aa r X i v : . [ m a t h . C T ] S e p Centers of categorified endomorphism rings
Alexandru Chirvasitu
Abstract
We prove that for a large class of well-behaved cocomplete categories C the weak and strongDrinfeld centers of the monoidal category E of cocontinuous endofunctors of C coincide. Thisgeneralizes similar results in the literature, where C is the category of modules over a ring A and hence E is the category of A -bimodules. Key words: Drinfeld center; weak center; locally presentable category; 2-abelian group; 2-ring;dualizable; linearly reductive
MSC 2020: 18D15; 18N10; 18E50; 16B50; 16T15
Introduction
The present note is motivated by the following result from [2] (see Theorem 2.10 therein):
Theorem 0.1
For a ring A , the weak and strong centers of the monoidal category A M A of A -bimodules coincide. We give a refresher on the terminology in § A ⊗ V → V ⊗ A of A ⊗ A -bimodules (for a bimodule V ∈ A M A underlying a weak-center object) is automaticallyan isomorphism. The proofs of [2, Propositions 2.5 and 2.6] make it clear that this is the type ofrigidity phenomenon familiar from the theory of descent in ring theory and / or algebraic geometry[13]. In the latter setup one typically starts with commutative rings R → S and an S -module M and seeks to recover an R -module M R such that M ∼ = S ⊗ R M R ;in other words, the goal is to descend the S -module to an R -module. The sort of structure necessaryto achieve this in good cases (e.g. S is faithfully flat over R [13, Th´eor`eme 3.2]) is a descent datum (see [13, discussion preceding Proposition 3.1]): an S ⊗ S -module morphism g : S ⊗ M → M ⊗ S (where ‘ ⊗ ’ means ‘ ⊗ R ’) such that 11) the diagram S ⊗ S ⊗ M S ⊗ M ⊗ S M ⊗ S ⊗ S g g g commutes (with the indices indicating the tensorands on which g operates), and(2) the morphism M → S ⊗ M g −→ M ⊗ S → M is the identity, where the leftmost arrow is the natural inclusion obtained by tensoring the unit R → S of S with id M and the rightmost arrow is multiplication by scalars in S .Under these circumstances it turns out [13, Proposition 3.1] that in fact g is automatically anisomorphism. This is essentially the same phenomenon as that captured in Theorem 0.1 in thebroader context of non-commutative rings.In attempting to isolate precisely what it is about categories of bimodules that occasions suchrigidity results one is led to consider the celebrated Eilenberg-Watts theorem ([22, Theorem 1] or[9]): • A M A is equivalent to the category of cocontinuous (i.e. colimit-preserving [15, § V.4]) endo-functors of the category A M of left A -modules (or its right-handed version M A ), • such that the monoidal structure given by ‘ ⊗ A ’ is identified with endofunctor composition.This is the starting point for the generalization of Theorem 0.1 appearing as Theorem 2.1 below.The pattern we extrapolate can be summarized as follows (with a forward reference to § • One can substitute other “well-behaved” cocomplete categories C for A M ; • and their duals C ∗ ∼ = consisting of cocontinuous functors C → (some “base” category) for M A ; • and their endomorphism 2-rings E := C ⊠ C ∗ ∼ = cocontinuous endofunctors of C (0-1)for A M A .For (0-1) to be both meaningful and valid C needs to be what in Definition 1.6 (and [5, Definition1.1]) we refer to as dualizable (this is what ‘well-behaved’ means in the above discussion). With allof this behind us, Theorem 2.1 reads more or less as follows. Theorem 0.2 If C is a dualizable locally presentable category then the weak center of its categoryof cocontinuous endofunctors coincides with its strong center. In addition to recovering Theorem 0.1, this applies to categories C going beyond modules, aswe recall in Section 2: C can be, for instance, • the category M C of right-comodules over a right-semiperfect ([14, p.369]) coalgebra over afield; • the category Qcoh ([ X/G ]) of quasicoherent sheaves over the quotient stack [
X/G ] where X is affine and G is a virtually linearly reductive [8, §
1] linear algebraic group acting on X .2 cknowledgements This work was partially supported by NSF grant DMS-2001128.I am grateful for illuminating comments by Ana Agore.
