Kernels for noncommutative projective schemes › farmanb › files › 2019 › 12 › KNCP.pdf · mutative graded rings has proven itself fruitful as well. Indeed, moduli spaces

Kernels for noncommutative projectiveschemes

Matthew Ballard Blake Farman

Abstract

We give a noncommutative geometric description of the inter-nal Hom dg-category in the homotopy category of dg-categories be-tween two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutativeprojective derived Morita statement along the lines of Rickard andOrlov.

Keywords. noncommutative algebra, noncommutative projectiveschemes, derived categories, Fourier-Mukai transforms

1 Introduction

Derived categories in algebraic geometry have proven themselves to be anenormously useful tool in studying birational geometry [7, 10, 13], modulitheory [14, 28, 29], and have relations to other fields like representationtheory of finite dimensional algebra [8, 18] and symplectic geometry [33,34], through mirror symmetry [24].

M. Ballard: Department of Mathematics, University of South Carolina, Columbia,SC 29208 USA; e-mail: [email protected]

B. Farman: Department of Mathematics, Lafayette College, Easton, PA 18042 USA;e-mail: [email protected]

Mathematics Subject Classification (2010): Primary 14A22; Secondary 16E35

1

2 Matthew Ballard, Blake Farman

At the same time, as conceived by Artin and Zhang [4] under the nameof noncommutative projective schemes, using tools of category theory and(commutative) algebraic geometry to understand the landscape of noncom-mutative graded rings has proven itself fruitful as well. Indeed, modulispaces of point modules form the key technical tool in the classificationof noncommutative P2’s [2,3,36,37]. Furthermore, a natural conjecture ofArtin classifies the noncommutative surfaces up to birational classificationof noncommutative surfaces [1].

If one is seriously interested in using algebro-geometric techniquesto study noncommutative graded rings, then focusing on and exploitingderived categories of noncommutative projective schemes is an obviousand seemingly fertile avenue. The structure of noncommutative projec-tive schemes is shaped by moduli theory and birational geometry, almostthe exact areas where derived categories realize their full power in (com-mutative) algebraic geometry. Additionally, Calabi-Yau noncommutativeprojective schemes can provide geometric interpretations for some non-geometric N = 2 superconformal field theories compactifying IIB strings[5]. Some hints of this are already in the literature. Two glimpses of thisare: Li and Zhao [25] show how Bridgeland stability for noncommuta-tive P2’s provides access to the Minimal Model Program for commuta-tive deformations of Hilbert schemes of points; Harder and Katzarkov [19]describe Homological Mirror Symmetry for four-dimensional quadraticSklyanin algebras.

In a first course on derived categories in algebraic geometry, one learnsthat the power and influence of derived categories lies in the geometric no-tion of a kernel. Consider the derived categories of quasi-coherent sheaves,D(X) and D(Y ), for two varieties X and Y . For a general exact functorF : D(X) → D(Y ), one has almost no control; it is pure abstract smoke.However, in the vast majority of problems, we are lucky to not be inter-ested in such a general functor, but one of a specific provenance. From an

Noncommutative kernels 3

object K ∈ D(X × Y ) we can construct an exact functor

ΦK : D(X)→ D(Y )

E 7→ q∗ (K ⊗ p∗E)

called an integral transform, in obvious analogy with analysis, where p :

X×Y → X and q : X×Y → Y are the projections. Continuing the anal-ogy, we call K the kernel of the integral transform. Kernels and integraltransforms categorify the notion of correspondences between varieties andthey naturally arise in moduli theory as universal objects and in birationalgeometry as objects on resolutions of rational maps.

Moreover, if we work with dg-enhancements, then thanks to a theoremof Toen [38] we know that integral transforms are all we need to study.Precisely, in his seminal work [38], Toen showed that

1. (existence of internal Hom) the localization of the category of dg-categories at quasi-equivalences admits an internal Hom, RHom,and

2. (geometric recognition) the subcategory of the Hom between thedg-enhancements of D(X) and D(Y ) consisting of quasi-functorscommuting with coproducts is isomorphic in Ho (dgcat) to the en-hancement of the derived category of the product X × Y ,

RHomc (D(X),D(Y )) ∼= D(X × Y ).

It is important to not confuse the two issues; knowing (1) in no way helpswith establishing (2). However, if we know that the functor in which wehappen to be interested admits a lift to a dg quasi-functor, then by (2) itmust be an integral transform and it must be geometric.

Predating Toen’s result, Orlov had showed that any equivalence F :

Db(cohX)→ Db(cohY ) between smooth and projective varieties is actu-ally isomorphic to ΦK for some kernel K [31]. Being revisionist, we cansay that, for varieties, equivalences lift to quasi-equivalences at the differ-ential graded level. That is to say, if there is an exact equivalence of triangu-lated categories of homotopy categories, then there is a quasi-equivalence


of the dg-categories. Post-Toen, Lunts and Orlov exhibited this lifting invastly more generality for exact equivalences between derived categoriesof abelian categories [26].

If we are to study derived categories of noncommutative projectiveschemes, we are then at the intersection of Artin-Zhang and Kontsevichstyle noncommutative geometry. The most basic issue is to know what thekernels and integral transforms are, and whether the pleasant results in thecontext of (commutative) algebraic geometry persist in this noncommuta-tive setting.

We can ask, for example, if we have an analogue of Orlov’s result on thegeometric nature of equivalences. Combining Toen’s and Lunts-Orlov’sresults, one immediately concludes that if X and Y are noncommutativeprojective schemes and one has an equivalence F : D(X) → D(Y ), thenin fact one has a quasi-equivalence F : D(X)→ D(Y ).

This is great, however, we are still stuck staring at the abstract smoke.To reap the benefits, we need to know that F is (noncommutative) geomet-ric. That is, we need the noncommutative projective analogue of Toen’sgeometric recognition. Existence of the internal Hom and/or uniqueness ofenhancements provides no guidance towards geometric recognition in anycontext. As such we encounter the following basic questions:

Question. For noncommutative projective schemes X and Y , what non-commutative projective scheme is X × Y ? Does geometric recognitionhold for X and Y (and X × Y )?

An intermediate issue is to provide a definition of integral transformin noncommutative projective geometry, which is entirely separate fromthe differential graded structure. No such creature has been observed inthe literature. Encouragingly, one notes that geometric recognition holdsin other settings beyond schemes:

• for higher derived stacks (using machinery of Lurie in place of Toen)in [9]

• and for various versions of matrix factorizations [6, 16, 32].


However, a look at the simpler question of graded Morita theory shows thatanswers for noncommutative projective schemes are already more compli-cated [43]. For commutative graded rings, Morita equivalence is the samething as isomorphism.

Fortunately, there is really only one noncommutative projective schemethat deserves to be called X ×Y , the Segre product of X and Y . However,the answer to geometric recognition is clearly only positive with some ho-mological restrictions on X and Y . It does not hold in general. Let us nowstate a version of the main result of this article.

Let A and B be connected graded k-algebras. We say that A and B

form a delightful couple if they are both Ext-finite in the sense of [40],both are left and right Noetherian, and both satisfy χ◦(R) for R = A,Aop

forA andR = B,Bop forB [4]. One can think of this requirement as Serrevanishing for the twisting sheaves plus some finite-dimensionality over k.LetX and Y be the associated noncommutative projective schemes, whichwe also say form a delightful couple.

Theorem 1.1. Let X and Y be noncommutative projective schemes asso-ciated to a delightful couple, A and B, over a field k. Assume that both Aand B are both generated in degree one. Then geometric recognition holdsfor X and Y . That is, there exists a quasi-equivalence

RHomc (D(X),D(Y )) ∼= D(X × Y ).

For a general delightful couple, geometric recognition holds, howeverone must step slightly outside the realm of noncommutative projectiveschemes, without losing the (noncommutative) geometry, to get the cor-rect product. See Theorem 4.15 for the precise statement of the generalresult.

As an immediate corollary to Theorem 1.1, we get the following state-ment which is a geometricity of equivalences statement along lines ofOrlov or Rickard.

Theorem 1.2. Let X and Y be noncommutative projective schemes asso-ciated to a delightful couple over a field k, both of which are generated in


degree one. If there is an exact equivalence F : D(X)→ D(Y ), then thereexists an object K of D(X × Y ) whose associated integral transform ΦK

is an equivalence. That is, X and Y are Fourier-Mukai partners.

Recall that ΦK is introduced in this paper. For this, see the statement ofTheorem 4.15. Note that this statement makes no reference to dg-categoriesand by restricting to commutative projective varieties recovers the analo-gous result there.

1.1 Conventions

We let k denote a field. Often, for ease of notation, C(X, Y ) will be usedto refer to the morphisms, HomC(X, Y ), between objects X and Y of acategory C. We shall also use an undecorated Hom again depending on thecomplexity of the notation. Whenever C has a natural enrichment over acategory, V , we will denote by C(X, Y ) the V-object of morphisms. Forexample, the category of complexes of k-vector spaces, C (k), can be en-dowed with the structure of a C (k)-enriched category using the Hom totalcomplex, C (k) (C,D) := C(k)(C,D) which has in degree n the k-vectorspace

C (k) (C,D)n =∏m∈Z

Mod k(Cm, Dm+n

)and differential

d(f) = dD ◦ f + (−1)n+1f ◦ dC .

It should be noted that Z0(C (k) (C,D)) = C (k) (C,D).

2 Background on DG-Categories

Recall that a dg-category, A, over k is a category enriched over the cat-egory of cochain complexes, C (k), a dg-functor, F : A → B is a C (k)-enriched functor, a morphism of dg-functors of degree n, η : F → G,is a C (k)-enriched natural transformation such that η(A) ∈ B (FA,GA)n

for all objects A of A, and a morphism of dg-functors is a degree zero,


closed morphism of dg-functors. We will denote by dgcatk the 2-categoryof small C (k)-enriched categories, and by dgcat

k(A,B) the dg-category

of dg-functors from A to B.Recall also that for A and B small dg categories, we may define a dg-

category A⊗ B with objects ob(A)× ob(B) and morphisms

(A⊗ B) ((X, Y ), (X ′, Y ′)) = A(X,X ′)⊗k B(Y, Y ′).

It is well known that there is an isomorphism

dgcatk(A⊗ B, C) ∼= dgcatk(A, dgcatk(B, C)),

endowing dgcatk with the structure of a symmetric monoidal closed cate-gory.

