Balmer spectra and Drinfeld centers

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category ${\mathbf{C}}$ to ${\mathbf{C}}$ extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson--Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.


Introduction
Tensor triangular geometry, initiated by Balmer [2005;2010], has proven to be a useful prism through which modular representation theory, algebraic geometry, commutative algebra, algebraic topology, and homotopy theory may all be studied (for a few examples, see [Balmer and Sanders 2017;Boe et al. 2017a;2017b;Matsui and Takahashi 2017;Balmer 2020]).The uniting feature is the existence, in each case, of a braided monoidal triangulated category; the braiding condition implies that there is a natural isomorphism for all objects X and Y .A noncommutative analogue of Balmer's theory (that is, one with no assumption of a braiding) was initiated and explored in [Buan et al. 2007;Nakano et al. 2022a;2022b], motivated by the abundance of examples of nonbraided monoidal triangulated categories arising in representation theory.This theory defines a topological space, called the Balmer spectrum, for any monoidal triangulated category T .This space is denoted Spc T , and is defined as the collection of prime ideals of T , reflecting the usual notion of prime spectrum from ring theory.
Nonbraided monoidal triangulated categories arise naturally as the stable categories of finite tensor categories.Broadly speaking, if C is a finite tensor category, then the stable category of C, denoted st(C), is the category obtained by factoring out the projective objects of C. One motivation for factoring 1082 Kent B. Vashaw out projectives comes from the theory of support varieties, where the support variety of an object only distinguishes an object up to direct sums with projective objects.The stable category is a monoidal triangulated category, where the monoidal product of st(C) is an extension of the monoidal product of C.
An important tool in the study of tensor categories is the Drinfeld center, a categorical analogue of the center of a ring; it is a generalization of the quantum or Drinfeld double construction for Hopf algebras, originally introduced by Drinfeld [1987].For any tensor category C, its Drinfeld center Z(C) is a braided tensor category equipped with a functor F : Z(C) → C. The Drinfeld center satisfies the universal property: if G : D → C is a strict tensor functor between strict tensor categories, and D is braided, such that G is bijective on objects and surjective on morphisms, then there exists a strict tensor functor H : If C is abelian, then Z(C) is automatically abelian as well.We do not see an analogue for this argument in the triangulated case: if T is triangulated, it does not seem to follow immediately that Z(T ) is triangulated.This is a reflection of the fact that the morphism given in the extension axiom for triangulated categories is not necessarily unique.
However, if C is a finite tensor category, one can form its stable category st(C) on one hand; on the other hand, Z(C) is again a finite tensor category, and one can form its stable category st(Z(C)).The natural question that arises is, therefore: how are the Balmer spectra between these two categories connected?
This question is of particular interest because Balmer spectra are related intimately with cohomological support varieties (as in [Bergh et al. 2021]); for example, under a particular homological condition, the projectivization of the spectrum of the cohomology ring of the small quantum groups u ζ (b) of Borel subalgebras at roots of unity (as computed in [Ginzburg and Kumar 1993;Bendel et al. 2014]) identifies with the Balmer spectrum of its stable category [Nakano et al. 2022a], which can be used to show that the support varieties for the small quantum Borel possess the tensor product property [Nakano et al. 2022b;Negron and Pevtsova 2023].In many specific cases, for instance see [Friedlander and Negron 2018;Negron 2021;Negron and Plavnik 2022], the cohomology of Drinfeld doubles has been studied, and its relationship to the cohomology of the original finite tensor category explored.
Additionally, this project will provide tools to aid in thick ideal classification problems.Balmer spectra, which are defined as the collection of prime ideals of the category, are intimately related to these problems, since every thick ideal of a rigid monoidal triangulated category is equal to an intersection of prime ideals.Classifications of thick ideals in various settings have been undertaken in many different settings, for instance in various categories arising from (1) commutative algebra and algebraic geometry [Hopkins 1987;Thomason 1997;Matsui and Takahashi 2017]; (2) Lie superalgebras [Boe et al. 2017a]; (3) finite groups and finite group schemes [Benson et al. 1997;Friedlander and Pevtsova 2007]; (4) tilting modules for quantum groups and algebraic groups in positive characteristic [Ostrik 1997;Achar et al. 2019]; (5) Hopf algebras which are not necessarily commutative, cocommutative, or even quasitriangular [Benson and Witherspoon 2014;Boe et al. 2017a;Nakano et al. 2022a;2022b].
There are examples, for instance the small quantum groups of Borel subalgebras u ζ (b) at roots of unity, where, as mentioned above, the Balmer spectrum and thick ideals are known for its stable module category; however, it is an open question to classify the Balmer spectrum and thick ideals for the stable category of its Drinfeld center, that is, the stable module category of u ζ (g) ⊗ kT , where kT is the group algebra of the group of generators K i for u ζ (g).This motivates our central question, to reiterate: what relationship exists between the Balmer spectra of st(C) and st(Z(C))?
We answer this question by the following approach.
In Section 1, we give a brief background on tensor triangular geometry, compactly generated triangulated categories, stable categories and finite tensor categories, support data, and Drinfeld centers, and establish notation.
Next, in Section 2, we consider directly the relationship between the Balmer spectra Spc st(C) and Spc st(Z(C)).Since the prime ideals of the Balmer spectrum of a nonbraided monoidal triangulated category are a categorical analogue of the prime ideals in a noncommutative ring, we are motivated by prime ideal contraction, that is, the statement that if p is a prime ideal of a noncommutative ring R, then p ∩ Z (R) is a prime ideal in Z (R), the center of R. For general background on prime ideals for noncommutative rings, see [Goodearl and Warfield 2004, Chapter 3].Finding a categorical analogue to prime ideal contraction is complicated by the fact that we work with st(Z(C)) rather than Z(st(C)); the latter is equipped with a forgetful functor Z(st(C)) → st(C), but, as noted above, is not necessarily triangulated.Nevertheless, we verify that the forgetful functor Reflecting the analogous property for rings, Balmer spectra of braided monoidal triangulated categories are functorial; but in the nonbraided situation, a monoidal triangulated functor does not necessarily induce a continuous map between Balmer spectra.However, we show that F does induce a continuous map, and we obtain an analogue of prime ideal contraction.This is summarized by the following: Theorem A (See Propositions 2. 1.2 and 2.1.3).Let C be a finite tensor category.There exists a monoidal triangulated functor F : st(Z(C)) → st(C) extending the forgetful functor F : Z(C) → C, which induces a continuous map f : Spc st(C) → Spc st(Z(C)), defined by To study the image of the map f , we utilize the machinery of localization and colocalization functors.To apply these functors, one must work in the setting of compactly generated triangulated categories.For us, the role of compactly generated monoidal triangulated category will be filled by the stable category of the indization of C; this category will be referred to as St(C), and it contains st(C) as a triangulated subcategory.For the details of this setting, see Section 1.2.It is straightforward that the functor F extends to a functor St(Z(C)) → St(C); denote this extension again by F. We are then able to use the kernel of this functor to describe the image of f .the continuous map induced by F as above.Then there are containments Here, Loc( P) refers to the localizing subcategory (meaning triangulated and closed under set-indexed coproducts) of St(Z(C)) generated by P.
This implies that if C satisfies the following property, then f is surjective: Additionally, if C is a braided tensor category to begin with, then we prove that f is injective.This leads to the following theorem.