Some standard background on monoidal categories is needed, as covered for instance in [11, ChapterXI], [6, § We reprise some terminology from [7, § Definition 1.1 (a) A is a locally presentable category in the sense of [1, Definition1.17]. 2-abelian groups form a 2-category with left adjoints as 1-morphisms and naturaltransformations as 2-morphisms.(b) A is a 2-abelian group C which is in addition a monoidal category with tensor product‘ ⊗ ’, so that all functors of the form x ⊗ − and − ⊗ x are left adjoints. 2-rings similarly form a2-category with monoidal left adjoints as 1-morphisms.(c) A commutative 2-ring is a 2-ring additionally equipped with a symmetry (i.e. it is a symmetricmonoidal category). As before, these form the 2-category with symmetric monoidalleft adjoints as 1-morphisms. (cid:7) It turns out (e.g. [7, Corollary 2.2.5]) that is symmetric monoidal, being equipped with atensor product denoted by ‘ ⊠ ’. For 2-abelian groups A and B their tensor product A ⊠ B is theuniversal recipient of a bifunctor A × B → A ⊠ B that is separately cocontinuous (i.e. a “bilinear map” of 2-abelian groups). The symmetric monoidalstructure lifts to and in the sense that if A and B are 2-rings so is A ⊠ B in anatural fashion, etc.This machinery allows us to employ the usual language of rings and modules in the context of2-abelian groups: Definition 1.2
Let R be a 2-ring. A left (2-) R -module is a 2-abelian group X equipped with amorphism R ⊠ X → X in , satisfying the obvious unitality and associativity conditions. Right (2-) R -modules are defined analogously, as are bimodules, etc.The respective 2-categories of left or right or bimodules are denoted by R M , M R and R M S .respectively. (cid:7) As usual, we have tensor product 2-bifunctors R M S × S M T ⊠ S −→ R M T and in particular, for a commutative R , the 2-category R M ∼ = M R is symmetric monoidalunder ⊠ R . Definition 1.3
For a commutative 2-ring R an R -algebra (or 2- R -algebra for extra precision) isan algebra in the symmetric monoidal 2-category R M . (cid:7)
3t turns out that is not only symmetric monoidal but also
Cartesian-closed , i.e. has internalhoms . More precisely, we have the familiar hom-tensor adjunction in the present higher-categoricalsetting (see e.g. [12, § Lemma 1.4
Let R , S and T be three 2-rings.(a) For any two bimodules • X ∈ R M T • Y ∈ S M T the category Hom T ( Y , X ) := { left adjoints X → Y compatible with the 2-module structures } has a natural structure of a R - S -bimodule.(b) This gives us, for each bimodule Y ∈ S M T , a 2-adjunction R M S R M T− ⊠ S Y Hom T ( Y , − ) with the top arrow as the left (2-)adjoint. (cid:4) Remark 1.5
We leave it to the reader to formulate analogous versions for tensoring on the leftrather than right, etc. (cid:7)
Definition 1.6
Let R be a 2-ring and X a left R -module.(a) The dual X ∗ of X over R is R Hom ( X , R ); it is a right R -module. Similarly, duals of rightmodules are naturally left modules.(b) If R is commutative the 2- R -module X is (1-)dualizable over R if the canonical morphism X ⊠ R X ∗ can −→ End R ( X ) (1-1)is an isomorphism of 2- R -modules. (cid:7) Remark 1.7
For any 2-ring R and 2- R -module X End R ( X ) is naturally a 2-ring (and a 2- R -algebra when R is commutative), with composition as the tensor product and id X as the unit. (cid:7) Dualizable objects (typically over R = Vect K for some field K ) were the focus of [5], where wegive alternative characterizations of dualizability in [5, Lemma 3.1]. In particular, it is enough torequire that the identity id X ∈ End R ( X )belong to the image of (1-1). 4 .2 Centers Recall (e.g. [11, Definition XIII.4.1] or [10, Definition 3]):
Definition 1.8
Let ( C , ⊗ , ) be a monoidal category. The (Drinfeld) center Z ( C ) of C is thecategory of pairs ( x, θ ) where x ∈ C is an object and θ : − ⊗ x ∼ = −→ x ⊗ − (1-2)is a natural isomorphism satisfying the following conditions (suppressing the associativity con-straints in the monoidal category):(1) For y, z ∈ C the diagram y ⊗ z ⊗ x y ⊗ x ⊗ z x ⊗ y ⊗ z id y ⊗ θ z θ y ⊗ id z θ y ⊗ z commutes, and(2) the isomorphism θ : ⊗ x → x ⊗ is the canonical one attached to the monoidal structure ( C , ⊗ , ). (cid:7) Remark 1.9
In fact, in Definition 1.8 condition (2) follows from (1), but this uses the fact that θ is an isomorphism ; we have displayed both conditions with an eye towards Definition 1.10 below. (cid:7) Following [19, Definition 4.3] (where the notion seems to have been introduced) and [2, § Definition 1.10
