6, Prop 1.5] ) Let M be a ﬁnitely generated ˜ D ( G ( r ) ) -module.Then M is projective if and only if P ( H • ( ˜ D ( G ( r ) ) , k )) M = ∅ . Similarly, if M is a ﬁnitely generated O ( G ( r ) ) -module, them M is projective ifand only if P ( H • ( O ( G ( r ) ) , k )) M = ∅ . Proof.

As shown in [15] for any ﬁnite dimensional Hopf algebra with ﬁnitely gen-erated cohomology, ˜ D ( G ( r ) ) and O ( G ( r ) ) are Frobenius algebras. This enables theproof of [6, Prop 1.5] to apply, even though that result was stated only for Hopfalgebras which are restricted enveloping algebras. (cid:3) Combining Theorem 5.4, with Proposition 5.5, we conclude that the supporttheory M Π( ˜ D ( G ( r ) )) M provides a test for projectivity. Theorem 5.6.

Assume the hypotheses on G of Theorem 5.4 and consider a ﬁnitedimensional ˜ D ( G ( r ) ) -module M . Then M is projective as a ˜ D ( G ( r ) ) -module if andonly if Π( ˜ D ( G ( r ) )) M = ∅ . Proof.

Assume that M is projective as a ˜ D ( G ( r ) )-module. Then the pull-back along( α K + β K ) ∗ of M is projective (and thus free) as a K [ t ] /t p -module for any π -pointpair ( α K , β K ), so that Π( ˜ D ( G ( r ) )) M = ∅ .Conversely, assume that Π( ˜ D ( G ( r ) )) M = ∅ . Since all the maps of the com-mutative square (4.6.1) are bijective (in fact, homeomorphisms), we conclude that P H • (( ˜ D ( G ( r ) ) , k ) M = ∅ . Thus, by Proposition 5.5, M is a projective ˜ D ( G ( r ) )-module. (cid:3) The arguments of Sections 4 and 5 apply to ˜ D ( G ′ ) ≡ k [ G ( r +1) ] G ′ ⊂ ˜ D ( G ( r ) )with merely a change of notation, as we state explicitly in the following corollary(and use in the next section). Corollary 5.7.

Let G ′ ֒ → G ( r ) be a subgroup scheme of G ( r ) . Theorems 4.2 and5.4 remain valid upon replacing i O : O ( G ( r ) ) → ˜ D ( G ( r ) ) by i O ′ : k [ G ( r +1) / G ( r ) ] kG ′ ֒ → k [ G ( r +1) ] kG ′ . Consequently, as in Proposition 5.5 and Theorem 5.6, a ﬁnite dimensional k [ G ( r +1) ] kG ′ -module M is projective if and only if its restriction along i O ′ is aprojective k [ G ( r +1) / G ( r ) ] kG ′ -module. Relating

Π( ˜ D ( G ( r ) )) M to P ( H • ( ˜ D ( G ( r ) ) , k )) M In this section, we verify for certain ﬁnite dimensional ˜ D ( G ( r ) )-modules M thatthe natural inclusion Ψ ˜ D,M : Π( ˜ D ( G ( r ) )) M ֒ → P ( H • ( ˜ D ( G ( r ) ) , k )) M is a bijectionand thus a homeomorphism.We begin with an important class of examples (due to J. Carlson; see [3]) ofmodules constructed to have given cohomological support. Let P ∗ → k be a minimal UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 19 resolution of k as a ˜ D ( G ( r ) )-module and deﬁne Ω n ˜ D ( G ( r ) ) ( k ) for n ≥ P n − → P n − (where P − is set equal to k ). Recall that ζ ∈ H n ( ˜ D ( G ( r ) ) , k )naturally corresponds to a map ˜ ζ : Ω n ˜ D ( G ( r ) ) ( k ) → k in the stable module category stmod ( ˜ D ( G ( r ) )). The ˜ D ( G ( r ) )-module L ζ is deﬁned by the short exact sequence of˜ D ( G ( r ) )-modules(6.0.1) 0 → L ζ → Ω n ˜ D ( G ( r ) ) ( k ) ˜ ζ → k → . For ξ ∈ H n ( O ( G ( r ) ) , k ) with corresponding map ˜ ζ : Ω nO ( G ( r ) ) k ) → k in the stablemodule category stmod ( O ( G ( r ) )), we similarly deﬁne the O ( G ( r ) )-module L ξ bythe short exact sequence of O ( G ( r ) )-modules0 → L ξ → Ω nO G ( r ) ) ( k ) ˜ ξ → k → . We restrict attention to n = 2 d >