Theorem C (See Theorem 2.5.1).Let C be a finite braided tensor category satisfying property ( * ).Then f is a homeomorphism Spc st(C) , and there is a bijection between the thick ideals of st(Z(C)) and the thick ideals of st(C), given by In Section 3, we illustrate the theory with concrete examples.We first consider C to be alternately mod(kG) and mod((k[G] cop ), for G a finite group and k an algebraically closed field of characteristic p dividing the order of G, where kG denotes the group algebra of G and k[G] denotes the dual group algebra to kG.Of these two examples, the first satisfies property ( * ) and the second does not.This allows us to classify the Balmer spectrum and classify the thick ideals for stmod(D(kG)), where D(kG) is the Drinfeld double of the group algebra kG.We are then able to generalize this example in the following way.
Theorem D (See Propositions 3. 1.2,3.2.5,and Theorem 3.3.4).For the following classes of Hopf algebras H , the Balmer spectrum of stmod(D(H )) is homeomorphic via the map f to the Balmer spectrum of stmod(H ), and the thick ideals of the two categories are in bijection, (1) finite-dimensional cosemisimple quasitriangular Hopf algebras (e.g., group algebras of finite groups G in characteristic dividing the order of G); (2) Benson-Witherspoon smash coproducts (k[G]#kL) * , where G and L are finite groups with dual group algebra and group algebra k[G] and kL respectively, k an algebraically closed field of characteristic p dividing the order of G and not dividing the order of L, such that L acts by group automorphisms on G.
1. Preliminaries 1.1.Tensor triangular geometry.We will recall some of the background of noncommutative tensor triangular geometry.Following the terminology of [Nakano et al. 2022a;2022b], a monoidal triangulated category T is a category such that the following conditions hold: (1) T is triangulated: it is an additive category equipped with an additive autoequivalence : T → T , called the shift functor, and a collection of distinguished triangles subject to the usual axioms (see [Happel 1988;Neeman 2001]).
(2) T is monoidal: it is equipped with a monoidal product ⊗ and unit 1 T , subject to the usual associativity and unit axioms (see [Kassel 1995;Bakalov and Kirillov 2001;Etingof et al. 2015]).
(3) The triangulated and monoidal structures on T are compatible: for any object A of T , the functors A⊗ − and − ⊗ A are triangulated functors.In other words, there exists a natural isomorphism is a distinguished triangle, then for any object D, the triangles Remark 1.1.1.In the terminology of [Balmer 2005;2010], a tensor triangulated category is a monoidal triangulated category such that the monoidal product is symmetric.Note that contrary to the definition of tensor category as in [Etingof et al. 2015], a tensor triangulated category is not required to have duals.
We will recall the definition of the Balmer spectrum of a monoidal triangulated category T , as in [Buan et al. 2007;Nakano et al. 2022a].
(1) A (two-sided) thick ideal I of T is a full subcategory such that the following hold.
(a) I is triangulated: it is closed under and −1 , and if The collection of thick ideals of T will be denoted by ThickId(T ), and the thick ideal generated by a collection of objects T will be denoted ⟨T ⟩.
(2) A thick ideal P of T is called prime if for all thick ideals I and J of T , a containment I ⊗ J ⊆ P implies either I or J ⊆ P; equivalently, P is prime if and only if for all objects A and B of T , a containment A ⊗ T ⊗ B ⊆ P implies either A or B is in P (see [Nakano et al. 2022a, Theorem 3.2.2]).
Here I ⊗ J refers to the collection of objects {A ⊗ B : A ∈ I, B ∈ J}, and A ⊗ T ⊗ B refers to the collection of objects {A ⊗ C ⊗ B : C ∈ T } for A and B in T .
(3) A thick ideal P of T is called completely prime if A ⊗ B ∈ P implies either A or B ∈ P, for all objects A and B of T .
(4) The Balmer spectrum of T , denoted Spc T , is the collection of prime ideals of T under the Zariski topology, where closed sets are defined as the sets for all collections of objects T of T .
(5) An arbitrary open set of Spc T , that is, the complement of a closed set V T (T ) for some collection T of objects of T , will be denoted Note that every completely prime ideal is prime.If T is a braided category then every prime ideal is completely prime, and so in that case the two notions coincide.
Remark 1.1.2.We emphasize that this choice of topology on the Balmer spectrum does not match what one might expect, by the analogy to ring theory.This reflects the fact that in natural examples when the Balmer spectrum of a monoidal triangular category T is realized concretely as the Proj or Spec of a commutative ring R, the bijection between prime ideals of T and the (homogeneous) prime ideals of R is containment-reversing.See Example 1.4.2below for concrete examples.
Remark 1.1.3.While we have only defined the Balmer spectrum as a topological space, Balmer's original definition [2005, Section 6] gives Spc the additional structure of a ringed space (which is actually locally ringed, by [Balmer 2010, Corollary 6.6]).Many of the classification theorems for Balmer spectra prove existence of isomorphisms of ringed spaces, rather than just homeomorphisms of topological spaces.However, the ringed space structures will not play a role in this paper, so we omit the precise definition.
We recall one topological property of the Balmer spectrum, for reference.This was proven by Balmer [2005, Corollary 2.17].
Theorem 1.1.4.Let T be a braided monoidal triangulated category.Then Spc T is Noetherian if and only if every closed subset of Spc T is of the form V T (A), for some object A of T .
Remark 1.1.5.In fact, if T is braided, or if there is an object of T which generates T as a thick subcategory, then Spc T is a spectral topological space.In other words, Spc T is T 0 , quasicompact, the quasicompact open sets form an open basis, and every nonempty irreducible closed subset has a generic point.It is a theorem of Hochster that this implies Spc T is homeomorphic to the prime spectrum of a commutative ring [Hochster 1969, Theorem 6].
Suppose T is rigid, in other words, every object is dualizable (see [Etingof et al. 2015, Section 2.10]).We then obtain the following facts, which we recall for reference.Both follow directly from the fact that if A is dualizable with dual A * , then A is a direct summand of A ⊗ A * ⊗ A. Proposition 1.1.6.Let T be a rigid monoidal triangulated category.Let A be an object of T with dual A * .Then (1) ⟨A⟩ = ⟨A * ⟩, and (2) every thick two-sided ideal I of T is semiprime, i.e., it is the intersection I = P∈Spc T , I⊆ P P of prime ideals over itself.Equivalently, for every ideal I of T , if the set of objects A ⊗ T ⊗ A ⊆ I for some object A in T , then A ∈ I, where A ⊗ T ⊗ A refers to the collection {A ⊗ B ⊗ A : B ∈ T }.

1.2.
Compactly generated triangulated categories.A powerful result in the theory of triangulated categories is Brown representability, which ensures the existence of adjoints to certain triangulated functors [Neeman 2001, Chapter 8].However, in order to apply these results, one must work in the setting of compactly generated triangulated categories.We recall the definition now.
(1) An object C in a triangulated category T is compact if the functor Hom T (C, −) commutes with arbitrary set-indexed coproducts.If T is a triangulated category, then T c will denote the subcategory of compact objects.
(2) A localizing subcategory of a triangulated category is a triangulated subcategory which is also closed under taking set-indexed coproducts.The smallest localizing category containing a collection T of objects will be denoted Loc(T ) and will be referred to as the localizing category generated by T .