For any dg-category, A, we denote by Z0(A) the category with objectsthose of A and morphisms

Z0(A)(A1, A2) := Z0(A(A1, A2)).

By H0(A) we denote the category with objects those of A and morphisms

H0(A)(A1, A2) := H0(A(A1, A2)).

Following [15], we say that two objects A1, A2 of a dg-category, A, aredg-isomorphic (respectively, homotopy equivalent) if there is a mor-phism f ∈ Z0(A)(A1, A2) such that f (respectively, the image of f inH0(A)(A1, A2)) is an isomorphism. In such a case, we say that f is a dg-isomorphism (respectively, homotopy equivalence).

2.1 The homotopy category of DG-Categories

We collect here some basic results on the model structure for dgcatk. Forany dg-functor F : A → B, we say that F is

(i) quasi-fully faithful if for any two objects A1, A2 ofA the morphism

F (A1, A2) : A(A1,A2)→ B(FA1, FA2)

is a quasi-isomorphism of chain complexes,


(ii) quasi-essentially surjective if the induced functor

H0(F ) : H0(A)→ H0(B)

is essentially surjective,

(iii) a quasi-equivalence if F is quasi-fully faithful and quasi-essentiallysurjective,

The localization of dgcatk at the class of quasi-equivalences is the homo-topy category, Ho (dgcatk). We will denote by [A,B] the morphisms ofHo (dgcatk).

2.2 dg Modules

For any small dg-category,A, denote by dgMod (A) the dg-category of dg-functors dgcat

k(Aop, C (k)), where C (k) denotes the dg-category of chain

complexes equipped with the internal Hom from its symmetric monoidalclosed structure. The objects of dgMod (A) will be called dg A modules.Since one may view the dg Aop modules as what should reasonably becalled left dg A modules, the terms right and left will be dropped in favorof dg A modules and dg Aop modules, respectively. We note here that thesomewhat vexing choice of terminology is such that we can view objectsof A as dg A modules by way of the enriched Yoneda embedding

YA : A → dgMod (A) .

As a special case, we define for any two small dg-categories, A and B,the category of dg A-B-bimodules to be dgMod (Aop ⊗ B). We note herethat the symmetric monoidal closed structure on dgcatk allows us to viewbimodules as morphisms of dg-categories by the isomorphism

dgMod (Aop ⊗ B) = dgcatk

(A⊗ Bop, C (k))

∼= dgcatk(A, dgcat

k(Bop, C (k)))

= dgcatk

(A, dgMod (B)) .

The image of a dgA-B-bimodule, E, is the dg-functor ΦE(A) = E(A,−).


2.2.1 h-Projective dgModules

We say that a dgAmodule, N , is acyclic if N(A) is an acyclic chain com-plex for all objects A of A. A dg A module M is said to be h-projectiveif

H0(dgMod (A))(M,N) := H0(dgMod (A) (M,N)) = 0

for every acyclic dg A module, N . The full dg-subcategory of dgMod (A)

consisting of h-projectives will be called h-Proj (A).We always have a special class of h-projectives given by the representa-

bles, hA = A(−, A) for if M is acyclic, then from the enriched YonedaLemma we have

H0(dgMod (A))(hA,M) := H0(dgMod (A) (hA,M))

∼= H0(M(A)) = 0.

Noting that the Yoneda Lemma applied to H0(dgMod (A)) immediatelyimplies h-Proj (A) is closed under homotopy equivalence, we denote thefull dg-subcategory of h-Proj (A) consisting of the dg A modules homo-topy equivalent to representables by A.

An h-projective dg A-B-bimodule, E, is right quasi-representable iffor every object A of A the dg B module ΦE(A) is an object of B, and wewill denote the full subcategory of h-Proj (Aop ⊗ B) consisting of all rightquasi-representables by h-Proj (Aop ⊗ B)rqr.

The dual notion, h-injective, is defined by reversing all the relevant ar-rows.

2.2.2 The Derived Category of a DG-Category

By definition, a degree zero closed morphism

η ∈ Z0(dgMod (A))(M,N)

satisfies

η(A) ∈ Z0(C (k) (M(A), N(A))) = C (k) (M(A), N(A))

for all objects A of A. Hence we are justified in the following definitions:


(i) η is a quasi-isomorphism if η(A) is a quasi-isomorphism of chaincomplexes for all objects A of A, and

(ii) η is a fibration if η(A) is a degree-wise surjective morphism of com-plexes for all objects A of A.

EquippingC (k) with the standard projective model structure (see [21, Sec-tion 2.3]), these definitions endow Z0(dgMod (A)) with the structure of aparticularly nice cofibrantly generated model category (see [38, Section3]). In analogy with the definition of the derived category of modules for aring A, the derived category of A is defined to be the homotopy category

D(A) = Ho(Z0(dgMod (A))

)= Z0(dgMod (A))[W−1]

that is obtained by localizing Z0(dgMod (A)) at the class, W , of quasi-isomorphisms.

It can be shown (see [23, Section 3.5]) that for every dg A module, M ,there exists an h-projective, N , and a quasi-isomorphism N → M , whichone calls an h-projective resolution of M . Moreover, it is not difficult tosee that any quasi-isomorphism between h-projective objects is in fact ahomotopy equivalence. It follows that there is an equivalence of categoriesbetween H0(h-Proj (A)) and D(A) for any small dg-category, A.

It should be noted that this generalizes the notion of derived categoriesof modules over a commutative ring. Indeed, for a commutative ring, A,one associates to A the ringoid, A, with one object, ∗, and morphisms,A(∗, ∗), the complex with A in degree zero. One identifies the chain com-plexes of A modules enriched by the Hom total complex with dgMod (A),which is simply the full dg-subcategory of Fun(A, C (k)) comprised ofall dg-functors. From this viewpoint it is easy to recognize the categoriesZ0(dgMod (A)), H0(dgMod (A)), and D(A), as the categories C (A),K(A), the usual category up to homotopy, and the derived category ofModA, respectively. In the language of [26], we say that h-Proj (A) is adg-enhancement of D(ModA).


2.3 Tensor Products of dg Modules

Let M be a dg A module, let N be a dg Aop module, and let A,B beobjects of A. For ease of notation, we drop the functor notation M(A) infavor ofMA and writeAA,B for the morphismsA(A,B). We have structuremorphisms

MA,B ∈ C (k) (AA,B, C (k) (MB,MA)) ∼= C (k) (MB ⊗k AA,B,MA)

and

NA,B ∈ C (k) (AA,B, C (k) (NA, NB)) ∼= C (k) (AA,B ⊗k NA, NB) ,

which give rise to a unique morphism

MB ⊗k AA,B ⊗k NA →MA ⊗k NA ⊕MB ⊗k NB

induced by the universal properties of the biproduct. The two collectionsof morphisms given by projecting onto each factor induce morphisms

Ξ1,Ξ2 :⊕

A,B∈Ob(A)

MB ⊗k AA,B ⊗k NA →⊕

C∈Ob(A)

MC ⊗k NC ,

and we define the tensor product ofM andN to be the coequalizer inC (k)⊕(i,j)∈Z2 Mj ⊗k AA,B ⊗k NA

⊕`∈ZM` ⊗k N` M ⊗A N

Ξ1

Ξ2

.

It is routine to check that a morphism M → M ′ of right dg A modulesinduces by the universal property for coequalizers a unique morphism

M ⊗A N →M ′ ⊗A N

yielding a functor

−⊗A N : dgMod (A)→ C (k) .

One extends this construction to bimodules as follows. Given objectsE of dgMod (A⊗ B) and F of dgMod (Bop ⊗ C), we recall that we haveassociated to each a dg-functor

ΦE : Aop → dgMod (B) and ΦF : Cop → dgMod (Bop)


by the symmetric monoidal closed structure on dgcatk. For each pair ofobjects A of A and C of C, we obtain dgModules

ΦE(A) = E(A,−) : Bop → C (k) and ΦF (C) = F (−, C) : B → C (k)

and hence one may define the object E ⊗B F of dgMod (A⊗ C) by

(E ⊗B F ) (A,C) = ΦE(A)⊗B ΦF (C).

One can show that by a similar argument to the original that a morphismE → E ′ of dgMod (A⊗ B) induces a morphism E ⊗B F → E ′ ⊗B F ofdgMod (A⊗ C), and a morphism F → F ′ of dgMod (Bop ⊗ C) induces amorphism E ⊗B F → E ⊗B F ′ of dgMod (A⊗ C).

Remark 2.1. Denote by K the dg-category with one object, ∗, and mor-phisms given by the chain complex

K(∗, ∗)n =

{k n = 00 n 6= 0

with zero differential. This category serves as the unit of the symmetricmonoidal structure on dgcatk, so for small dg-categories,A and C, we canalways identifyA withA⊗K and C with Kop⊗C. With this identificationin hand, we obtain from taking B = K in the latter construction a specialcase: Given a dg Aop module, E, and a dg C module, F , we have a dgA-C-bimodule defined by the tensor product

(E ⊗ F ) (A,C) := (E ⊗K F ) (A,C) = E(A)⊗k F (C).

2.4 Extensions of Morphisms Associated to Bimodules

Let E be a dg A-B-bimodule. Following [15, Section 3], we can extendthe associated functor ΦE to a dg-functor

ΦE : dgMod (A)→ dgMod (B)

defined by ΦE(M) = M ⊗A E. Similarly, we have a dg-functor in theopposite direction

ΦE : dgMod (B)→ dgMod (A)


defined by ΦE(N) = dgMod (B) (ΦE(−), N).For any dg-functor G : A → B we denote by IndG the extension of the

dg-functorA→ B YB→ dgMod (B)

and its right adjoint by ResG. By way of the enriched Yoneda Lemma wesee that for any object A of A and any dg B module, N ,

ResG(N)(A) = dgMod (B) (hGA, N) ∼= N(GA).

We record here some useful propositions regarding extensions of dg-functors.

Proposition 2.2 ( [15, Prop 3.2]). LetA and B be small dg-categories. LetF : A → dgMod (B) and G : A → B be dg-functors.