0. For such n = 2 d , Ω dk [ t ] /t p ( k ) is stablyisomorphic to k . Proposition 6.1.

Consider the ˜ D ( G ( r ) ) -module L ζ associated to a cohomologyclass ζ ∈ H d ( ˜ D ( G ( r ) ) , k ) . Then a π -point pair ( α K , β K ) of ˜ D ( G ( r ) ) satisﬁes thecondition ( α K + β K ) ∗ ( L ζ ) is projective as a K [ t ] /t p -module if and only if ( α K + β K ) ∗ ( ζ ) = 0 ∈ H d ( K [ t ] /t p , K ) .Consequently, Π( ˜ D ( G ( r ) )) L ζ consists of equivalence classes of π -point pairs ( α K , β K ) such that ( α K + β K ) ∗ ( ζ ) = 0 ∈ H d ( K [ t ] /t p , K ) .Proof. Carlson’s identiﬁcation of the support variety of L ζ in the context of a groupalgebra of a ﬁnite group applies to ˜ D ( G ( r ) ), for his argument merely requires thatthe algebra ˜ D ( G ( r ) ) be a Frobenius algebra and that α K + β K : K [ t ] /t p → ˜ D ( G ( r ) )be ﬂat. The key observation of Carlson’s proof is that the pullback along α K + β K of (6.0.1) is split if and only if ( α K + β K ) ∗ ( ζ ) = 0 ∈ H d ( K [ t ] /t p , K ). (cid:3) Using Proposition 4.6 and Theorem 5.4, we verify that Ψ ˜ D,L ζ is a homeomor-phism. Proposition 6.2.

Assume the hypotheses on G of Theorem 5.4. Consider ζ ∈ H • ( ˜ D ( G ( r ) ) , k ) and denote by ζ O the restriction of ζ to H • ( O ( G ( r ) ) , k ) . Then (1) The restriction of the ˜ D ( G ( r ) ) -module L ζ to O ( G ( r ) ) is stably equivalent to L ζ O . (2) P i ∗ O restricts to a bijection P O : P ( H • ( O ( G ( r ) ) , k )) L ζO ∼ → P ( H • ( ˜ D ( G ( r ) ) , k )) L ζ . (3) Ψ ˜ D restricts to a homeomorphism Ψ ˜ D,L ζ : Π( ˜ D ( G ( r ) )) L ζ ∼ → P ( H • ( ˜ D ( G ( r ) ) , k )) L ζ . Proof.

To prove the ﬁrst assertion, observe that the O ( G ( r ) )-module Ω dO ( G ( r ) ( k ) isstably equivalent to the restriction of Ω d ˜ D ( G ( r ) ) ( k ) because the restriction of ˜ D ( G ( r ) )-projectives to O ( G ( r ) ) are projective O ( G ( r ) )-modules. Thus, restricting the short exact sequence (6.0.1) of ˜ D ( G ( r ) )-modules along i O : O ( G ( r ) ) → ˜ D ( G ( r ) ) takes theform of a distinguished triangle L ζ O → Ω dO ( G ( r ) ( k ) ζ O → k in the stable module category stmod ( O ( G ( r ) ).Using Proposition 6.1, we see that P ( H • ( O ( G ( r ) ) , k )) L ζO ⊂ P ( H • ( O ( G ( r ) ) , k ))is the zero locus of the homogeneous “function” ζ O on P ( H • ( O ( G ( r ) ) , k )), andsimilarly that P ( H • ( ˜ D ( G ( r ) ) , k )) L ζ ⊂ P ( H • ( ˜ D ( G ( r ) ) , k )) is the zero locus of thehomogeneous “function” ζ on P ( H • ( ˜ D ( G ( r ) ) , k )). The second assertion thus followsfrom the fact that ζ O = i ∗ O ( ζ ) and the fact that P O is a homeomorphism byTheorem 5.4.The last assertion now follows from the commutative square (4.6.1) of Proposi-tion 4.6 and the fact that Ψ O restricts to a bijection Ψ O,L ζO : Ψ( ˜ O ( G ( r ) )) L ζO ∼ → P ( H • ( ˜ O ( G ( r ) ) , k )) L ζO by Proposition 4.5. (cid:3) The following proposition gives some insight into the topology on Π( ˜ D ( G ( r ) ))beyond knowing that it is the topology inherited from P ( H • ( ˜ D ( G ( r ) ) , k )). Proposition 6.3.