(3) A compactly generated triangulated category is a triangulated category T which contains arbitrary set-indexed coproducts such that Loc(T c ) = T .
Note that any localizing subcategory I of T is thick by a version of the Eilenberg swindle: if A ⊕ B is in I, then we have a distinguished triangle where the first map sends the i-th copy of B in the first object to the i-th copy of B in the second object, and sends the i-th copy of A in the first object to the (i+1)-th copy of A in the second object.Since I is localizing, the first and second objects are in I, and hence A is in I as well.For additional background on compactly generated triangulated categories; see [Benson et al. 2012, Section 1.3.9].
The following theorem, due to Rickard [1997], is the primary technical reason we need to move to the compactly generated setting.For details; see [Boe et al. 2017a, Theorems 3.1.1, 3.1.2;Benson et al. 2008, Section 3;2012, Section 2].
Theorem 1.2.1.Let T be a compactly generated triangulated category.Given a thick subcategory S of T c , there exist functors Γ S and L S from T → T , which gives for every object M of T a distinguished triangle such that (1) L S and Γ S are unique up to isomorphism, , that is, there are no nonzero maps from Loc(S) → L S (M), and The functors Γ S and L S are called colocalizing and localizing functors, respectively.They are constructed by first taking a Verdier quotient of T by Loc(S), that is, forming a category where all morphisms with cones in Loc(S) are formally inverted, which one may do using the calculus of roofs.This quotient is a triangulated category where the objects isomorphic to 0 are precisely those from Loc(S), and Brown representability guarantees that there are right adjoint functors i ! and j * to the inclusion i * : Loc(S) → T and quotient j * : T → T / Loc(S) functors, giving a diagram Loc(S) The functor L S is then defined as j * • j * , and Γ S is defined as 1.3.Stable categories and finite tensor categories.The monoidal triangulated categories that are the primary focus of this paper arise as stable categories.We first recall the construction of the stable category of any quasi-Frobenius category.Recall that a quasi-Frobenius category is an abelian category with enough projectives, such that projective and injective objects coincide.For any quasi-Frobenius category C, one may form the stable category st(C), which is triangulated (see [Happel 1988, Chapter I]).The stable category is constructed by factoring out the projective objects of C. In more detail, let PHom C (A, B) consist of the morphisms f : A → B in C such that f factors through a projective object.The stable category st(C) is the category where (1) objects are the same as the objects of C; (2) morphisms A → B are defined to be Hom There is a straightforward functor G : C → st(C) sending objects to themselves and morphisms to their image in the quotient.
If P is a projective object of C, note that the corresponding object G(P) in st(C) is isomorphic to 0, since id P factors through a projective; and the converse is also true, since G(P) ∼ = 0 in st(C) implies that the 0-morphism G(P) → G(P) is equal to id G(P) in Hom st(C) (G(P), G(P)), in other words, id P factors through a projective Q in C: Of course, this implies P is a summand of Q, and so P is projective.
We recall the triangulated structure on st(C), for reference.If A is an object of C, denote by (A) the kernel of the projective cover of A. The functor extends to the stable module category, and this in fact gives an autoequivalence on st(C).The shift is then defined to be (A) = −1 (A).For any short exact sequence of objects in C, say in st(C); the distinguished triangles of st(C) are then defined to be all triangles which are isomorphic to triangles of this form.
We now specialize to the case that C is a finite tensor category.Recall that a finite tensor category (following the notation given in [Etingof and Ostrik 2004;Etingof et al. 2015]) consists of a monoidal category C such that (1) C is abelian and k-linear for an algebraically closed field k; (2) the tensor product − ⊗ − is bilinear on spaces of morphisms; (3) every object of C has finite length; (4) Hom C (1, 1) ∼ = k; (5) for any pair of objects A and B, the vector space Hom C (A, B) is finite-dimensional over k; (6) C has enough projectives; (7) there are finitely many isomorphism classes of simple objects of C; (8) C is rigid, i.e., every object has a left and a right dual.
The prototypical example of a finite tensor category is the category of finite-dimensional modules of a finite-dimensional Hopf algebra H . Notation 1.3.1.Denote the category of finite-dimensional modules of an algebra H by mod(H ).Denote the category of all (not necessarily finite-dimensional) modules of H by Mod(H ).
Recall that if C is a finite tensor category, it is a consequence that the tensor product is biexact [Etingof et al. 2015, Proposition 4.2.1].Additionally, every finite tensor category is quasi-Frobenius [Etingof et al. 2015, Proposition 6.1.3].The stable category st(C) inherits a monoidal product directly from C: we define G(A) ⊗ G(B) := G(A ⊗ B), and similarly for morphisms f : A → B and g : factors through a projective P, and then f ⊗ g − f ⊗ g factors through P ⊗ D, which is projective by projectivity of P (see [Etingof et al. 2015, Proposition 4.2.12]).
Although the primary objects of focus in this paper are stable categories st(C) for finite tensor categories C, note that st(C) is not compactly generated, since in particular it does not contain arbitrary set-indexed coproducts.Thus, in order to apply Theorem 1.2.1, it is necessary to produce a compactly generated monoidal triangulated category which contains st(C) as a monoidal triangulated subcategory.In fact, this is possible, using the Ind-completion (see [Kashiwara and Schapira 2006, Chapter 6]) of C: Let C be a finite tensor category.Then its Ind-completion Ind(C) is a quasi-Frobenius abelian monoidal category, and its stable category st(Ind(C)) is a compactly generated monoidal triangulated category, with st(Ind(C)) c ∼ = st(C) via the stabilization of the natural inclusion functor C → Ind(C).1.4.Support data.Suppose that T is a monoidal triangulated category and S a topological space.We will denote the collection of subsets of S by X (S), closed subsets of S by X cl (S), and specialization-closed subsets of S by X sp (S); recall that by definition, a set is specialization-closed if it is a union of closed sets.When the underlying space is clear from context, we will denote these collections by X , X cl , and X sp .
Given a monoidal triangulated category T and a topological space S, a support datum on T with value in S is a map σ : T → X cl (S) satisfying the following axioms: (1) σ (0) = ∅ and σ (1) = S; See [Nakano et al. 2022a, Section 4] for a more in-depth discussion of support data (although note that in that paper, a support datum is permitted to take value in X (S) rather than X cl (S)).For any monoidal triangulated category T , the map V T (A) = { P ∈ Spc T : A ̸ ∈ P} defined above is a support datum T → X cl (Spc T ), since by definition, V T (A) is a closed set in Spc T .We will refer to this support datum as the Balmer support.Indeed, the Balmer support satisfies a universal property in the category of support data, see [Nakano et al. 2022a, Theorem 4.2.2].
Theorem 1.4.1.If σ : T → X cl (S) is a support datum with value in S, then there exists a unique continuous map For any support datum σ , we have a map where T is any collection of objects of T .If σ takes values in X cl , then by definition σ takes values in X sp .The map σ in fact only depends on thick ideals rather than arbitrary subsets, since by [Nakano et al. 2022a, Lemma 4.3.2]we have σ (T ) = σ (⟨T ⟩).