(i) F is left adjoint to F ,

(ii) F ◦ YA is dg-isomorphic to F and H0(F ) is continuous,

(iii) F (h-Proj (A)) ⊆ h-Proj (B) if and only if F (A) ⊆ h-Proj (B),

(iv) ResG(h-Proj (B)) ⊆ h-Proj (A) if and only ifResG(B) ⊆ h-Proj (A); moreover, H0(ResG) is always continuous,

(v) IndG : h-Proj (A) → h-Proj (B) is a quasi-equivalence if G is aquasi-equivalence.

Remark 2.3. 1. We note that for dg A- and Aop modules, M and N ,part (i) implies that the dg-functors

−⊗A N : dgMod (A)→ C (k)

andM ⊗A − : dgMod (Aop)→ C (k)

have right adjoints

N(C) = C (k) (N(−), C) and M(C) = C (k) (M(−), C),


respectively. As an immediate consequence of the enriched YonedaLemma

hA ⊗A N ∼= N(A) and M ⊗A hA ∼= M(A)

holds for any object A of A.

2. Let ∆A denote the dg A-A-bimodule corresponding to the Yonedaembedding, YA, under the isomorphism

dgMod (Aop ⊗A) ∼= dgcatk

(A, dgMod (A)) .

It’s clear that we have a dg-functor

∆A ⊗A − : dgMod (Aop ⊗A)→ dgMod (Aop ⊗A)

and for any dg A-A-bimodule, E, we see that

(∆A ⊗A E)(A,A′) = hA ⊗A E(−, A′) ∼= E(A,A′)

implies that ∆A ⊗A E ∼= E.

When starting with an h-projective we have a very nice extension ofdg-functors:

Proposition 2.4 ( [15, Lemma 3.4]). If E is any h-projective dg A-B-bimodule, then the associated functor

ΦE : A → dgMod (B)

factors through h-Proj (B).

As a direct consequence of the penultimate proposition, this means thatwe can view the extension of ΦE as a dg-functor

ΦE = −⊗A E : h-Proj (A)→ h-Proj (B) .

Put another way, tensoring with an h-projective A-B-bimodule preservesh-projectives.

One essential result about dgcatk comes from Toen’s result on the ex-istence, and description of, the internal Hom in its homotopy category.


Theorem 2.5 ( [38, Theorem 1.1], [15, 4.1]). Let A, B, and C be objectsof dgcatk. There exists a natural bijection

[A, C] 1:1←→ Iso(H0(h-Proj (Aop ⊗ C)rqr)

)Moreover, the dg-category RHom (B, C) := h-Proj (Bop ⊗ C)rqr yields anatural bijection

[A⊗ B, C] 1:1←→ [A,RHom (B, C)]

proving that the symmetric monoidal category Ho (dgcatk) is closed.

Corollary 2.6 ( [38, 7.2], [15, Cor. 4.2]). Given two dg categories Aand B, RHom (A, h-Proj (B)) and h-Proj (Aop ⊗ B) are isomorphic inHo (dgcatk). Moreover, there exists a quasi-equivalence

RHomc (h-Proj (A) , h-Proj (B))→ RHom (A, h-Proj (B)) .

To get a sense of the value of this result, let us recall one applica-tion from [38, Section 8.3]. Let X and Y be quasi-compact and separatedschemes over Spec k. Recall the dg model for D(QcohX), Lqcoh(X), isthe C (k)-enriched subcategory of fibrant-cofibrant objects in the injectivemodel structure on C (QcohX).

Theorem 2.7 ( [38, Theorem 8.3]). LetX and Y be quasi-compact, quasi-separated schemes over k. There exists an isomorphism in Ho (dgcatk)

RHomc (LqcohX,LqcohY ) ∼= Lqcoh(X ×k Y )

which takes a complex E ∈ Lqcoh(X ×k Y ) to the exact functor on thehomotopy categories

ΦE : D(QcohX)→ D(QcohY )

M 7→ Rπ2∗

(E

L⊗ Lπ∗1M

)Proof. The first part of the statement is exactly as in [38]. The second partis implicit.


3 Details on noncommutative projectiveschemes

3.1 Recollections and conditions

Noncommutative projective schemes were introduced by Artin and Zhangin [4]. We recall the definition.

Definition 3.1. Let N be a finitely-generated abelian group. We say that ak-algebra A is N -graded if there exists a decomposition as k modules

A =⊕n∈N

An

with AnAm ⊂ An+m. One says that A is connected graded if it is Z-graded with A0 = k and An = 0 for n < 0.

For algebraic geometers, the most common example is the homoge-neous coordinate ring of a projective scheme. These are of course commu-tative. One has a plenitude of noncommutative examples.

Example 3.2. Let us take k = C and consider the following quotient ofthe free algebra

Aq := C〈x0, . . . , xn〉/(xixj − qijxjxi)

for qij ∈ C× with qij = q−1ji . These give noncommutative deformations of

Pn.

Example 3.3. Building off of Example 3.2, we recall the following class ofnoncommutative algebras of Kanazawa [22]. Pick φ ∈ C and qij accordingto [22, Theorem 2.1] with

∏ni=1 qij = 1 for all j. And set

Aφq := Aq/

(n∑i=0

xn+1i − φ(n+ 1)(x0 · · ·xn)

).

This is the noncommutative version of the homogeneous coordinate ringsof the Hesse (or Dwork) pencil of Calabi-Yau hypersurfaces in Pn.


Definition 3.4. Let M be a graded A module. We say that M has rightlimited grading if there exists a d such Md′ = 0 for d′ ≥ d. We define leftlimited grading analogously.

In general, good behavior requires some homological assumptions onthe ring A. We recall two common such ones.

Definition 3.5. Let A be a connected graded k-algebra. Following Vanden Bergh [40], we say that A is Ext-finite if for each n ≥ 0 the ungradedExt-groups are finite dimensional

dimk ExtnA(k, k) <∞.

Remark 3.6. The Ext’s are taken in the category of left A modules, apriori.

Definition 3.7. Following Artin and Zhang [4], given a graded left moduleM , we say A satisfies χ◦(M) if ExtnA(k,M) has right limited grading foreach n ≥ 0.

We recall some basic results on Ext-finiteness, essentially from [40,Section 4].

Proposition 3.8. Assume that A and B are Ext-finite. Then

1. the ring A⊗k B is Ext-finite.

2. the ring Aop is Ext-finite.

Furthermore, if A is Ext-finite then A is finitely presented as a k-algebra.

Proof. See [40, Lemma 4.2] and the discussion preceding it. For the finalstatement, see the opening paragraphs of [12, Section 4.1].

For a connected graded k-algebra, A, one has the two-sided ideal

A≥` :=⊕n≥`

An.


Definition 3.9. Let A be a finitely generated connected graded algebra.Recall that an element, m, of a module, M , is torsion if there is an n suchthat

A≥nm = 0.

We let τ denote the functor that takes a module, M , to its torsion submod-ule. The module M is torsion if τM = M .

The functor τ is right adjoint to the inclusion functor and we denote thecounit η : τ → 1GrA.

The following is likely well-known (under the assumption of a Noethe-rian ring the conclusion is contained in [4]). However the authors wereunable to locate a convenient reference under the assumption of finitelygeneration, so we include the following

Proposition 3.10. Let A be a connected graded k-algebra. If A is gener-ated as a k-algebra by a finite set of elements of positive degree, then foreach ` ∈ Z the tails A≥` are finitely generated A modules.

Proof. Let S = {xi}gi=1 be a set of generators for A as a k-algebra and letdi = deg(xi). By possibly relabeling, we may assume that 1 ≤ d1 ≤ d2 ≤. . . ≤ dg.

Fix `. First we show that A≥` is generated by the sum

`+dg−1⊕r=`

Ar.

Take a ∈ An with ` ≤ n. We induct on n. If n ≤ ` + dg − 1, then a isgenerated by

`+dg−1⊕r=`

Ar.

If n > `+ dg − 1, then we can write

a =∑

aibi


with n > deg(ai), deg(bi) > 0. So a is generated by

n−1⊕r=n−dg

Ar

By the induction hypothesis, we can generate any element of Ar for n −dg ≤ r ≤ n− 1 using

`+dg−1⊕r=`

Ar.

It suffices to show that`+dg−1⊕r=`

Ar

is a finite dimensional k-vector space. Hence it is enough to show that Aris a finite dimensional k-vector space for each r. Consider the free algebrak〈S〉 as a graded algebra. By assumption there is a surjection

k〈S〉 → A

so it suffices to show that there are only finitely many words of degree r ink〈S〉.

Consider a word w of length N . We can write

w = αX1 · · ·Xn

where Xi ∈ S and α ∈ k, so

Nd1 ≤ deg(w) ≤ Ndg.

If r/d1 < N , then w 6∈ k〈S〉r. This implies

{w | deg(w) = r} ⊆ {w | 0 ≤ len(w) ≤ r/d1} .

The latter has only finitely many elements since S has only finitely manyelements. Hence k〈S〉r is a finite dimensional vector space.


Proposition 3.11. Let A be a connected graded k-algebra. Denote byTorsA, the full subcategory of GrA consisting of all torsion modules. If Ais finitely generated in positive degree, then TorsA is a Serre subcategory.

Proof. Consider a short exact sequence

0→M ′ →Mp→M ′′ → 0.

It’s clear that if M is an object of TorsA, then so are M ′ and M ′′. Henceit suffices to show that if M ′ and M ′′ are both objects of TorsA, then so isM .

Fix an element m ∈ M . Since M ′′ is an object of TorsA, there existssome n such that A≥np(m) = 0 and hence A≥nm ∈ M ′. By Proposi-tion 3.10 the latter is finitely generated and so we can choose generatorsm′1, . . . ,m

′t and integers n1, . . . , nt such that A≥ni

m′i = 0 for i = 1, . . . , t.Take nm = maxi=1,...,t{n+ ni} so that A≥nmm = 0, as desired.

As such, we can form the quotient.

Definition 3.12. Let A be connected graded and finitely generated as a k-algebra. Then denote the quotient of the category of graded A modules bythe subcategory of torsion modules as

QGrA := GrA/TorsA

Letπ : GrA→ QGrA

denote the quotient functor. By Proposition 3.11 and [17, Cor. 1, III.3], πadmits a fully faithful right adjoint which we denote by

ω : QGrA→ GrA.

Finally, we denote the composition Q := ωπ and the unit of adjunctionε : 1GrA → Q.

The category QGrA is defined to be the quasi-coherent sheaves on thenoncommutative projective scheme X .