Assume the hypotheses on G of Theorem 5.4 so that the mapsof the commutative square (4.6.1) are homeomorphisms. Then { Π( ˜ D ( G ( r ) )) L ζ , ζ ∈ H • ( ˜ D ( G ( r ) ) , k ) h omogeneous } is a closed base for the topological space Π( ˜ D ( G ( r ) )) .Proof. Observe that { P ( H • ( O ( G ( r ) ) , k )) L ξ ξ ∈ H • ( O ( G ( r ) ) , k ) h omogeneous } is a closed base for the topology of P ( H • ( O ( G ( r ) ) , k )). Thus, the fact that P O : P ( H • ( O ( G ( r ) ) , k )) → P ( H • ( ˜ D ( G ( r ) ) , k )) is a homeomorphism (by Theorem 5.4)together with Proposition 6.2(2) implies that { P ( H • ( ˜ D ( G ( r ) ) , k )) L ζ , ζ ∈ H • ( D ( G ( r ) ) , k ) h omogeneous } is a closed base for the topology of P ( H • ( ˜ D ( G ( r ) ) , k )). Since Ψ ˜ D is a homeomor-phism, Proposition 6.2(3) implies that { Π( ˜ D ( G ( r ) )) L ζ , ζ ∈ H • ( ˜ D ( G ( r ) ) , k ) h omogeneous } is a closed base for the topological space Π( ˜ D ( G ( r ) )). (cid:3) We next consider the class of ˜ D ( G ( r ) )-modules given as the inﬂation along˜ D ( G ( r ) ) → G ( r ) of ﬁnite dimensional G ( r ) -modules (equivalently, as ﬁnite dimen-sional ˜ D ( G ( r ) )-module whose restriction to k [ G ( r +1) ] ⊂ ˜ D ( G ( r ) ) is trivial). Proposition 6.4.

Assume the hypotheses on G of Theorem 5.4. Let M be a ﬁnitedimensional ˜ D ( G ( r ) ) -module whose restriction to k [ G ( r +1) ] ⊂ ˜ D ( G ( r ) ) is trivial.Then (1) P O,M : P ( H • ( O ( G ( r ) ) , k )) M → P ( H • ( ˜ D ( G ( r ) ) , k )) M is a homeomorphism. (2) P ( H • ( O ( G ( r ) ) , k )) M can be identiﬁed as the “geometric join” of Proj( S • ( g ( r +1) )) and P ( H • ( G ( r ) , k )) M . (3) Ψ ˜ D,M : Π( ˜ D ( G ( r ) )) M → P ( H • ( ˜ D ( G ( r ) ) , k )) M is also a homeomorphism. UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 21

Proof.

We consider the commutative square(6.4.1) H • ( ˜ D ( G ( r ) ) , k ) i ∗ O / / (cid:15) (cid:15) H • ( O ( G ( r ) ) , k ) (cid:15) (cid:15) H • ( ˜ D ( G ( r ) ) , M ∗ ⊗ M ) i ∗ O / / H • ( O ( G ( r ) ) , M ∗ ⊗ M ) . We observe that there are multiplicative spectral sequences for M ∗ ⊗ M replacing k corresponding to the spectral sequences (4.2.2) and (5.4.1). Equipping M ∗ ⊗ M with internal grading degree 0, (6.4.1) determines a commutative square ofmultiplicative spectral sequences with internal grading whose E -page has the form(6.4.2) H s ( G ( r ) , H t ( k [ G ( r +1) ] , k )) / / (cid:15) (cid:15) H s ( G ( r ) , H t ( k [ G ( r +1) / G ( r ) ] , k )) (cid:15) (cid:15) H s ( G ( r ) , H t ( k [ G ( r +1) ] , k ) ⊗ M ∗ ⊗ M ) / / H s ( G ( r ) , H t ( k [ G ( r +1) / G ( r ) ] , k ) ⊗ M ∗ ⊗ M ) . The restriction map H ∗ ( k [ G ( r +1) ] , k ) → H ∗ ( k [ G ( r +1) / G ( r ) ] , k ) has nilpotent ker-nel and cokernel which are nilpotent, G ( r ) -summands of H ∗ ( k [ G ( r +1) / G ( r ) ] , k ).Thus, arguing as in the proof of Theorem 5.4, classes at the E -level with internalgrading not congruent to 0 modulo p r +1 are torsion; moreover, the lower horizon-tal arrow of (6.4.2) restricted to internal degrees congruent to 0 modulo p r +1 hasnilpotent kernel and cokernel which are nilpotent, G ( r ) -summands.Since P O is a homeomorphism, to prove (1) it suﬃces to verify that if p ⊂ H • ( O ( G ( r ) ) , k ) is a homogeneous prime ideal whose inverse image q ≡ i − O ( p ) con-tains the kernel of the left vertical map of (6.4.1), then p contains the kernel ofthe right vertical map. The validity of this statement is equivalent to the samestatement of the commutative square obtained from (6.4.1) by replacing the lowerhorizontal arrow by its associated map on reduced algebras (i.e., by dividing outbe the nilradicals of the domain and range of this map). By the preceding discus-sion, this latter map is an isomorphism. Thus, having made this replacement, therequired statement about i − O ( p ) is immediate.Assertion (2) follows from the familiar interpretation of the projectivization ofthe product of two aﬃne varieties.Assertion (3) follows from the commutativity of (4.6.1) and assertion (1). (cid:3) We consider a third class of ﬁnite dimensional ˜ D ( G ( r ) )-modules, those with theproperty that their restrictions to k [ G ( r +1) ] are projective. We ﬁrst recall thefollowing detection theorem of A. Suslin whose formulation we quote from [7, Thm4.10]. Theorem 6.5. [22]