For a support datum σ , we have a second map σ : X sp → ThickId(T ) defined by for any specialization-closed subset S ′ of S. For any specialization closed set S ′ , the collection σ (S ′ ) is a thick ideal of T .Hence, we have the following collection of maps, given a support datum σ on T : Classifications of thick ideals are obtained in many cases (see [Balmer 2005;2010 where  5), referred to as the tensor product property, and was proven by Friedlander and Pevtsova [2007].It is a theorem that for finite group schemes, the cohomological support is classifying, and the map f : Proj H • (G, k) → Spc stmod(G) is a homeomorphism [Benson et al. 1997;Balmer 2005;Friedlander and Pevtsova 2007].Cohomological support exists for arbitrary finite tensor categories [Bergh et al. 2021], but is not known to be classifying in general, see [Nakano et al. 2023, Conjecture E].
1.5.The Drinfeld center.Let C be a strict monoidal category.Then the Drinfeld center or center of C, which we will denote by Z(C), is defined as the following braided monoidal category.
(1) Objects are pairs (A, γ ) where A is an object of C and γ is a natural isomorphism γ B : B⊗ A for all B and C. Such a natural isomorphism γ is called a half-braiding of A. ( (3) The monoidal product (A, γ ) ⊗ (A ′ , γ ′ ) is defined as (A ⊗ A ′ , γ ) where γ is defined as being a valid map in Z(C) amounts to checking the commutativity of the diagram This diagram commutes by the naturality of γ ′ , since it can be rewritten, using the defining diagram for γ ′ , as Here if H is a Hopf algebra with multiplication µ, unit η, comultiplication , counit ϵ, and antipode S, then (H op ) * is the Hopf algebra with multiplication * , unit ϵ * , comultiplication (µ op ) * , counit η * , and antipode (S −1 ) * .
The following result of Etingof and Ostrik [2004] will be important in extending the forgetful functor Z(C) → C to the stable categories.
Proposition 1.5.1.If C is a finite tensor category, then its Drinfeld center Z(C) is a finite tensor category, and the forgetful functor F is exact and sends projective objects to projective objects.
The fact that F preserves projectivity is a generalization of the classical Nichols-Zoeller theorem [1989] for Hopf algebras, which states that a finite-dimensional Hopf algebra is free as a module over any Hopf subalgebra.

Drinfeld Centers and Balmer Spectra
In this section, we prove general results relating the Balmer spectrum of st(C) to the Balmer spectrum of st(Z(C)), under the assumption that C is an arbitrary finite tensor category.
2.1.Construction of a continuous map between Balmer spectra.Recall the stable categories defined in Section 1.3.For the rest of this section, let C be a finite tensor category, st(C) its stable category, Z(C) its Drinfeld center, st(Z(C)) the stable category of its Drinfeld center (which may be formed by Proposition 1.5.1), and St(C) and St(Z(C)) the respective "big" stable categories, recall Notation 1.3.3.We have a forgetful functor F : Z(C) → C, and we have functors G : C → st(C) and H : Z(C) → st(Z(C)).The functor F extends to a functor Ind(Z(C)) → Ind(C), which we again denote by F, by [Kashiwara and Schapira 2006, Proposition 6.1.9].We have the respective Balmer support data associated to st(C) and st(Z(C)), defined in their respective categories by sending A → {primes not containing A}.
Notation 2.1.1.For readability, when C is a finite tensor category we will denote . The corresponding maps (recalling the construction from Section 1.4) associated to these support data will similarly be denoted C and Z , respectively.We will similarly denote open sets in the Balmer spectrum on these respective categories by U C := U st(C) and U Z := U st(Z(C)) , recall the notation from Section 1.1.
The following proposition is probably well-known to experts, but we record it for completeness.Proof.Since the objects of St(Z(C)) are the in bijection with those of Ind(Z(C)), the functor F is well-defined on objects, namely by defining Let f : X → Y be a morphism in Ind(Z(C)).Then for F(H ( f )) := G F( f ) to be well-defined, we need G F(g) = 0 for each morphism g which factors through a projective in Ind(Z(C)).In other words, we need F(g) to factor through a projective in Ind(C).Hence, to define F, it is enough to know that G • F sends all projective objects of Ind(Z(C)) to 0, which is true by Proposition 1.5.1.
Let H (X ) ∈ St(Z(C)) an arbitrary object, where X ∈ Ind(Z(C)).Then H (X ) is defined as H (Z ), such that there exists a short exact sequence 0 → X → P → Z → 0 in Ind(Z(C)), where P is a projective object in Ind(Z(C)).The object H (Z ) is well-defined in St(Z(C)), by Schanuel's lemma.Since F is exact and sends projectives to projectives, 0 → F(X ) → F(P) → F(Z ) → 0 is an exact sequence in C with F(P) projective; therefore, (G F(X )) ∼ = G F(Z ) in st(C), and so we have for some short exact sequence in Ind(Z(C)).Since F is exact, and G sends exact sequences to triangles, we have that the composition G F is exact and hence is a triangle as well, and so F is a triangulated functor.□ For braided tensor triangulated categories, the Balmer spectrum Spc is functorial, as Balmer [2005, Proposition 3.6] has shown.This is a categorical reflection the ring-theoretic fact that Spec is functorial for commutative rings.On the other hand, for noncommutative rings, Spec is not a functor (for an in-depth exploration of the extent of the failure of functoriality of Spec for noncommutative rings, see [Reyes 2012]).It is not surprising, then, that for generic monoidal triangulated categories, the Balmer spectrum is also not functorial; in other words, a monoidal triangulated functor between monoidal triangulated categories does not necessarily induce a map between their Balmer spectra.
However, reflecting the classical prime ideal contraction for noncommutative rings, the forgetful functor F does induce a map between the Balmer spectra of st(C) and st(Z(C)).Proof.Let P be a prime ideal of st(C).We must first show that f ( P) is a prime ideal of st(Z(C)).
We first check that f ( P) is a thick ideal of st(Z(C)).This necessitates checking four properties: Triangulated.Suppose X ∈ f ( P), in other words, F( X ) ∈ P. Since F is triangulated, this is true if and only if F(X ) ∈ P, which is true if and only if F(X ) ∈ P, in other words, X ∈ f ( P).Now, suppose is a distinguished triangle with X and Y in f ( P).This means that F(X ) and F(Y ) are in P. Since F is triangulated, the triangle is distinguished in st(C).Now since the first two objects are in P, so is F(Z ), and so Z ∈ f ( P).
Thick.If X ⊕ Y is in f ( P), then F(X ⊕ Y ) ∈ P; F is an additive functor, and so F(X ) ⊕ F(Y ) ∈ P.This implies that both F(X ) and F(Y ) are in P, and so X and Y are both in f ( P).
Ideal.Suppose X ∈ f ( P) and Y ∈ st(Z(C)).Since F is monoidal, we have Since F(X ) ∈ P, so is F(X ) ⊗ F(Y ), and thus F(X ⊗ Y ) ∈ P as well.Hence X ⊗ Y ∈ f ( P).The symmetric argument shows that Y ⊗ X is in f ( P) as well, so f ( P) is a two-sided ideal.
Prime.Let A ⊗ B ∈ f ( P).Then F(A) ⊗ F(B) ∈ P.But F(A) and F(B) commute with every object of st(C): by the ideal property of P, we have with the last step following by primeness of P.This implies that either A or B is in f ( P), which means that f ( P) is prime.