Remark 3.13. Note that, traditionally speaking, X is not a space, in gen-eral. In the case A is commutative and finitely-generated by elements ofdegree 1, then a famous result of Serre says that there is an equivalencebetween QcohX and QGrA, so that X is effectively ProjA.

One can give more explicit descriptions of Q and τ .

Proposition 3.14. Let A be a connected graded k-algebra and let M be agraded A module. If A is finitely generated, then

τM = colimn GrA(A/A≥n,M)

QM = colimn GrA(A≥n,M).

and ε(M)(m) ∈ Q(M) is the class of the morphism

ϕm : A M

a a ·m

Proof. This is standard localization theory, see [35].

3.2 Noncommutative Bi-projective Schemes

In studying questions of kernels and bimodules, we will have to move out-side the realm of Z-gradings. While one can generally treat N -graded k-algebras in our analysis, we limit the scope a bit and only consider Z2-gradings of the following form.

Definition 3.15. Let A and B be connected graded k-algebras. The tensorproduct A⊗k B will be equipped with its natural bi-grading

(A⊗k B)n1,n2 = An1 ⊗k Bn2 .

A bi-bi module for the pair (A,B) is a Z2-graded A⊗k B module.

From bi-bi modules, we can produce A or B modules by taking slicesof the gradings. For fixed v ∈ Z we have a functor

(−)∗,v : Gr(A⊗k B) GrA

P⊕

m∈Z Pm,v


and for fixed u ∈ Z a functor

(−)u,∗ : Gr(A⊗k B) GrB

P⊕

n∈Z Pu,n

In the case that A = B, there is a particular bi-bi module of interest.

Definition 3.16. For A a finitely generated, connected graded k-algebra,we define ∆A to be the A-A bi-bi module with

(∆A)i,j = Ai+j

and the natural left and rightA actions. If the context is clear, we will oftensimply write ∆.

Notice that the forgetful functor UA : Gr(A ⊗k B) → GrA sends abi-bi module P to the Z-graded sum

UA(P ) =⊕u∈Z

Pu,∗

and similarly the forgetful functor UB : Gr(A ⊗k B) → GrB. Allowingfor some repetition of notation we define for any bi-bi module

QA(P ) := QA ◦ UA(P ) and QB(P ) := QB ◦ UB(P )

and also

τA(P ) := τA ◦ UA(P ) and τB(P ) := τB ◦ UB(P ).

In general, these will no longer be bi-bi modules, although under mildassumptions we can guarantee they will.

If A is finitely generated as a k-algebra, then we have a functor

Q′A : Gr(A⊗k B)→ Gr(A⊗k B)

that takes a bi-bi module P to the bi-bi module⊕

v∈ZQA(P∗,v), where theZ2 grading is given by

Q′A(P )u,v := QA(P∗,v)u.

The two constructions are naturally isomorphic.


Lemma 3.17. Assume A is finitely generated as a k-algebra. Let P be abi-bi module. The natural map Q′A(P ) → QA(P ) is an isomorphism andthus QA(P ) is also a bi-bi module. Similarly, τA(P ) is also a bi-bi module.

Symmetrically, ifB is finitely generated as a k-algebra, then the naturalmap Q′B(P )→ QB(P ) is an isomorphism and thus QB(P ) is also a bi-bimodule. Similarly, τB(P ) is also a bi-bi module.

Proof. By the universal property for coproducts we have a morphism

Q′A(P ) =⊕v∈Z

QA(P∗,v)→ QA

(⊕v∈Z

P∗,v

)= QA(P )

so it suffices to check that QA commutes with coproducts. Since coprod-ucts commute with colimits it suffices to check that the natural morphism⊕

v∈Z

GrA (A≥n, P∗,v)→ GrA

(A≥n,

⊕v∈Z

P∗,v

)is an isomorphism. This holds providedA≥n is finitely generated as a mod-ule, but this follows from the assumption that A is finitely generated as analgebra. This is Proposition 3.10. A similar argument works for τA.

There are a couple notions of torsion for a bi-bi module that one candream up. We use the following.

Definition 3.18. Let A and B be finitely generated, connected graded k-algebras, and let M be a bi-bi A-B module. We say that M is torsion if itlies in the smallest Serre subcategory containing A-torsion bi-bi modulesand B-torsion bi-bi modules.

Lemma 3.19. Let A and B be finitely generated, connected graded k-algebras. A bi-bi module P is torsion if and only if there exists a pair ofintegers, n1, n2, such that

(A⊗B)≥n1,≥n2p = 0 (1)

for all p ∈ P .Furthermore

τA⊗kB(P ) = colimn1,n2 Gr(A⊗k B) ((A⊗k B) / (A≥n1 ⊗k B≥n2) , P )


Proof. For ease of notation, denote by T the full subcategory of bi-bi mod-ules satisfying Equation 1. Assume that the bi-bi modules P ′ and P ′′ bothbelong to T and that we have a short exact sequence of bi-bi modules

0 P ′ P P ′′ 0.

Since P ′′ belongs to T , we may choose for any p ∈ P integers `1 and `2

such that(A⊗k B)≥`1,≥`2p ⊆ P ′,

which is finitely generated by Proposition 3.10 by, say, p1, p2, . . . , pk. Aselements of P ′, there exists for each i integers ni1 and ni2 such that

(A⊗k B)≥ni1,≥ni

2pi = 0.

Taking n1 = maxi{ni1} and n2 = maxi{ni2} implies that P belongs to T ,and thus T is a Serre subcategory.

If P is either A-torsion, in which case (A ⊗ B)≥n,≥0p = 0 for somen, or B-torsion, in which case (A ⊗ B)≥0,≥np = 0 for some n, then Pbelongs to T . By minimality, T necessarily contains Tors(A⊗k B) and soit suffices to show that if

(A⊗B)≥n1,≥n2p = 0, ∀p ∈ P

then P lies in Tors(A⊗k B). Denote by τBP the B-torsion submodule ofP and observe from the short exact sequence of bi-bi modules

0 τBP P P/τBP 0.

that it suffices to show P/τBP belongs to Tors(A⊗kB). For every p ∈ P ,there is some n1 such that A≥n1p is B-torsion. Hence, A≥n1p = 0 and p isA-torsion. Consequently, P/τBP is itself a torsionAmodule and thereforebelongs to Tors(A⊗k B), as desired.

The final statement now follows immediately.

We can compare our notion, τA⊗kB, of torsion to the torsion of [40]

τVdBA⊗kB

(P ) = colimm,n Gr(A⊗k B) (A/A≥m ⊗k B/B≥n, P ) .


Lemma 3.20. There exists a natural inclusion

τVdBA⊗kB

(P )νP→ τA⊗kB(P ).

Proof. The surjections A→ A/A≥m and B → B/B≥n induce morphisms

A⊗k B/A≥m ⊗k B≥n → A/A≥m ⊗k B/B≥n

which in turn induce νP .

Remark 3.21. In general, this inclusion is strict. For example, take P =

A/A≥m ⊗k N . One has

τA⊗kB (A/A≥m ⊗k N) = A/A≥m ⊗k N

whileτVdBA⊗kB

(A/A≥m ⊗k N) = A/A≥m ⊗k τBN.

One can form the quotient category

QGr(A⊗k B) := Gr(A⊗k B)/Tors(A⊗k B).

Lemma 3.22. The quotient functor

π : Gr(A⊗k B)→ QGr(A⊗k B)

has a fully faithful right adjoint

ω : QGr(A⊗k B)→ Gr(A⊗k B)

with

QM := ωπM = colimn1,n2 Gr(A⊗k B)(A≥n1 ⊗k B≥n2 ,M)

Proof. This is just an application of [17, Cor. 1, III.3].

For a given bi-bi module, a natural question one might ask is how QA,QB, and QA⊗kB relate, as well as how τA, τB, and τA⊗kB relate.


Lemma 3.23. Let A and B be finitely generated, connected graded k-algebras. For any complex of bi-bi modules, P , there exist natural mor-phisms of complexes

α`P : QAP → QA⊗kBP

αrP : QBP → QA⊗kBP.

Moreover, the diagram

P P P

QAP QA⊗kBP QBP

εA(P )

1P

εA(P ) εA⊗kB(P )

1P

εB(P )

α`P αr

P

commutes

Proof. We handle the case of α` and note that αr follows from the sameargument, mutatis mutandis.

First observe that it suffices to prove the result for P simply a bi-bimodule and apply the result in each degree, for then all faces of the cube

P n P n

P n+1 P n+1

QA(P n) QA⊗kB(P n)

QA(P n+1) QA⊗kB(P n+1)

dn

1nP

εnA(P )

dnP

εnA(P )1n+1P

εn+1A (P ) εn+1

A⊗kB(P )α`Pn

dnQA(P )

dnQA⊗kB(P )

α`Pn+1

obviously commute for all n. Hence we assume that P is a bi-bi module.Now fix an integer m and observe that GrA(A≥m, P ) is a graded bi-bi

module with Z2 grading given by

GrA(A≥m, P )u,v = GrA(A≥m, P∗,v(u)).


We have

ϕmu,v : GrA(A≥m, P )u,v Gr(A⊗k B)(A≥m ⊗k B,P )u,v

f{a⊗ b 7→ f(a) · b := (1⊗ b)f(a)

}since f(a) ∈ P∗,v(u)r = Pr+u,v for any a ∈ Ar and hence

f(a) · b ∈ Pr+u,v+s = P (u, v)r,s

for any b ∈ Bs. Since the choice of m was arbitrary, we have by the uni-versal property for colimits the commutative diagram

GrA(A≥m, P ) Gr(A⊗k B)(A≥m ⊗k B,P )u,v

QA(P ) QA⊗kB(P )

ϕm

∃!α`P

For naturality, it suffices to show that

P1 GrA(A≥m, P1) QA(P1) QA⊗kB(P1)

P2 QA(P2) QA⊗kB(P2)

f

α`P1

QA(f) QA⊗kB(f)

α`P2

commutes for all m. It’s clear from the colimit definition that QA(f) ap-plied to the class of a morphism ϕ : A≥m → P1 is the class of the compo-sition, [f ◦ ϕ], which is mapped by α`P2

to the class of the morphism

a⊗ b 7→ f ◦ ϕ(a) · b.