Let G be a ﬁnite group scheme, Λ a unital associative G -algebra,and ζ ∈ H • ( G, k ) be a homogeneous cohomology class of even degree. Then ζ isnilpotent if and only if ζ K restricts to a nilpotent class in the cohomology of everyquasi-elementary subgroup scheme E K ⊂ G K for any ﬁeld extension K/k . Let M be a ﬁnite dimensional ˜ D ( G ( r ) )-module with the property that the re-striction of M to k [ G ( r +1) ] is projective. If M satisﬁes this property, then thespectral sequences (4.2.2) and (4.2.3 degenerate so that the edge homomorphisms provide the isomorphisms of algebras(6.5.1) H ∗ ( G ( r ) , H ( k [ G ( r +1) ] , M ∗ ⊗ M )) ∼ → H ∗ ( ˜ D ( G ( r ) ) , M ∗ ⊗ M ) ,H ∗ ( G ( r ) , H ( k [ G ( r +1) / G ( r ) ] , M ∗ ⊗ M )) ∼ → H ∗ ( O ( G ( r ) ) , M ∗ ⊗ M ) . We use (6.5.1) in conjunction with Theorem 6.5 to prove thatΨ ˜ D M : Π( ˜ D ( G ( r ) )) M → P ( H • ( ˜ D ( G ( r ) ) , k )) M is a homeomorphism for these modules. Proposition 6.6.

Assume the hypotheses on G of Theorem 5.4. Let M be a ﬁnitedimensional ˜ D ( G ( r ) ) -module with the property that the restriction of M to k [ G ( r +1) ] is projective. Then the restriction of Ψ ˜ D is a homeomorphism Ψ ˜ D,M : Π( ˜ D ( G ( r ) )) M ∼ → P ( H • ( ˜ D ( G ( r ) ) , k )) M . .Proof. Let Λ( M ) denote unital associative G ( r ) -algebra H ( k [ G ( r +1) ] , M ∗ ⊗ M ) andlet Λ O ( M ) denote H ( k [ G ( r +1) / G ( r ) ] , M ∗ ⊗ M ). Using the isomorphisms (6.5.1),we rewrite (6.4.1) as(6.6.1) H • ( ˜ D ( G ( r ) ) , k ) i ∗ O / / (cid:15) (cid:15) H • ( O ( G ( r ) ) , k ) (cid:15) (cid:15) H • ( G ( r ) , Λ( M )) i ∗ O / / H • ( G ( r ) , Λ O ( M )) . We set I M to be the kernel of the left vertical map and J M the kernel of the rightvertical map of (6.6.1).Since we know that Ψ ˜ D,M is well deﬁned and injective and that Ψ ˜ D is bijec-tive, to prove the proposition we must show for any homogeneous prime ideal q ⊂ H • ( O ( G ( r ) ) , k ) with inverse image p ⊂ H • ( ˜ D ( G ( r ) ) , k ) that J M ⊂ q (i.e., p is a pointof P ( H • ( O ( G ( r ) ) , k )) M ) whenever I M ⊂ p . Assume that ζ ∈ p ⊂ H • ( ˜ D ( G ( r ) ) , k ) isa homogeneous element which is not in I M , so that its image ζ ∈ H • ( G ( r ) , Λ( M ))is not nilpotent. Using Theorem 6.5, let γ : G a ( r ) ,K → G ( r ) ,K be a 1-parametersubgroup with the property that γ ∗ ( ζ ) ∈ H • ( G a ( r ) ,K , γ ∗ (Λ( M K ))) is not nilpotent.In other words, γ ∗ ( ζ ) ∈ H • ( K [ G ( r +1) ] G a ( r ) , k ) has image γ ∗ ( ζ ) ∈ Ext ∗ K [ G ( r +1) ] G a ( r ) ,K ] ( γ ∗ ( M K ) , γ ∗ ( M K ))which is not nilpotent. By Corollary 5.7, the image of γ ∗ ( ζ ) in Ext ∗ K [ G ( r +1) / G ( r ) ] G a ( r ) ,K ] ( γ ∗ ( M K ) , γ ∗ ( M K ))is not nilpotent as well. Consequently, i ∗ O ( ζ ) does not lie in J M . Thus, we concludeif p contains I M then q must contain J M . (cid:3) Further remarks