We can also check directly that f is continuous: an arbitrary closed set of Spc st(Z(C)) is of the form V Z (T ) = { P ∈ Spc(st(Z(C))) : T ∩ P = ∅} (recalling Notation 2.1.1)for some collection of objects T of st(Z(C)).Then where by F(T ) we mean the collection {F(X ) : X ∈ T }. □ Remark 2.1.4.Recall the construction of the Drinfeld double from Section 1.5.If R is a finite-dimensional Hopf algebra, then Z(mod(R)) ∼ = mod(D(R)).In this case, D(R) ∼ = D((R op ) * ), and so we have two functors, which then give two maps between Balmer spectra,

2.2.
A support data interpretation.We can interpret the map f in the context of support data (recalling the definition from Section 1.4), by first defining a new support datum given as the composition of the functor F with the Balmer support V C on st(C).
Proposition 2.2.1.Define a map W : This map is a support datum.
Proof.The first four conditions follow directly from the facts that F is a triangulated functor and V C is itself a support datum, since To check the last condition, we need to show that By the ideal condition, if P is a prime ideal which does not contain F(X ) ⊗ F(Z ) ⊗ F(Y ) for some object Z , then it must also not contain F(X ) or F(Y ).Hence, is automatic.
For the reverse containment, suppose P is a prime ideal which does not contain F(X ) or F(Y ).By the prime condition, that means P does not contain the entire collection of objects F(X ) ⊗ st(C) ⊗ F(Y ).But since F(X ) and F(Y ) commute up to isomorphism with all elements of st(C), if F(X ) ⊗ F(Y ) ∈ P, that would imply there is a containment F(X ) ⊗ F(Y ) ⊗ st(C) ⊆ P, which would then imply F(X ) ⊗ st(C) ⊗ F(Y ) ⊆ P, a contradiction.Hence, P ∈ W (X ⊗ Y ), and we have the claimed equality. □ By the universal property of the Balmer spectrum as in Theorem 1.4.1, the support datum W induces a continuous map Spc st(C) → Spc st(Z(C)).This map is defined as by [Nakano et al. 2022a, Theorem 4.2.2].One may observe that this map is the same as the map defined in Proposition 2.1.3.We have the following diagram, which commutes by definition: On the level of ideals, we have the following induced maps, recall and associated to a support datum as constructed in Section 1.4: X sp (Spc(st(C))) Here, for thick ideals I of st(Z(C)) and J of st(C), the maps and are defined by By definition, the inner and outer triangles commute: in other words, W = C • , and W = • C .

2.3.
Recovering ideals from their supports.In [Nakano et al. 2022a, Theorem 6.2.1], conditions were given under which an arbitrary support datum σ : T → X (S) has the property that σ is a left, right, and two-sided inverse to σ .If σ is a left inverse to σ , this means that all thick ideals can be recovered from their supports; when σ and σ are a mutually inverse bijection, the ideals are completely classified by the topological space S. Since the support datum W (−) defined above might not satisfy conditions under which every ideal may be recovered from their support (see Section 3 for examples), in this section we discuss precisely which ideals can be recovered in this way; this allows us to describe the image of the map f defined above.We now introduce some terminology, which will be useful for our reconstruction theory.
Notation 2.3.1.When the finite tensor category C is clear by context, we will denote by K the kernel of the functor An equivalent characterization of the kernel of F can be given by This follows from the fact that the objects of St(C) isomorphic to 0 correspond precisely to the projective objects of Ind(C), as we saw in Section 1.3.
Lemma 2.3.2.The kernel of F is a thick localizing ideal of St(Z(C)).
Proof.Since F is a monoidal triangulated functor, it is straightforward to verify that the collection of objects X such that F(X ) ∼ = 0 is closed under taking cones, shifts, direct summands, and by tensoring on the left or right by arbitrary objects of St(Z(C)).The functor F commutes with arbitrary coproducts by [Kashiwara and Schapira 2006, Proposition 6.1.9],and so the kernel of F is closed under arbitrary coproducts, i.e., K is localizing.
Proof.First, note that if A ∈ K , then by definition F(A) ∼ = 0, and so For the other direction, recall that by the rigidity of C, all thick ideals of st(C) are semiprime, i.e., intersections of prime ideals, by Proposition 1.1.6.This implies in particular that the ideal ⟨0⟩ is semiprime; in other words, the only object contained in all prime ideals of st(C) is 0. By definition, this means that if X is an object of st(C) such that V C (X ) = ∅, then X ∼ = 0. Hence, we have Using the localization and colocalization functors defined in Section 1.2, we are now able to prove the following, which is the critical step in determining which ideals can be recovered from their W -support and determining the image of the map f : Spc st(C) → Spc st(Z(C)) defined in Proposition 2.1.3.Theorem 2.3.4.Let I be a thick ideal of st(Z(C)) such that Loc(I) contains K .Suppose that X is an object of st(Z(C)) such that F(X ) ∈ ⟨F(I)⟩, that is, the thick ideal of st(C) generated by all F(Y ) for Y ∈ I. Then X is in I.
Proof.By Theorem 1.2.1, we have a distinguished triangle in St(Z(C)), using the localization and colocalization functors associated to the thick ideal I.We know that there are no morphisms from I to L I (X ); in other words, if Y ∈ I and Z is any compact object in St(Z(C)), then Since this holds for all compact objects Z , this implies that L I (X ) ⊗ Y * ∼ = 0. Since all compact objects are rigid, and by Proposition 1.1.6all thick ideals are closed under taking duals, we have L I (X ) ⊗ Y ∼ = 0 for all Y ∈ I. Since F is a monoidal functor, this additionally implies that Now, consider the thick ideal ⟨F(I)⟩.This is formed successively by taking shifts, cones, direct summands, and tensor products with arbitrary elements of st(C), starting from the collection of objects of the form F(Y ) for Y ∈ I.This allows us to conclude inductively that F(L I (X )) ⊗ A ∼ = 0 for all A in ⟨F(I)⟩, since inductively each step by which we construct ⟨F(I)⟩ preserves the property that tensoring with F(L I (X )) gives 0. To be more explicit, if 0 as well, using the commutativity of F(L I (X )).
To reiterate, the upshot of the previous paragraph is that A ⊗ F(L I (X )) ∼ = 0 for all A ∈ ⟨F(I)⟩.But by assumption, we have F(X ) ∈ ⟨F(I)⟩.Hence, Therefore, X ⊗ L I (X ) is an object in K , the collection of objects of St(Z(C)) mapped to 0 by F. By assumption, Loc(I) contains K , and so X ⊗ L I (X ) ∈ Loc(I).Now, consider the distinguished triangle obtained by tensoring the triangle We have just finished showing that the third object of this triangle is in Loc(I).The first object is in Loc(I) as well, by Theorem 1.2.1.Since Loc(I) is triangulated, this implies X ⊗ X is in Loc(I).But by [Neeman 1992, Lemma 2.2], since I is a thick subcategory of compact objects, the compact objects in Loc(I) are precisely the objects of I. Thus, X ⊗ X ∈ I, and by semiprimeness of I (Proposition 1.1.6)so is X ; this completes the proof.□ We can now give a condition under which an ideal I can be recovered from its support W (I).
Corollary 2.3.5.Let I be an ideal such that Loc(I) contains K .Then W • W (I) = I.