Along the other side of the square we see that α`P1maps [ϕ] to the class of

a⊗ b 7→ 1⊗ ϕ(a) · b

whose image under QA⊗kB(f) is the class of the morphism

a⊗ b 7→ f(ϕ(a) · b) = f ◦ ϕ(a) · b


since f is a morphism of bi-bi modules.By identifying P with GrA(A,P ) we can see that image of an element

p of P under the unit of adjunction εA : 1GrA → QA is the class of the mor-phism ϕp(a) = a · p in QA(P ). Similarly, the unit of adjunction εA⊗kB(P )

takes an element p of P to the class of the morphism

ψp(a⊗ b) = a⊗ b · p

in QA⊗kB(P ). From this observation it is clear that the result of applyingα`P to the image of p in QA(P ) is the class of the morphism

a⊗ b 7→ ϕp(a) · b = (a · p) · b = a⊗ b · p = ψp(a⊗ b).

As a result, we see that α`P factors the unit of adjunction

εA⊗kB = α`P ◦ εA.

Lemma 3.24. Let A and B be finitely generated, connected graded k-algebras. There exist natural isomorphisms

QA⊗kBP

QB ◦QA(P ) QA ◦QB(P )γ`P

γrP

making the diagram

QAP QA⊗kBP QBP

QB ◦QA(P ) QA ◦QB(P )

α`P

εB(QAP ) γ`PγrP

αrP

εA(QBP )

commute.

Proof. We handle the left case and note the other is symmetric. Since thetails, B≥n, of B are all finitely generated, the natural map

γlP := colimn,m Gr(A⊗k B) (A≥m ⊗k B≥n, P )→colimn GrB (B≥n, colimm GrA (A≥m, P ))


induced by the isomorphisms

Gr(A⊗k B) (A≥m ⊗k B≥n, P ) GrB (B≥n,GrA (A≥m, P ))

ϕ (b 7→ ϕ(−⊗ b))

is also an isomorphism.For an element x of QA(P ) we can always choose a representative

ψ : A≥n → P . The image of x under α`P is represented by the morphism

a⊗ b 7→ ψ(a) · b

and its image under γ`P is represented by

b 7→ αlPψ(−⊗ b) = ψ(−) · b.

The image of x under εA(QAP ) is represented by the morphism

b 7→ ψ(−) · b

and the diagram commutes.

Lemma 3.25. Let A and B be finitely generated, connected graded k-algebras. There exist a commutative diagram

τAP τVdBA⊗kB

P τBP

τB ◦ τA(P ) τA ◦ τB(P )

ξrPξ`P

ηB(τAP ) κ`PηA(τBP )κrP

with κ`P and κrP natural isomorphisms.

Proof. See [40, Lemma 4.5].

3.3 Derived functors

For a general Grothendieck category, we equip the category of chain com-plexes with the injective model structure. We compute the total right de-rived functors of a left exact functor, F , on an object, M , by applying F tothe fibrant replacement, RM ,

RF (M) = F (RM).


When necessary to distinguish between multiple fibrant replacement func-tors, we will decorate with the relevant ring, e.g. RF (M) = F (RAM).

Many of the relevant statements, e.g. checking that a morphism is aquasi-isomorphism, can be verified by passing to the level of the homotopycategory, i.e. the derived category.

Definition 3.26. We say J is an F -acyclic if the natural morphism

F (J)→ RF (J)

is a quasi-isomorphism.

A special class of F -acyclics are the h-injectives, which are all homo-topy equivalent to a fibrant object. As such, one can use h-injective resolu-tions to compute derived functors when convenient.

We say that RF commutes with coproducts if the natural map⊕γ∈Γ

F (RXγ)→ FR

(⊕γ∈Γ

Xγ

)is a quasi-isomorphism for any coproduct.

Lemma 3.27. Let C and D be abelian categories and let F : C → D beleft exact. If C is Grothendieck, then RF commutes with coproducts if andonly if

1. F commutes with arbitrary coproducts of objects in C, and

2. Arbitrary coproducts of F -acyclics are F -acyclic.

Proof. First assume that RF commutes with coproducts. Since F is leftexact,

H0 (RF (A)) ∼= F (A)

for any object of C. So (1) is satisfied. Now assume that Jγ are F -acyclic.Then, we have the commutative diagram⊕

γ F (Jγ)⊕

γ RF (Jγ)

F(⊕

γ Jγ

)RF

(⊕γ Jγ

)


with vertical arrows quasi-isomorphisms. The upper horizontal arrow is acoproduct of quasi-isomorphisms so is also a quasi-isomorphism. Conse-quently, the lower horizontal arrow is a quasi-isomorphism and thus

⊕Jγ

is F -acyclic.Conversely, assume that (1) and (2) hold. Given a collection {Xγ} of

objects of C, the map ⊕γ

Xγ →⊕γ

RXγ

is a quasi-isomorphism and⊕

γ RXγ is F -acyclic. We can factor the nat-ural map

⊕γ F (RXγ) FR

(⊕γ Xγ

)

F(⊕

γ RXγ

)Condition (1) says that the downward arrow is a quasi-isomorphism whilecondition (2) says that the upward arrow comes from applying F to a mapof F -acyclic objects. As the F ∼= RF on F -acyclic objects, F preservesquasi-isomorphisms between acyclic objects. Thus, the upward arrow isalso a quasi-isomorphism. Consequently, the horizontal arrow is a quasi-isomorphism and RF commutes with coproducts.

For a given complex of bi-bi modules, P , the complex RQAP will notgenerally be a bi-bi module due to the fact that RA does not commute withcoproducts. However, the assumption that RQA commutes with coprod-ucts says exactly that ⊕

v

QARA(P∗,v)→ RQAP

is a quasi-isomorphism and the source is a complex of bi-bi modules. Wetherefore set

R′QAP :=⊕v

QARA(P∗,v).


Lemma 3.28. If I is an h-injective complex of bi-bi modules, then I∗,v isan h-injective complex of A modules for any v ∈ Z.

Proof. We observe from the isomorphism of A modules

I∗,v ∼= GrB(B(−v), I)

that one just needs to check that applying GrB(B(−v),−) in each degreepreserves h-injectivity. For the complex I∗,v of A modules to be h-injectivewe need only show that

K(GrA)(D, I∗,v) = 0

holds for all acyclic A modules, D. This follows from the tensor-hom ad-junction

K(GrA)(D, I∗,v) ∼= K(GrA)(D,GrB(B(−v), I))

∼= K(Gr(A⊗k B))(D ⊗k B(−v), I) = 0

because I is a h-injective bi-bi module and D ⊗k B(−v) is acyclic.

Lemma 3.29. The functor

ω : h-Inj (QGrA)→ h-Inj (GrA)

is well-defined. Moreover, H0(ω) is an equivalence with its essential im-age.

Proof. For the first statement, we just need to check that ω takes h-injectivecomplexes to h-injective complexes. This is clear from the fact that ω isright adjoint to π, which is exact.

To see this is fully faithful, we recall that πω ∼= Id so

h-Inj (GrA) (ωM,ωN) ∼= h-Inj (QGrA) (πωM,N)

∼= h-Inj (QGrA) (M,N).


Remark 3.30. Using Lemma 3.29, we can either use h-Inj (QGrA) or itsimage under ω in h-Inj (GrA) as an enhancement of D(QGrA).

Corollary 3.31. Let A and B be finitely generated, connected graded k-algebras. If RQA and RQB both commute with coproducts, then, for a bi-bi module P , RQA⊗kBP is naturally quasi-isomorphic to R′QA(R′QBP )

and to R′QB(R′QAP ).

Proof. We handle the quasi-isomorphism between the first two. The re-maining part is analogous.

Let P be an object of C (Gr(A⊗k B)). First observe that we have

RQA⊗kB(P ) = QA⊗kB(RA⊗kBP ) ∼= QA ◦QB(RA⊗kBP )

via γrRA⊗kBP. Note that we have two homotopy equivalences

QBRBPv,∗ → QBRB(RA⊗kBP )v,∗ ← QB(RA⊗kBP )v,∗

which induce a quasi-isomorphism between R′QBP andQBRA⊗kBP . Ap-plying RQA preserves quasi-isomorphisms. Since R′QA is naturally quasi-isomorphic to RQA, there is an induced natural quasi-isomorphism be-tween R′QA(R′QBP ) and R′QA(QBRA⊗kBP ).

Therefore it suffices to show that

QAQBI → R′QA(QBI)

is a quasi-isomorphism for any h-injective, I . Since R′QA and RQA arenaturally quasi-isomorphic, we can instead show that

QAQBI → RQA(QBI)

is a quasi-isomorphism, which is another way of saying that QBI is QA-acyclic. Note that there is a short exact sequence of complexes

0→⊕n

Gr(B)(B≥n, I)→⊕n

Gr(B)(B≥n, I)→ QBI → 0


which induces an exact triangle in the derived category. Because the tailsA≥m are finitely generated, any map

A≥m → QBI

lifts to a mapA≥m → Gr(B)(B≥n, I).

This says that the sequence

0→⊕n

QAGr(B)(B≥n, I)→⊕n

QAGr(B)(B≥n, I)→ QAQBI → 0

remains exact. Using the fact that RQA commutes with coproducts, weget a map of triangles⊕

nQAGr(B)(B≥n, I)⊕

nQAGr(B)(B≥n, I) QAQBI

⊕nRQAGr(B)(B≥n, I)

⊕nRQAGr(B)(B≥n, I) RQA(QBI)

where the first displayed vertical maps are quasi-isomorphisms since eachGr(B)(B≥n, I) is h-injective for all n. Consequently, the map

QAQBI → RQA (QBI)

is also a quasi-isomorphism for any h-injective I .

We can strengthen the previous statement if we assume some finite-dimensionality on Rτ .

Proposition 3.32. Assume that A is Ext-finite and there exists a fixed Nsuch that RnτA = 0 for all n ≥ N . Then, the natural maps

RτA(RτAM)→ RτAM

RQA(M)→ RQA(QAM)

are quasi-isomorphisms for any complex of graded A modules, M . More-over, both RτA(RQAM) and RQA(RτAM) are acyclic.


Proof. It suffices to show that, for a cofibrant M , the object τAM is τA-acyclic. [12, Lemma 4.1.3] implies this for injective graded modules. Anappeal to the spectral sequence whose E1-page is RpτA(τAM

q) plus theassumption that Rτ pA vanishes for sufficiently large p gives the acyclicity ingeneral. A similar argument demonstrates the second quasi-isomorphism.