We begin this ﬁnal section with a natural question.

Question 7.1.

Assume that G satisﬁes the conditions of Theorem 5.4. Can oneextend the results of Section 6 to prove that Ψ ˜ D ( G ( r ) ) ,M : Π( ˜ D ( G ( r ) )) M → P ( H • ( ˜ D ( G ( r ) ) , k ) M is a homeomorphism for all ﬁnite dimensional ˜ D ( G ( r ) ) -modules? UPPORT THEORY FOR EXTENDED DRINFELD DOUBLES 23

We proceed to brieﬂy introduce Hopf subalgebras ˜ D ( s ) ( G ( r ) ) ⊂ ˜ D ( G ( r ) ) (gen-eralizing O ( G ( r ) ) ⊂ ˜ D ( G ( r ) )) and quotient maps of Hopf algebras k [ G r + s ] ։ D ( G ( r ) ) (generalizing ˜ D ( G ( r ) ) ։ D ( G ( r ) )). These generalizations behave much asthe algebras we have consider in previous sections. Proposition 7.2.

The G -equivariant quotient map G ( r +1) ։ G ( r +1) / G (1) ≃ ( G (1) ) ( r ) determines a Hopf algebra embedding of smash products (7.2.1) i D : ˜ D (1) ( G ( r ) ) ≡ k [( G (1) ) ( r ) ] k G ( r ) ֒ → k [ G ( r +1 ] k G ( r ) ≡ ˜ D ( G ( r ) ) . This is a map of free (right) O ( G ( r ) ) -modulesThe composition ˜ D (1) ( G ( r ) ) ֒ → ˜ D ( G ( r ) ) ։ D ( G ( r ) ) is the map F id : k [( G (1) ) ( r ) ] k G ( r ) → k [ G ( r ) ] k G ( r ) , where F is the Frobeniusmap.Proof. As in the justiﬁcation of (1.1.2), the map of (7.2.1) is an embedding of Hopfalgebras. We choose a k -linear section σ : k [ G ( r ) / G (1) ] → k [ G ( r +1) / G (1) ] of thequotient map induced by the embedding G ( r ) / G (1) ⊂ G ( r +1) / G (1) and consider the k [( G ( r +1) / G (1) ) / ( G ( r ) / G (1) )] ≃ k [ G ( r +1) / G ( r ) ] ≃ k [( G ( r ) ) (1) ]bilinear map k [( G (1) ) ( r − ] ⊗ k [( G ( r ) ) (1) ] → k [( G (1) ) ( r ) ]analogous to (1.5.1). Since k [( G (1) ) ( r − )] ⊗ D (1) ( G ( r ) ), we concludeas in the proof of Proposition 1.5 that this provides ˜ D (1) ( G ( r ) ) with the structureof a free (right) O ( G ( r ) )-module.To verify that the asserted composition equals F id , we use the identiﬁcation G ( r +1) / G (1) ≃ ( G ( r ) ) (1) . (cid:3) We can view ˜ D (1) ( G ( r ) ) as a twisted form of D ( G ( r ) ) for which the action of G ( r ) on k [( G (1) ) ( r ) ] equals the Frobenius twist of the coadjoint action of G ( r ) on k [ G ( r ) ].The proof of the following theorem is a basically identical to the proofs given inprevious sections of the corresponding properties for ˜ D ( G ( r ) )-modules. Proposition 7.3.