Proof.By definition, The last equality follows from Proposition 1.1.6.The corollary now follows directly from Theorem 2.3.4.
□ 2.4.The image of prime ideal contraction.We now describe the relationship of the image of the map f to the kernel K of F.
(2) For every nonzero X in K , there exists a compact object Y in K which has some nonzero map Y → X in St(Z(C)).
In particular, to prove that ( 2) ⇒ (1), one simply observes that if X ∈ K , then the distinguished triangle given by Theorem 1.2.1 implies that L K ∩st(Z(C)) X ∈ K .But by definition, it is in the perpendicular space to K ∩ st(Z(C)), which by the assumption of ( 2) means that it is 0. Hence If these conditions are satisfied, then we can sharpen (2.4.1), as well as Corollary 2.3.5.
Corollary 2.4.3.Suppose the kernel K of F satisfies the equivalent conditions of Lemma 2.4.2.
(1) The image of f is precisely the collection of prime ideals of st(Z(C)) which contain K ∩ st(Z(C)), that is, the collection of objects X in st(C) such that F(X ) ∼ = 0. ( In other words, the image of f can be written as an intersection of open sets.If K ∩ st(Z(C)) is generated (as a thick ideal) by a finite collection of objects, say (recall the notation of U Z from Section 1.1 and Notation 2.1.1).
Remark 2.4.5.In the situation of Corollary 2.4.3(2),we have Corollary 2.3.5 sharpened from a one-way implication to a two-way implication.We note on the other hand that if the conditions of Lemma 2.4.2 are not satisfied, then Corollary 2.3.5 can never be an if-and-only-if, for the following reason.The collection of objects K ∩ st(Z(C)) is itself a thick ideal of st(Z(C)), since it is in particular the kernel of the monoidal triangulated functor F restricted to compact objects.But now note that Here the first equality is by the definition of W , the second and third equalities are by Lemma 2.3.3 and the definition of W , and the last equality by the definition of the kernel K .In other words, the thick ideal K ∩ st(Z(C)) can be recovered from its support.But plainly, since we are assuming the conditions of Lemma 2.4.2 are not satisfied, we have and so Corollary 2.3.5 cannot be sharpened to an if-and-only-if statement.

2.5.
Conditions under which f is injective, surjective, or a homeomorphism.We now give conditions under which W and W are inverses, and f is surjective, injective, and a homeomorphism.Theorem 2.5.1.Let C be a finite tensor category.
(1) The following conditions are equivalent.
(a) For all X ∈ K , there exists an isomorphism X ∼ = 0 in St(Z(C)).
(b) The map f is surjective and K is generated as a localizing category by its subcategory K ∩ st(Z(C)).Proof.Suppose (1a) holds, and so K consists only of objects isomorphic to 0, in other words, for all objects X ∈ Z(C), In particular this means that K is generated by K ∩ st(Z(C)), since all objects of K are isomorphic to 0. Then (1c) follows from Theorem 2.3.4,and the conditions (1b) and (1d) follow directly from Corollary 2.4.3.Now, suppose (1b) is satisfied.By Proposition 2.4.1, this means that every prime ideal of st(Z(C)) contains K ∩ st(Z(C)).But since every ideal is semiprime, the zero ideal is equal to the intersection of all primes of st(Z(C)), and so K ∩ st(Z(C)) is contained in the zero ideal.Since K is generated by K ∩ st(Z(C)), i.e., the zero ideal, this implies that (1a) holds.
Note that Hence (1c) implies (1a).Lastly, suppose condition (1d) holds.This implies by Corollary 2.3.5 that K ⊆ Loc(I) for every thick ideal I; in particular, this means that K is contained in the localizing category generated by 0, which consists only of objects isomorphic to 0. Hence, (1a) holds.
To show (2), first note that if C is braided with a braiding γ , then F is essentially surjective, since for any object X in C, the pair (X, γ X ) is an object of Z(C) and F sends H (X, γ X ) to G(X ).Now, we note that if P and Q are prime ideals of st(C), then Hence, if C is braided then (2a) follows.
Condition (2b) also follows directly from the fact that F is essentially surjective.For (2c), recall that by Theorem 1.1.4,Spc(st(C))is Noetherian if and only if every closed set is of the form V C (A) for some object A ∈ st(C).If S is a specialization-closed set in Spc(st(C)), then by definition For the other direction, we can write S as a union of closed sets, say S = i∈I S i , and by the Noetherianity of Spc(st(C)), there exist objects A i of st(C) such that S i = V C (A i ).Since F is essentially surjective, we can pick we have by definition each X i is in W (S). Therefore, Thus S = W ( W (S)).
Suppose the assumptions of (3).Then (3b) and (3c) follow immediately from parts ( 1) and ( 2).To show (3a), it is enough to show that f is a closed map, by (1a) and (2a).Take an arbitrary closed set V C (T ) in Spc st(C).We claim that the image of V C (T ) under f is precisely V Z ( T ), where For the first direction, suppose P ∈ V C (T ), in other words, P ∩ T = ∅.Since f ( P) = {X : F(X ) ∈ P}, this implies that for all X ∈ f ( P), we have X ̸ ∈ T .Therefore f ( P) ∩ T = ∅, and so f ( P) ∈ V Z ( T ).This shows f (V C (T )) ⊆ V Z ( T ).
For the other containment, suppose Q is a prime ideal of st(Z(C)) in V Z ( T ).Then F(X ) ̸ ∈ T for all X ∈ Q.Since f is surjective, we can pick P ∈ Spc st(C) with f ( P) = Q, and for all F(X ) ∈ P, we must have F(X ) ̸ ∈ T .Since F is essentially surjective, this implies A ̸ ∈ T for all A ∈ P, and so P ∩ T = ∅, i.e., P ∈ V C (T ).This shows the other containment f (V C (T )) ⊇ V Z ( T ), and so we have equality.
Hence, f sends the closed set V C (T ) to the closed set V Z ( T ), and so it is a continuous, bijective, closed map, and therefore a homeomorphism.□

Applications
The time has come for concrete applications of our theory.Proof.A module for D(kG) is a kG module M which is also a G-graded vector space, such that if m ∈ M is a homogeneous element of degree g, then h.m is homogeneous of degree hgh −1 .Suppose we have a short exact sequence is a split short exact sequence of G-modules.We claim that the original sequence splits as D(kG)-modules.Pick a homogeneous basis {c i } of C under the G-grading, where c i has degree g i .Now pick a splitting s : C → B. Define ŝ(c i ) = p g i s(c i ).This map is homogeneous with respect to the G-grading, and it is still a G-module map: Since on the basis {c i } we have we have that ŝ is a splitting of D(kG)-modules.Now, to prove the original claim, suppose F(P) is projective as a G-module.Since F is exact, this means that for every short exact sequence is split as G-modules.Therefore, the original sequences are all split, and so P is projective.Proof.Since kG is cocommutative, mod(kG) is braided symmetric.By Lemma 3.1.1,we have X ∼ = 0 in StMod(D(H )) for all X ∈ K , and so we are in the situation given of Theorem 2.5.1(3).Additionally, since cohomology rings of groups are finitely generated (for instance by the more general result of [Friedlander and Suslin 1997], in which finite generation of cohomology rings for finite-dimensional cocommutative Hopf algebras in positive characteristic was proven), we know that Proj H 3.2.Cosemisimple Hopf algebras.In fact, we are able to generalize Lemma 3.1.1and Proposition 3.1.2from the group algebra case to the case certain finite-dimensional cosemisimple Hopf algebras.Recall that a finite-dimensional Hopf algebra is called cosemisimple if its Hopf dual is semisimple, as an algebra.