Remark 3.33. It seems like the assumption of finite dimensionality of RτAis unnecessary.

Definition 3.34. We say an object M of C(GrA) is RτA-torsion free ifRτAM is acyclic.

We also have the following standard triangles of derived functors.

Lemma 3.35. Let A and B be finitely generated connected graded alge-bras. Then, we have natural transformations

RτA R RQA

which when applied to any graded A module gives an exact triangle in thederived category. Analogous statements hold for graded B modules andbi-graded A⊗k B modules.

Proof. The natural transformations are η ◦ R and ε ◦ R. For the case ofgraded A modules (or graded B modules), this is well-known, see [12,Property 4.6]. For a bi-bi module, P , the sequence

0→ τA⊗kBP → P → QA⊗kBP

is exact. It suffices to prove that if P = I is injective, then the wholesequence is actually exact. Here one can use the system of exact sequences

0→ A≥n1 ⊗k B≥n2 → A⊗k B → (A⊗k B)/A≥n1 ⊗k B≥n2 → 0

and exactness of Gr(A ⊗k B)(−, I) plus Lemmas 3.19 and 3.22 to getexactness.


Proposition 3.36. Assume that A is Ext-finite. Then RτA and RQA bothcommute with coproducts.

Proof. See [40, Lemma 4.3] for RτA. Since coproducts are exact, usingthe triangle

RτAM → RM → RQAM

we see that RτA commutes with coproducts if and only if RQA commuteswith coproducts.

Corollary 3.37. Assume that A and B are left Noetherian, and that RQA

and RQB both commute with coproducts. There exist natural diagrams ofcomplexes of bimodules

R′QAP QARA⊗kBP

R′QA(RA⊗BP ) RQA⊗kBP

∼ ∼α`RA⊗kBP

and

R′QBP QBRA⊗kBP

R′QB(RA⊗BP ) RQA⊗kBP

∼ ∼αrRA⊗kBP

where the arrows labeled with ∼ are quasi-isomorphisms.

Proof. Because the canonical morphismA→ A⊗kB is flat, the associatedadjunction is Quillen by [20, Proposition 2.13], and hence the fibrant re-placementRA⊗kBP is also fibrant when regarded as an object of C (GrA).Since fibrancy is closed under retracts, (RA⊗kBP )∗,v is also fibrant for anyv.

Therefore, the map

(RA⊗kBP )∗,v → RA(RA⊗kBP )∗,v


is a homotopy equivalence and remains such after application of QA. Con-sequently,

QARA⊗kBP⊕

vQA(RA⊗kBP )∗,v

⊕vQARA(RA⊗kBP )∗,v R′QARA⊗kBP

∼

is a quasi-isomorphism. Similarly, the map

RAP∗,v → RA(RA⊗kBP∗,v)

is a homotopy equivalence and the map

R′QAP → R′QARA⊗kBP


Remark 3.38. We will denote the maps in the derived category resultingfrom the diagrams in Corollary 3.37 as

β`P : R′QAP → RQA⊗kBP

βrP : R′QBP → RQA⊗kBP.

We have an analogous definition for τ ,

R′τAP :=⊕v

τARAP∗,v,

and an analogous result to Corollary 3.37.

Corollary 3.39. Assume that A and B are left Noetherian, and that RτAand RτB both commute with coproducts. There exist natural diagrams ofcomplexes of bimodules

RτVdBA⊗kB

P R′τAP

R′τARA⊗kBP

∼


and

RτVdBA⊗kB

P R′τBP

R′τBRA⊗kBP .

∼

where the arrows labeled with ∼ are quasi-isomorphisms.

Proof. Establishing the quasi-isomorphism is completely analogous to theprevious corollary. The only ambiguity is the map

RτVdBA⊗kB

P → R′τARA⊗kBP.

This comes from the composition of ξrRA⊗kBPand the natural map

τARA⊗kBP⊕

v τA(RA⊗kBP )∗,v

⊕v τARA(RA⊗kBP )∗,v R′τARA⊗kBP.

∼

Remark 3.40. We denote the resulting maps in the derived category as

µ`P : RτVdBA⊗BP → R′τAP

µrP : RτVdBA⊗BP → R′τBP.

Proposition 3.41. Assume that A and B are left Noetherian and Ext-finite.Assume further that RτA and RτB are finite-dimensional. Then, we havenatural isomorphisms in the derived category

δ`P : R′QB(R′QAP )→ RQA⊗kBP

δrP : R′QA(R′QBP )→ RQA⊗kBP.

Consequently, β`P (respectively βrP ) is an isomorphism if and only if RQAP

(respectively RQBP ) is RτB (respectively RτA) torsion-free.


Proof. Applying R′QA to the diagram from Corollary 3.37 gives

R′QAR′QBP R′QAQBRA⊗kBP

R′QAR′QB(RA⊗BP ) R′QARQA⊗kBP

∼ ∼R′QA(αr

RA⊗kBP )

Since QA⊗kB = QA ◦QB an argument analogous to the proof of Proposi-tion 3.32 shows that the natural map

λ`P : RQA⊗kB(P )→ R′QA (RQA⊗kB(P ))

is a quasi-isomorphism. We set

δ`P :=(λ`P)−1

R′QA(β`P ).

Now we check that R′QA(αrRA⊗kBP) is a quasi-isomorphism. By acyclic-

ity, this reduces to checking that the natural map

QAQB(RA⊗kBP )→ QA(QA⊗kBRA⊗kBP ) ∼= QA(QA(QB(RA⊗kBP )))

is a quasi-isomorphism. But the natural map

QA → QA ◦QA

is always an isomorphism.Assuming that

R′QAP → R′QB(R′QAP )

is a quasi-isomorphism, i.e. assuming that R′QAP is RτB torsion free, weimmediately get that the original map β`P is a quasi-isomorphism.

By using the standard homological assumptions above, one has betterstatements for P = ∆.

Proposition 3.42. Let A be left and right Noetherian and assume that theconditions χ◦(A) holds as left and right A modules. Furthermore, assumethat RτA and RτAop are finite dimensional. Then the maps


R′QA∆ R′QB∆

RQA⊗kB∆

β`∆ βr

∆

are quasi-isomorphisms.Furthermore, the maps

RτVdBA⊗kAop∆

R′τA∆ R′τAop∆

µ`∆ µr∆

are also quasi-isomorphisms.

Proof. We have a triangle in D(Gr(A⊗k Aop))

R′τAop(R′QA∆)→ R′QA∆→ R′QAop(R′QA∆)→ R′τAop(R′QA∆)[1]

By Proposition 3.41, R′QAop(R′QA∆) ∼= R′QA⊗kAop∆, so it suffices toshow that we have R′τAop(R′QA∆) = 0.

First we note that for any bi-bi module, P , the natural morphism

R′τAopP → P

is a quasi-isomorphism if and only if the natural morphism

RτAopPx,∗ → Px,∗

is a quasi-isomorphism for each x ∈ Z. Moreover, for a right A mod-ule, M , if Hj(M) is right limited for each j then RτAopM → M is aquasi-isomorphism. So it suffices to show that (RjτA∆)x,∗ has right lim-ited grading for each x and j. We have(

RjτA∆)x,y∼=(RjτA∆∗,y

)x∼=(RjτAA(y)

)x.

By [4, Cor. 3.6 (3)], we have (RjτAA(y))x = 0 for sufficiently large y andhence is right limited. This implies that

RτAop

((RτA∆)x,∗

)→ (RτA∆)x,∗


is a quasi-isomorphism, as desired. As this sits in a triangle,

R′τAop(R′τA∆)→ R′τA∆→ R′QAop(R′τA∆)→ R′τAop(R′τA∆)[1]

we see that R′τAop(R′QA∆) is acyclic as desired.

Hypotheses similar to those of Proposition 3.42 will appear often so weattach a name.

Definition 3.43. Let A and B be connected graded k-algebras. If A is Ext-finite, left and right Noetherian, satisfies χ◦(A) and χ◦(Aop), and has bothRτA and RτAop finite-dimensional, then we say that A is delightful. If Aand B are both delightful, then we say that A and B form a delightfulcouple.

3.4 Segre Products

Definition 3.44. Let A and B be connected graded k-algebras. The Segreproduct of A and B is the graded k-algebra

A×k B =⊕i∈Z

Ai ⊗k Bi.

Proposition 3.45. If A and B are connected graded k-algebras that arefinitely generated in degree one, thenA×kB is finitely generated in degreeone.

Proof. If {xi}ri=1 ⊆ A1 and {yi}si=1 ⊆ B1 are generators for A and B,respectively, then A⊗k B is finitely generated by {xi ⊗ yj}i,j .

As a nice corollary, we can relax the conditions on [42, Theorem 2.4]to avoid the Noetherian conditions on the Segre and tensor products.

Theorem 3.46 ( [42, Theorem 2.4]). Let A and B be finitely generated,connected graded k-algebras, and let S = A ×k B, T = A ⊗k B. If Aand B are both generated in degree one, then there is an equivalence ofcategories

V : QGrS QGrT

E πT (T ⊗S ωSE)


Proof. As noted in Van Rompay’s comments preceding the Theorem, thehypothesis is necessary only to ensure that QGrS and QGrT are well-defined. Thanks to Proposition 3.11 and Lemma 3.19, the equivalence fol-lows by running the same argument.

3.5 A Comparison with the Commutative Situation

To provide a touchstone for the reader, we interpret the definitions and re-sults when A and B are commutative and finitely-generated by elementsof weight 1. Then, A = k[x1, . . . , xn]/IA and B = k[y1, . . . , ym]/IB forsome homogeneous ideals IA, IB. So SpecA is a closed Gm-stable sub-scheme of affine space An and similarly for SpecB. Let X and Y be theassociated projective schemes. Then,

SpecA⊗k B ⊂ An+m

is G2m-stable. The category Gr(A⊗kB) is equivalent to the G2

m-equivariantquasi-coherent sheaves on SpecA ⊗k B with Tors(A ⊗k B) being thosemodules supported on the subscheme corresponding to

(x1, . . . , xn)(y1, . . . , ym).

Descent then gives that

QGr (A⊗k B) ∼= Qcoh(X × Y ).