For any ˜ D (1) ( G ( r ) ) -module M , we can deﬁne M Π( ˜ D (1) ( G ( r ) )) M in strict analogy with M Π( ˜ D ( G ( r ) )) M . So deﬁned, π -point pairs of ˜ D (1) ( G ( r ) ) are π -point pairs of O ( G ( r ) ) with the equivalence relation determined by considera-tion of ﬁnite dimensional ˜ D (1) ( G ( r ) ) -modules. For any ﬁnite dimensional ˜ D (1) ( G ( r ) ) -module M , we deﬁne Π( ˜ D (1) ( G ( r ) )) M ⊂ Π( ˜ D (1) ( G ( r ) )) to consist of those equiva-lence classes of π -point pairs ( α K , η K ) with the property that ( α K + β K ) ∗ ( M K ) isnot projective.There is a natural continuous map Ψ ˜ D (1) ( G ( r ) : Π( ˜ D (1) ( G ( r ) )) → P ( H • ( ˜ D (1) ( G ( r ) ) , k ) .Assuming that G admits a quasilogarithm and p r +1 > dim ( G ) , this map is a home-omorphism and restricts to a homeomorphism Ψ ˜ D (1) ( G ( r ) ,M : Π( ˜ D (1) ( G ( r ) )) M ∼ → P ( H • ( ˜ D (1) ( G ( r ) ) , k ) M provided that M is in one of the classes analogous to those classes of ˜ D ( G ( r ) ) -modules considered in Section 6. Remark 7.4.

The triple of Hopf algebras over kO ( G ( r ) ) ֒ → ˜ D (1) ( G ( r ) ) ֒ → ˜ D ( G ( r ) )can be reﬁned to a sequence of embeddings of Hopf algebras which are free over O ( G ( r ) ) O ( G ( r ) ) ≡ D ( r ) ( G ( r ) ) ֒ → · · · ֒ → ˜ D (1) ( G ( r ) ) ֒ → ˜ D (0) ( G ( r ) ) ≡ ˜ D ( G ( r ) ) , where ˜ D ( s ) ( G ( r ) ) ≡ k [( G ( s )( r +1 − s ) ] k G ( r ) , ≤ s ≤ r. One readily checks that replacing ˜ D (1) ( G ( r ) ) in Proposition 7.3 by ˜ D ( s ) ( G ( r ) ) givesanalogous properties for ˜ D ( s ) ( G ( r ) )-modules. Remark 7.5.

For any s ≥

1, we deﬁne the Hopf subalgebra k [ G ( r + s ) ] k G ( r ) ⊂ k [ G ( r + s ) ] k G ( r + s ) ≡ D ( G ( r + s ) . This Hopf algebra admits a quotient map of Hopf algebras k [ G ( r + s ) ] k G ( r ) ։ D ( G ( r )) determined by the quotient map k [ G ( r + s ) ] ։ k [ G ( r + s ) ] → k [ G ( r ) ] as in (1.4.1). Weobtain a sequence of quotient Hopf algebras · · · ։ k [ G ( r + s ) ] k G ( r ) ։ k [ G ( r + s − ] k G ( r ) ։ · · · → ˜ D ( G ( r ) ) ։ D ( G ( r ) ) . We leave to the reader to check that our results for ˜ D ( G ( r ) )-modules extend to k [ G ( r + s ) ] k G ( r ) for s ≥ Remark 7.6.

We recall that a theorem of Pevtsova, Suslin, and the author [10,Thm 4.10] asserts that if M is a ﬁnite dimensional G -module for a ﬁnite groupscheme G over k and if α K : K [ t ] /t p → G K is a π -point of G at which the Jordantype α ∗ K ( M K ) is maximal among all Jordan types β ∗ L ( M L ) as β L : L [ t ] / p → G L varies among all π -points of G , then the Jordan type of α ′∗ K ( M K ) equals the Jordantype of α ∗ K ( M K ) whenever the π -point α ′ K is equivalent to α K .This immediately applies to O ( G ( r ) )-modules, since O ( G ( r ) ) is isomorphic asa k -algebra to the group algebra of a ﬁnite group scheme as seen in Proposition1.6. By the deﬁnition of the equivalence relation on π -point pairs of ˜ D ( G ( r ) ), thisremains valid for a ˜ D ( G ( r ) )-module M whose maximal Jordan type is that of aprojective module. Question 7.7.

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Department of Mathematics, University of Southern California, Los Angeles, CA

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