We first record the following straightforward lemma.
is an exact functor (recalling that projectives are also injective), and so D(H ) ⊗ H − preserves projectivity.Therefore, if P is a D(H )-module such that F(P) is projective, then D(H ) ⊗ H F(P) is a projective D(H )-module.But then, we have where the last isomorphism here can be seen from, e.g., [Garland and Lepowsky 1976, Proposition 1.7] and the remark following it, which notes that although the proposition is stated for certain universal enveloping algebras, in fact the proof uses only the Hopf algebra structure, and so the result holds for arbitrary Hopf algebras.Note that it holds not just for finite-dimensional modules, but for arbitrary modules, which we need since in this case P may be infinite-dimensional.Now, since Let H be a finite-dimensional unimodular cosemisimple Hopf algebra with Drinfeld double D(H ) and forgetful functor F : Mod(D(H )) → Mod(H ).Then F(P) is projective as an H -module if and only if P is projective as a D(H )-module.
Proof.This follows from Lemma 3.2.1 and the proof of [Etingof et al. 2015, Proposition 7.18.15].In the course of the proof of the latter, it is shown that if H is unimodular and cosemisimple, then 1 D(H ) is a direct summand of D(H ) ⊗ H 1 H as D(H )-modules (note that here, we are reversing the roles of H and H * given in their proof).Although this proposition assumes a stronger condition-that H itself is also semisimple-this assumption is not used for the part of the proof by which D(H ) ⊗ H 1 H has 1 D(H ) as a summand.By Lemma 3. Gelaki [1997, Theorem 1.3.6]has shown that every quasitriangular cosemisimple Hopf algebra is unimodular.Hence, again by Corollary 3.2.2 and Theorem 2.5.1, we conclude: Proposition 3.2.5.Let H be a finite-dimensional quasitriangular cosemisimple Hopf algebra.
(1) The map f constructed in Section 2.1 is a homeomorphism Spc stmod(H ) ∼ = −→ Spc stmod(D(H )), and the maps and as in Section 2.2 give inverse bijections between the thick ideals of stmod(H ) and stmod(D(H )).
(2) If Spc stmod(H ) is topologically Noetherian, then the W and W constructed in Section 2.2 are inverse maps, and so we have the following bijections of thick ideals: Of course, if H itself is also semisimple, then Propositions 3.2.4 and 3.2.5 are not particularly illuminating, since this implies that D(H ) is also semisimple, and then the Balmer spectra of stmod(H ) and stmod(D(H )) are both ∅.It is a classical theorem of Larson and Radford [1988a] that in characteristic 0, all cosemisimple finite-dimensional Hopf algebras are also semisimple.Hence, Propositions 3.2.4 and 3.2.5only provide interesting examples in positive characteristic.

3.3.
Benson-Witherspoon smash coproduct Hopf algebras.We will now consider the Benson-Witherspoon smash coproducts which were originally studied in [Benson and Witherspoon 2014], with generalizations studied in [Montgomery et al. 2016;Plavnik and Witherspoon 2018]; their Balmer spectra and thick ideals were classified in [Nakano et al. 2022a].We recall the general construction of these algebras.Let G and L be finite groups, such that L acts on G by group automorphisms, and let k be an algebraically closed field of characteristic dividing the order of G.We then define H G,L to be the Hopf algebra dual of the smash product k[G] # kL, where k[G] is the coordinate ring of G, and kL is the group algebra of L.
As an algebra, H G,L is isomorphic to kG ⊗ k[L].We will denote by { p x : x ∈ L} the standard dual basis for k[L], as in Section 3.1.Denote by e the identity element of L. The additional Hopf algebra structures of comultiplication, counit, and antipode on A are defined by (g ⊗ p x ) = y∈L (g ⊗ p y ) ⊗ (y −1 .g⊗ p y −1 x ), ϵ(g ⊗ p x ) = δ x,1 , S(g ⊗ p x ) = x −1 .(g−1 ) ⊗ p x −1 , for all g ∈ G and x ∈ L.
Since as an algebra H G,L ∼ = kG ⊗ k[L], an H G,L -module is the same as a G-module with an L-grading, such that the action of G preserves the L-grading.That is, every H G,L -module M may be decomposed By Noetherianity, this set is compact, and hence has a finite subcover where T ′ ⊆ T is some finite collection of objects.Enumerate the objects of T ′ by A 1 , . . ., A n .Choose Â1 , . . ., Ân as constructed in Lemma 3.3.3:they are in the image of F, and for any thick ideal I, we have A j ∈ I if and only if Â j ∈ I. Using this property, it is clear that V C (A j ) = V C ( Â j ) for all j.Now we claim that The last equality (more specifically, the containment ⊇) uses the fact that each Â j is in the image of F, and hence commutes with all objects of st(C) up to isomorphism, since this implies that Â1 ⊗ • • • ⊗ Ân ∈ P ⇒ A j ∈ P for some j.
Our claim is now shown: every closed set in Spc st(C) is of the form V C (A) for some object A.
In particular, each of the closed sets S i , for i ∈ I , can be written as V C (M i ) for some object M i ∈ st(C).As above, we can replace M i by M i , which is in the image of F, i.e., we can pick an object X i in st(Z(C)) with F(X i ) = M i .Since we have X i ∈ W (S) by definition.Hence, we now have Since we have both containments, we can conclude that W ( W (S)) = S for any specialization-closed set S in Spc st(C).□ We also note that if p does not divide the order of L, then we can apply the results of the previous section to obtain: Theorem 3.3.4.Let G, L, k, H G,L , C = mod(H G,L ), and Z(C) = mod(D(H G,L )) be as above, and assume additionally that p does not divide the order of L. Then we have the following.Proof.First, note that H G,L is cosemisimple: its dual is the smash product k[G] # kL.Since p does not divide the order of L, the group algebra kL is semisimple, and by [Cohen and Fishman 1986, Theorem 6], as the smash product of two semisimple algebras, k[G] # kL is semisimple as well.
Next, we claim that H G,L is unimodular.This can be observed directly, by noting that the element is both a left and a right integral in H G,L .By application of Propositions 3.3.1 and 3.2.4,f is bijective and the maps W and W are inverse bijections.To conclude the proof, we must just prove that f is closed, and hence a homeomorphism.This follows similarly to the proof of Theorem 2.5.1(3a), except that we must again use Lemma 3.3.3as a substitute for commutativity of the tensor product.Let V C (M) an arbitrary closed set, and, just as before, we may assume (by replacing M with M as in Lemma 3.3.3if need be) that M is in the image of F, and so we can pick X ∈ Z(st(C)) with F(X ) = M.We now have f (V C (M)) = { f ( P) : P ∈ Spc st(C), M ̸ ∈ P} = { Q ∈ Spc st(Z(C)) : X ̸ ∈ Q} = V Z (X ).