For ( C , ⊗ , ) as in Definition 1.8 the weak right center W Z r ( C ) is the category ofpairs ( x, θ ) as above, satisfying conditions (1) and (2), but requiring only that (1-2) be a naturaltransformation.One defines the weak left center W Z ℓ ( C ) analogously, requiring a natural transformation x ⊗ − → − ⊗ x instead.Unless specified otherwise weak center means weak right center, and we simply write W Z for
W Z r . (cid:7) Theorem 2.1
Let R be a commutative 2-ring, X ∈ R M a dualizable R -module, and E := X ⊠ R X ∗ ∼ = End R ( X ) (2-1) its endomorphism ring. Then, the canonical fully faithful inclusion Z ( C ) → W Z ( C ) (2-2) is an equivalence. C where the canonical functor (2-2) is not an equivalence. Example 2.2
Consider the category C of modules over the ring of dual numbers R := k [ ε ] / ( ε )for some field k , equipped with its usual symmetric monoidal category structure given by ⊗ := ⊗ R .We have an object ( R, θ ) ∈ W Z ( C ) with y ∼ = y ⊗ R R θ y −→ R ⊗ R y ∼ = y defined to be • the standard isomorphism when y is projective (i.e. a direct sum of copies of R ), and • zero otherwise.Clearly, the object ( R, θ ) ∈ W Z ( C ) is not in the image of (2-2). (cid:7) The preceding example is in fact one of a broad family:
Example 2.3
Let ( C , ⊗ , ) be any symmetric, additive monoidal category and J ⊆ C a primeideal (e.g. [17, § § • an ideal, i.e. a full subcategory which along with an object z contains all objects z ⊗ z ′ and z ′ ⊗ z obtained by tensoring, and • prime, i.e. it does not contain the monoidal unit, and along with x ⊗ y it contains one of theobjects x and y .We can now define an object ( , θ ) ∈ W Z ( C ) that is not in the image of (2-2) as follows: for y ∈ C set (cid:18) y ∼ = y ⊗ θ y → ⊗ y ∼ = y (cid:19) = ( the identity if y
6∈ J y ∈ J This specializes back to Example 2.2 for C = R M and J the ideal of non-projective modules, i.e.those in which ε ∈ R = k [ ε ] / ( ε ) has non-trivial annihilator. (cid:7) Since R is our “base ring” throughout the discussion we henceforth abbreviate ‘ ⊠ R ’ to simply‘ ⊠ ’, and similarly for Hom := Hom R . Recall also our notation (2-1) for the endomorphism 2-ring E of X .Because X is assumed dualizable over R , the canonical morphism X → X ∗∗ is an isomorphism (of abelian 2-groups, i.e. an equivalence of categories). It follows from this that E is also dualizable and in fact self-dual, and we can identify End R ( E ) ∼ = E ⊠ E ∗ ∼ = E ⊠ E ∼ = X ⊠ X ∗ ⊠ X ⊠ X ∗ . (2-3)Given that is a symmetric monoidal 2-category, there is some choice in how we identify theright and left-hand sides of (2-3). In the sequel, it will be convenient to make this identification6y pairing the two middle tensorands on the right-hand side of (2-3) against E ∼ = X ⊠ X ∗ in theobvious fashion (by pairing each X to a X ∗ ).Now fix an object ( e, θ ) ∈ W Z ( E ) of the weak right center and consider the two endomorphisms − ⊗ e and e ⊗ − ∈ End ( E ) ∼ = E ⊠ E . With the above convention in mind, they are identifiable, respectively, with ⊠ e and e ⊠ (2-4)where := id X = E ∈ E = End ( X ) ∼ = X ⊠ X ∗ is the identity functor on X (i.e. the monoidal unit of E ). We caution the reader that the tensorproduct in (2-4) is external , i.e. it is not to be confused with the internal tensor product ‘ ⊗ ’ of E .Indeed, under the latter we of course have ⊗ e ∼ = e ∼ = e ⊗ (as in any monoidal category).The natural transformation − ⊗ e → e ⊗ − that constitutes the structure of a weak-center element (Definition 1.10) translates to a morphism ⊠ e θ −→ e ⊠ (2-5)in E ⊠ E (denoted slightly abusively by the same symbol ‘ θ ’ we used for the natural transformationin Definition 1.10). The conditions (1) and (2) can then be recast in terms of (2-5) as we explainpresently.To express condition (1) we need to work in the triple tensor product E ⊠ . To that end, weconsider morphisms between tensor products of e and two copies of , with two indices among 1,2 and 3 indicating where θ operates. Thus: θ := θ ⊠ id : ⊠ e ⊠ → e ⊠ ⊠ ,θ := id ⊠ θ : ⊠ ⊠ e → ⊠ e ⊠ , and similarly, θ : ⊠ ⊠ e → e ⊠ ⊠ is the morphism acting identically on the middle tensorand and as θ on the two outer ones. (1) inDefinition 1.8 can now be recovered simply as θ = θ ◦ θ : ⊠ ⊠ e → e ⊠ ⊠ . (2-6)Next, denote by m : E ⊠ E → E the “multiplication” morphism, imparting on E = End ( X ) its monoidal category structure. Interms of the decomposition E ⊠ E ∼ = X ⊠ X ∗ ⊠ X ⊠ X ∗ is simply the evaluation of the two middle tensorands X and X ∗ against each other. With thisin place, condition (2) in Definition 1.8 simply asks that m ( θ ) : ⊗ e → e ⊗ be the canonical isomorphism, i.e. the identity once we have made the usual identifications ⊗ e ∼ = e ∼ = e ⊗ . In short, for future reference: m ( θ ) = id e : e ∼ = ⊗ e → e ⊗ ∼ = e. (2-7) Proof of Theorem 2.1