The quotient Gr(A⊗kB)/Tors(A⊗kB) is equivalent to QcohG2m(U×V )

for the quasi-affines U = SpecA \ 0 and V = SpecB \ 0. Since U × V isa G2

m torsor over X × Y we have QcohG2m(U × V ) ∼= Qcoh(X × Y ).

3.6 Graded Morita Theory

This section demonstrates how the tools of dg-categories yield a nice per-spective on derived graded Morita theory. One can compare with the well-known graded Morita statement in [43].


In order to utilize the machinery of dg-categories, we must first trans-late chain complexes of graded modules into dg-categories. While one cannaıvely regard this category as a dg-category by way of an enriched Homentirely analogous to the ungraded situation, the relevant statements of [38]are better suited to the perspective of functor categories. As such, we adaptthe association of a ringoid with one object to a ring from Section 2.2 tothe graded situation, considering instead a ringoid with multiple objects.This notion is standard, see eg [41].

Throughout this section, we will let G = (G,+) be an abelian group,and let A and B be not necessarily commutative G-graded algebras overk. We will generally be concerned with the groups Z and Z2. In the sequel,there will be many instances where there are two simultaneous gradingson an object: homological degree and homogeneous degree. We avoid thelatter term, preferring weight, and use degree solely when referring to ho-mological degree.

For clarity, consider the example of a complex of G-graded left A mod-ules, M . The degree n piece of M is the G-graded left A module Mn. Theweight g piece of the graded module Mn is the A0 module of homoge-neous elements of (graded) degree g, Mn

g . Note that in this terminology,the usual morphisms of graded modules are the weight zero morphisms.

Definition 3.47. Denote by C (GrA) the dg-category with objects chaincomplexes of G-graded left A modules and morphisms defined as follows.

We say that a morphism f : M → N of degree p is a collection of mor-phisms fn : Mn → Nn+p of weight zero. We denote by C (GrA) (M,N)p

the collection of all such morphisms, which we equip with the differential

d(f) = dN ◦ f + (−1)p+1f ◦ dM

and define C (GrA) (M,N) to be the resulting chain complex. Composi-tion is the usual composition of graded morphisms.

We denote by C (Gr (Aop)) the same construction with G-graded rightA modules, which are equivalently left modules over the opposite ring,Aop.


Remark 3.48. One should note that the closed morphisms are precisely themorphisms of complexes M → N [p] and, in particular, the closed degreezero morphisms are precisely the usual morphisms of complexes.

Definition 3.49. To each G-graded k-algebra, A, associate the category Awith objects the group G and morphisms given by

A(g1, g2) = Ag2−g1

and composition defined by the multiplication Ag2−g1Ag3−g2 ⊆ Ag3−g1 .We regard A as a dg-category by considering the k module of mor-

phisms, A(g1, g2), as the complex with Ag2−g1 in degree 0 and zero differ-ential.

Lemma 3.50. Let G be an abelian group. If A is a G-graded algebra overk and A the associated dg-category, then there is an isomorphism of dg-categories

C (GrA) ∼= dgMod (A) .

Proof. We first construct a dg-functor F : C (GrA) → dgMod (A). Foreach g ∈ G, denote by A(g)[0] the complex with A(g) in degree zero andconsider the full subcategory of C (GrA) of all such complexes. We seethat a morphism

f ∈ C (GrA) (A(g)[0],M)n

is just the data of a morphism f 0 : A(g)→Mn which gives

C (GrA) (A(g)[0],M)n ∼= GrA(A(g),Mn) ∼= Mn−g

and hence M−g := C (GrA) (A(g)[0],M) is the complex with Mn−g in

degree n. In particular, when M = A(h)[0], we have

C (GrA) (A(g)[0], A(h)[0]) := A(h)[0]−g = A(g, h),

which allows us to identify this subcategory withA via the Yoneda embed-ding, A(h)[0] corresponding to the representable functor A(−, h). Using


this identification, we can define the image of M in dgMod (A) to be thedg-functor that takes an object g ∈ G to

M−g = C (GrA) (A(g)[0],M)

with structure morphism

A(g, h) ∼= C (GrA) (A(g)[0], A(h)[0])→ C (k) (M−h,M−g)

induced by the representable functor C (GrA) (−,M). The image of amorphism f ∈ C (GrA) (M,N) is defined to be the natural transforma-tion given by the collection of morphisms

hA(−g)[0](f) : C (GrA) (A(−g)[0],M)→ C (GrA) (A(−g)[0], N)

indexed by G.Conversely, we note that the data of a functor M : Aop → C (k) is a

collection of chain complexes,Mg := M(g), indexed byG and morphismsof complexes

· · · Ag−h 0 · · ·

· · · C (k) (Mg,Mh)0 C (k) (Mg,Mh)

1 · · ·

The non-zero arrow factors through Z0(C (k) (Mg,Mh)), so the structuremorphism is equivalent to giving a morphism

Ag−h → C (k) (Mg,Mh)

and thus M determines a complex of graded A modules

M =⊕g∈G

M−g.

A morphism η : M → N is simply a collection of natural transformationsηp such that for each g ∈ G we have ηp(g) ∈ C (k) (Mg, Ng)

p and the


naturality implies that ηp(g) isA-linear. The natural transformation ηp thusdetermines a morphism⊕

g∈G

ηp(−g) ∈ C (GrA)(M, N

)p,

and hence η determines a morphism in C (GrA)(M, N

), which is the

collection of all such homogeneous components. This defines a dg-functordgMod (A)→ C (GrA) which is clearly the inverse of F .

Remark 3.51. 1. It is worth noting that it is natural from the ringoidperspective to reverse the weighting on the opposite ring in that, for-mally,

Aopg = Aop(0, g) = A(g, 0) = A−g

so that Aop(−, h) = A(h,−) is the representable functor corre-sponding to the left module Aop(h) by⊕

g∈G

Aop(−g, h) =⊕g∈G

A(h,−g) =⊕g∈G

A−(g+h) =⊕g∈G

Aopg+h = Aop(h).

With this convention, when considering right modules, one can dis-pense with the formality of the opposite ring by constructing froma complex, M , the dg-functor A → C (k) mapping g to Mg :=

C (Gr (Aop)) (A(−g)[0],M).

2. For A the category associated to the k-algebra A, M an object ofGrA, N an object of Gr(Aop), the k-vector space

N ⊗AM

is usually called the Z-algebra tensor product. See [11, Section 4].

When G = Z2, and A, B are Z-graded algebras over k, we denote byC (GrAop ⊗k B) the dg-category of chain complexes of G-graded B-A-bimodules. We associate to the Z2-graded k-algebra Aop ⊗k B the tensor


product of the associated dg-categories, Aop ⊗B. Note that in the identifi-cation

C (Gr (Aop ⊗k B)) ∼= dgMod (Aop ⊗ B)

the weighting coming from the A module structure is reversed, as in theremark above.

From this construction, we have a dg-enhancement, h-Proj (A), of thederived category of graded modules, D(GrA). Passing through the ma-chinery of Corollary 2.6, we have an isomorphism in Ho (dgcatk)

RHomc (h-Proj (A) , h-Proj (B)) ∼= h-Proj (Aop ⊗ B) .

This allows us to identify an object,

F ∈ RHomc (h-Proj (A) , h-Proj (B))

with a dg A-B-bimodule, P , which in turn corresponds to a morphismΦP : A → h-Proj (B) by way of the symmetric monoidal closed structureon dgcatk.

Following Section 3.3 of [15], we identify the homotopy equivalenceclass, [P ]Iso, of P with [ΦP ] ∈ [A, h-Proj (B)]. The extension of ΦP ,

P ⊗A − = ΦP : h-Proj (A)→ h-Proj (B)

descends to a morphism [ΦP ] ∈ [h-Proj (A) , h-Proj (B)] and induces atriangulated functor that commutes with coproducts

H0(ΦP ) : D(GrA) D(GrB)

M PL⊗AM.

In particular, given an equivalence f : D(GrA) → D(GrB), we obtainfrom [26] a quasi-equivalence

F : h-Proj (A)→ h-Proj (B) .

Tracing through the remarks above, we obtain an object

P ∈ h-Proj (Aop ⊗ B)


providing an equivalence

H0(ΦP ) : D(GrA)→ D(GrB).

4 Derived Morita Theory for NoncommutativeProjective Schemes

Let A and B be left Noetherian, connected graded k-algebras. We want toextend the ideas from Section 3.6 to cover dg-enhancements of D(QGrA).

4.1 Repeated triangulated justifications

We recall a particularly nice type of property of objects in the setting ofcompactly generated triangulated categories. Many important propertiesare of this type, so we name it.

Definition 4.1. Let T be a compactly generated triangulated category. LetP be a property of objects of T . We say that P is RTJ if it satisfies thefollowing three conditions.

• Whenever A → B → C is a triangle in T and P holds for A and B,then P holds for C.

• If P holds for A, then P holds for the translate A[1].

• Let I be a set and Ai be objects of T for each i ∈ I . If P holds foreach Ai, then P holds for

⊕i∈I Ai.

We say that P is rtj if it satisfies the following three conditions.

• Whenever A → B → C is a triangle in T and P holds for A and B,then P holds for C.

• If P holds for A, then P holds for the translate A[1].

• Let I be a finite set and Ai be objects of T for each i ∈ I . If P holdsfor each Ai, then P holds for

⊕i∈I Ai.


Proposition 4.2. Let P be an (rtj) RTJ property that holds for a set ofcompact generators of T . Then P holds for all objects of (T c) T .

Proof. Let P be the full triangulated subcategory of objects for which P

holds. Then P

• contains a set of compact generators,

• is triangulated, and

• is closed under formation of (finite) coproducts.

Thus, P is all of (T c) T .

4.2 Vanishing of a tensor product

Recall from Section 2.3 that the tensor product of M,N over A is thetruncation of the standard bar complex

(ML⊗A N)` = M ⊗k A⊗k · · · ⊗k A︸︷︷︸

l

⊗kN.

As k is a field, everything is k-flat. As a consequence,L⊗A preserves quasi-

isomorphisms in each entry. This justifies the notation and gives a particu-lar model for the derived tensor product. In particular, for an h-projectiveE, the natural map

−L⊗A E → −⊗A E


Definition 4.3. Let M be a complex of graded left A modules and let Nbe a complex of graded right A modules. We say that the pair satisfiesF(M,N) if the tensor product

RτAopNL⊗A RQAM

is acyclic. IfF(M,N) holds for all M and N , then we say that A satisfiesF.