The second equality follows from the fact that f is bijective.Hence, f is closed, and the theorem is complete. □ Theorem B (See Proposition 2.4.1).Denote by K the kernel of F : St(Z(C)) → St(C), and f : Spc st(C) → Spc st(Z(C)) then if any two of A, B, and C are in I, then so is the third.(b) I is thick: if A ⊕ B is in I, then so are A and B. (c) I is an ideal: if A ∈ I, then so are A ⊗ B and B ⊗ A, for any object B.
For the details of the categorical localization and Verdier quotient, as well as additional details on the formation of the localization and colocalization functors, see [Neeman 2001, Section 2.1, Theorem 8.4.4;Krause 2010; Stevenson 2018, Section 3].
under the action induced by the functor M ⊗ − [Benson 1998, Section 5.7].Cohomological support is a support datum; the most nontrivial property is (

A⊗B
We will denote by F : Z(C) → C the forgetful functor sending (A, γ ) → A. If H is a Hopf algebra and C is the category of H -modules, it is well-known that the Drinfeld centerZ(C) of C isequivalent to the category of modules of D(H ) the Drinfeld (or quantum) double of H .For the details of Drinfeld doubles, see [Montgomery 1993, Section 10.3], [Chari and Pressley 1994, Section 4.2.D], [Kassel 1995, Section IX.4], or [Etingof et al. 2015, Section 7.14].The Drinfeld double D(H ) is isomorphic as a vector space to (H op ) * ⊗ H , and contains both H and (H op ) * as Hopf subalgebras.

Proposition 2 . 1 . 2 .
There is a functor F : St(Z(C)) → St(C) which extends the forgetful functor F, i.e.functor F is monoidal and triangulated.
Hence the collection of inequalities of(2.4.1)  becomes an equality, and we are done.For(2), similarly, we have by Corollary 2.3.5 that if Loc(I) contains K , then W • W (I) = I.Since K = Loc(K ∩ st(Z(C))), we have K ⊆ Loc(I) if and only if there is containment K ∩ st(Z(C)) ⊆ I.For the other direction, we note that for any ideal I, we have K ∩ st(Z(C)) ⊆ W • W (I), and so any thick ideal satisfying W • W (I) = I must have K ∩ st(Z(C)) ⊆ I as well.□ Remark 2.4.4.Corollary 2.4.3 implies that if C satisfies the conditions of Lemma 2.4.2, then the image of f : Spc st(C) → Spc st(Z(C)) is automatically the complement of a specialization-closed set, since we have im

( a )
The map f is injective.(b) As maps ThickId(st(C)) → ThickId(st(C)), we have • = id.(c) If additionally Spc st(C) is topologically Noetherian, then W • W = id.(3) If X ∼ = 0 in St(Z(C)) for all X ∈ K and C is braided, then the following hold.(a) The map f is a homeomorphism.(b) The maps and define mutually inverse bijections between ThickId st(Z(C)) and ThickId(st(C)).(c) If additionally Spc st(C) is topologically Noetherian, then W and W are mutually inverse bijections between ThickId st(Z(C)) and X sp Spc(st(C)) .

3. 1 .
Group algebras and dual group algebras.Let G be a finite group, k be an algebraically closed field of characteristic p which divides the order of G, and kG the group algebra of G over k.Let C = mod(kG), a finite tensor category.The Drinfeld double D(kG) is a Hopf algebra containing kG and (kG op ) * as Hopf subalgebras.We will denote the dual of the group algebra by k[G], and in that case we can write(kG op ) * = k[G] cop .The collection { p g h : g, h ∈ G} is a k-basis of D(kG), where the elements { p g : g ∈ G} refer to the basis of k[G] cop dual to the standard basis of kG.The multiplication is determined by the relations hp g = p hgh −1 h, see for instance[Kassel 1995, Section IX.4.3].Lemma 3.1.1.Let G and k be as above and F : Mod(D(kG)) → Mod(kG) the forgetful functor.Then if F(P) is projective as a kG-module, then P is projective as a D(kG)-module.
st(C).Then the complement of V C (T ) is by definition U C (T ) = { P ∈ Spc st(C) : P ∩ T ̸ = ∅}, and has an open cover U C (T ) = A∈T U C (A) = A∈T { P ∈ Spc st(C) : A ∈ P}.

( 2 )⇄
The maps W and W constructed in Section 2.2 are mutually inverse, and so we have the following bijections of thick ideals:ThickId stmod(D(H G,L )) ThickId(stmod(H G,L )).
;Boe et al. 2017a;  2017b; Nakano et al. 2022a; 2022b]for examples) by constructing a support datum for which these maps are bijective and inverse to each other.In that case, the support datum σ is called classifying.For rigid braided monoidal triangulated categories T , the Balmer support V T is always classifying[Balmer 2005,  Theorem 4.10].
• (G, k) is a Noetherian topological space.Using Balmer's classification of thick ideals [2005, Theorem 4.10], the thick ideals of stmod(kG) are in bijection with specialization-closed sets in Spc stmod(kG).The rest of the theorem now follows directly as an application of Theorem 2.5.1.□ Now, note that since k[G] cop is a semisimple algebra, stmod(k[G] cop ) consists only of the zero object, up to isomorphism, and so Spc(stmod(k[G] cop )) is the empty set.Thus, the diagram from Remark 2.1.4becomes 2.1.Let H be a finite-dimensional Hopf algebra such that 1 D(H ) is a direct summand of D(H )⊗ H 1 H as D(H )-modules, and let F : Mod(D(H )) → Mod(H ) be the forgetful functor.Then F(P) is projective in Mod(H ) if and only if P is projective in Mod(D(H )).Proof.The functor D(H ) ⊗ H − is a left adjoint to the forgetful functor F [Kauffman and Radford 1995;Hennings 1996]ence P is projective as well, and the claim is proven.□RecallthataHopfalgebra(or,moregenerally, a tensor category) is called unimodular if its spaces of left and right integrals coincide (see[Montgomery 1993, Section 2.1; Etingof et al. 2015, Section 6.5]).Unimodular Hopf algebras are of particular interest due to their use in constructing Hennings-Kauffman-Radford invariants for 3-manifolds[Kauffman and Radford 1995;Hennings 1996].In light of Shimizu's result[2017, Theorem 4.10] on unimodular finite tensor categories, if H satisfies the conditions of Lemma 3.2.1 -that is, if 1 D(H ) is a direct summand of D(H ) ⊗ H 1 H -then H must be unimodular.The converse is not true; the dual of a finite group algebra is unimodular [Shimizu 2017, Corollary 5.5], but 1 Kaplansky [1975]ry follows.□Remark3.2.3.The condition that H is unimodular in Corollary 3.2.2 is not too restrictive.It is a long-standing conjecture ofKaplansky [1975]that finite-dimensional cosemisimple Hopf algebras are involutory (i.e., the square of the antipode is the identity).In view of results ofLarson [1971, Corollary 4.2], a weaker form of the Kaplansky conjecture is that all finite-dimensional cosemisimple Hopf algebras are unimodular[Aljadeff et al. 2002, Remark 3.9].This conjecture is still open.
Proposition 3.2.4.Let H be a finite-dimensional unimodular cosemisimple Hopf algebra.Then the map f : Spc stmod(H ) → Spc stmod(D(H )) constructed in Section 2.1 is surjective, and the maps • and W • W (as in Section 2.2) are each the identity, as maps from the collection of thick ideals of stmod(D(H )) to itself.