Since we already know that (2-2) is fully faithful (as is immediate fromDefinitions 1.8 and 1.10), it remains to show that it is essentially surjective: for an arbitrary object( e, θ ) ∈ W Z ( E ) the morphism (2-5) is an isomorphism in E ⊠ E . What we will in fact do is identifythe inverse of θ : it is precisely θ ′ := τ ◦ θ ◦ τ : e ⊠ → ⊠ e, where τ is the tensorand-reversal functor on E ⊠ E .Denote by m : E ⊠ E ⊠ E → E ⊠ E the functor that multiplies the outer (first and third) tensorands of the domain onto the secondtensorand of the codomain. We then have θ = m ( θ ) and θ ′ = m ( θ ) , meaning that m ( θ ) = m ( θ ◦ θ ) = θ ′ ◦ θ (2-8)(where the first equality uses (2-6) above). On the other hand though, (2-7) implies that theleft-hand side m ( θ ) of (2-8) is nothing but the identity, and thus θ ′ ◦ θ = id ⊠ e . The other composition θ ◦ θ ′ is treated similarly, so we do not repeat the argument. (cid:4) Remark 2.4
The proof of Theorem 2.1 given above is a paraphrase, in the present categorifiedcontext, of an argument familiar from descent theory. See e.g. [13, Proposition 3.1]. Where X would have been the category of modules over (in those authors’ notation) a ring S . (cid:7) In the context of k -linear 2-abelian groups (for some field k ) the examples of dualizable C in [5]are all, abstractly, of the form k -linear functors Γ op → k Vect (2-9)for small k -linear categories Γ. These are also • the k -linear abelian categories admitting a generating set of small projective objects [18, § the k -linear locally presentable admitting a strongly generating set of small projective objects[12, Theorem 5.26];recall that the small projective objects in an abelian category C are simply those x ∈ C for whichthe representable functor hom( x, − ) : C →
Set is cocontinuous. This is taken as the definition of the hyphenated term ‘ strong-projective ’ in [12, § all categories of the form (2-9) are dualizable 2-modulesover k Vect . The same argument goes through for arbitrary commutative 2-rings R (in place of k Vect ), so we have
Corollary 2.5
Let R be a commutative 2-ring and X a 2- R -module with a strong generating setof small-projective objects. Then, the weak center of the monoidal category X ⊠ R X ∗ ∼ = End R ( X ) coincides with its strong center. (cid:4) We end with some examples of categories falling under the scope of Corollary 2.5 (and henceTheorem 2.1).
Example 2.6
Throughout the present discussion we assume C is a coalgebra over a field.By [5, Theorem 1.3], Theorem 2.1 applies to categories of right comodules M C over right-semiperfect coalgebras C in the sense of [14, p.369]: every right C -comodule has a projective cover.This of course includes cosemisimple coalgebras (i.e. those with only projective modules orequivalently, direct sums of simple coalgebras; [21, Definition, p.290] or [16, Definition 2.4.1]). (cid:7) Example 2.7
Overlapping Example 2.6 to a degree, consider a linear algebraic group [4, § G acting on an affine scheme X and the category Qcoh ( X ) G ∼ = Qcoh([X/G]) (2-10)of G -equivariant quasicoherent sheaves on X , or equivalently, as (2-10) recalls ([20, Tag 06WV]),that of quasicoherent sheaves on the quotient stack [20, Tag 044O] [ X/G ].According to [5, Theorem 1.5] (2-10) is dualizable provided G is virtually linearly reductive inthe sense of [8, § G has a normal linearly reductive closed algebraic subgroup H E G such that G/H is a finite group scheme. This means that • the Hopf algebra O ( H ) is cosemisimple while O ( G/H ) is finite-dimensional; • equivalently by [8, Theorem, p.76], the Hopf algebra O ( G ) of regular functions on G is (leftand right) semiperfect in the sense of Example 2.6. (cid:7) Example 2.8
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Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail address : [email protected]@buffalo.edu