Proposition 4.4. Fix a finitely generated connected graded k-algebra, A.Assume that RτA and RτAop commute with coproducts. Then A satisfiesF if and onlyF(A(u), A(v)) holds for each u, v ∈ Z.

Proof. The necessity is clear, so assume thatF(A(u), A(v)) holds for eachu, v ∈ Z. Note that F(M,A(v)) holds for all v is an RTJ property ofM that holds for the set of compact generators A(u), u ∈ Z. Thus, byProposition 4.2, F(M,A(v)) holds for all v holds for all M in D(GrA).Similarly, we can consider the property of N in D(Gr(Aop)): F(M,N)

holds for all objects M of D(GrA). This is also RTJ so F(M,N) holdsfor all M and N .

There are also analogs of the projection formula in (commutative) al-gebraic geometry.

Proposition 4.5. Fix a finitely generated connected graded k-algebra, A.Let P be a complex of bi-bi A modules and let M be a complex of gradedleft A modules. Assume RτA commutes with coproducts. There is naturalquasi-isomorphism

(R′τAP )L⊗AM → RτA

(P

L⊗AM

).

Similarly, assume RQA commutes with coproducts. There is natural quasi-isomorphism

(R′QAP )L⊗AM → RQA

(P

L⊗AM

).

Proof. We treat theQ projection formula. The τ projection formula is anal-ogous. Levelwise, we have the natural map

QAP ⊗k A⊗k · · · ⊗k M → QP (P ⊗k A⊗k · · · ⊗k M)

which comes from the following. For any k module N , given

ψ ⊗k N ∈ GrA(A/A≥m, P )⊗k N


we naturally get

ψ : A/A≥m → P ⊗k Na 7→ ψ(a)⊗ n.

Taking the colimit gives the natural transformation. Note this also gives anatural map

QAP ⊗AM → QA (P ⊗AM) .

Since QA commutes with coproducts, if P is QA-acyclic, then so isP ⊗kN = P⊕ dimk N where we interpret dimkN as a set. Furthermore, thelevelwise map is an isomorphism since, again, QA commutes with coprod-ucts.

For the hypothesis, recall Definition 3.43.

Proposition 4.6. Assume A is delightful. ThenF holds for A.

Proof. By Proposition 4.4, it suffices to check F(M,A(v)) for each v,

which is equivalent to checkingF(M,⊕

v A(v)) because RτAop and−L⊗A

− both commute with coproducts. While computingF(M,⊕

v A(v)) onlydepends on the right A module structure of

⊕v A(v), if we recognize that

∆ with its natural bi-bi structure⊕v

A(v) =⊕v

(⊕u

Au+v

)=⊕u,v

Au+v = ∆,

restricts to⊕

v A(v) as a right A module, then it is equivalent to checkF(M,∆). This observation provides the advantage of being able to utilizethe projection formulas as follows.

Apply − ⊗LA RQAM to the diagram of Proposition 3.42 to obtain a

diagram

RτVdBA⊗kAop∆

L⊗A RQAM

R′τA∆⊗LA RQA(M) R′τAop∆⊗L

A RQA(M)


with all arrows quasi-isomorphisms. We may now apply Proposition 4.5to the left-hand side, because ∆ is a bi-bi module, to obtain the quasi-isomorphism

R′τA∆L⊗A RQAM ∼= RτA

(∆

L⊗A RQAM

)∼= RτA (RQAM) = 0,

as desired.

4.3 Duality

One can regard the complex of bi-bi modules RQA⊗kAop∆ as a sum ofcomplexes of A modules

RQA⊗kAop∆ =⊕x

(RQA⊗kAop∆)∗,x

and define for any object, M , of C (GrA) the object

RHomA(M,RQA⊗kAop∆) =⊕x

RHomA(M,RQA⊗kAop∆∗,x)

of C (Gr (Aop)). Consider the functor

(−)∨ : C (GrA)op → C (Gr (Aop))

M 7→ RHomA (M,RQA⊗kAop∆)

Note that evaluation provides a natural transformation

η : Id→ (−)∨∨.

We also have the usual duality between left and right modules

(−)∗ : C (GrA)op → C (Gr (Aop))

M 7→ RHomA (M,∆)

with the associated natural transformation

ν : Id→ (−)∗∗.


It is easy to see that (A(x))∗ is A(−x).For any bi-bi module P , Hom-tensor adjunction gives a natural isomor-

phism

HomAop(A≥n,HomA(M,P ))→ HomA(M,HomAop(A≥n, P )).

Passing to the colimit, we get a natural map

QAop (HomA(M,P ))→ HomA(M,QAopP )

and therefore a natural map

γP : RQAop (RHomA(M,P ))→ RHomA(M,R′QAopP ).

Note that the map γP is a quasi-isomorphism if we assume that M is com-pact.

Proposition 4.7. We have a commutative diagram

RQAM

RQARHomAop(RHomA(M,∆),∆)

RHomAop(RHomA(M,∆),R′QA∆)

RHomAop(RQAopRHomA(M,∆),R′QAopR′QA∆)

RHomAop(RQAopRHomA(M,∆),RQA⊗Aop∆)

RHomAop(RHomA(M,R′QAop∆),RQA⊗Aop∆)

RHomAop(RHomA(RQAM,R′QAR′QAop∆),RQA⊗kAop∆)

RHomAop(RHomA(RQAM,RQA⊗kAop∆),RQA⊗kAop∆)

ηRQAM

RQA(νM)

γ∆

RQAop(−)

δ∆ ◦ −

− ◦ γ∆

− ◦RQA(−)RHom(RQAM, δ∆)∨

Proof. The existence of this diagram follows from the existence of theunderived version. The underived version is straightforward to verify. We


suppress most the details but note that the image of a map φ : A≥n1 → M

from QAM is the map

HomAop(A≥n2 ,M∗)→ HomA⊗Aop(A≥n1 ⊗ A≥n2 ,∆)

ψ 7→[a1 ⊗ a2 7→ ψ(a2)(φ(a1))

].

For clarity: the latter is the evaluation of ψ(a2) at φ(a1).

Lemma 4.8. Assume that A is delightful and M is compact object ofD(Gr(A)). Then, ηRQAM is a quasi-isomorphism.

Proof. We check that all the other maps in the diagram of Proposition 4.7are quasi-isomorphisms.

First, since M is compact νM is a quasi-isomorphism and RQA pre-serves quasi-isomorphisms. Hence RQAνM is also a quasi-isomorphism.

Next since M is compact so is M∗. Thus, γ∆ is a quasi-isomorphism.Since A is delightful, we know that R′QA∆ is RτAop torsion-free.

Thus, applying RQAop to maps from any object to R′QA∆ yields a quasi-isomorphism of morphism spaces.

The map δ∆ is a quasi-isomorphism by Proposition 3.41. Derived mor-phism spaces preserve quasi-isomorphisms so it follows that δ∆ ◦ − is aquasi-isomorphism.

Since M is compact, γ∆ is a quasi-isomorphism and so is compositionwith it since derived morphism spaces preserve quasi-isomorphisms.

We again appeal to A being delightful to know that R′QAop∆ is RτA

torsion free. Thus, RQA acting on morphism spaces from anything toR′QAop∆ yields a quasi-isomorphism. Again composition with it remainsa quasi-isomorphism.

Finally (−)∨ and RHom(RQAM,−) preserve quasi-isomorphisms soRHom(RQAM, δ∆)∨ is also a quasi-isomorphism.

Lemma 4.9. Assume that A is delightful. Then, there is a natural quasi-isomorphism between RQAopA(−x) and (RQAA(x))∨.


Proof. We have the following sequence of maps:

RQAopRHomA(A(x),∆) RHomA(A(x),R′QAop∆)

RHomA(RQAA(x),R′QAR′QAop∆).

γ∆

RQA(−)

The former is a quasi-isomorphism because A(x) is compact. The latteris a quasi-isomorphism because A is delightful and so R′QAop∆ is RτA

torsion free.Using Proposition 3.41 gives a quasi-isomorphism

RHomA(RQAA(x),R′QAR′QAop∆) ∼=

RHomA(RQAA(x),RQA⊗Aop∆) = (RQAA(x))∨.

On the other side, we have

A(−x) ∼= HomA(A(x),∆) ∼= RHomA(A(x),∆).

Applying RQAop gives the final quasi-isomorphisms.

Definition 4.10. Let QA be the full dg-subcategory of C (GrA) with ob-jects given by

RωAπA(x) := ωARQGrAπAA(x)

for all x ∈ Z. HereRQGrA is the fibrant replacement functor in C(QGrA).Note that, since ωAπA ∼= Id, all objects ofQA satisfy the condition that

εM : M → QAM is an isomorphism of complexes. Similarly, since ωAhas an exact left adjoint, ωA preserves fibrations. Hence, each RωAπA(x)

is fibrant as a complex of graded A modules.

Lemma 4.11. There exists a map of chain complexes

RQAA(x)→ RωAπAA(x)

which is a quasi-isomorphism.

Proof. Consider the diagram


πAM RQGrAπAM

πARGrAM 0.

Since πA is exact, the map πAM → πARGrAM is a trivial cofibration.Hence there exists a lift

πAM RQGrAπAM

πARGrAM 0.

which must also be a quasi-isomorphism.

Corollary 4.12. Assume that A is delightful. There is a quasi-equivalencebetween (QA)op and Q(Aop) which is isomorphic to (−)∨ at the level ofderived categories.

Proof. We can choose a h-injective complex bi-bi modules I such thatthere is a homotopy equivalence RQA⊗Aop∆ → I and the natural mapsI → QAopI and I → QAI are isomorphisms. This comes from taking acofibrant replacement of πA⊗AopRQA⊗Aop∆ and applying ωA⊗Aop . Then,we have a homotopy equivalence

RHomA(M,RQA⊗Aop∆)→ HomA(M, I)

and we may replace (−)∨ by HomA(M, I). We do so but we keep the samenotation. Note the image of (−)∨ now consists of h-injective graded Aop

modules.From Lemma 4.8, we see (−)∨ is quasi-fully faithful on QA and from

Lemma 4.9 we see there is a quasi-isomorphism between (RQAA(x))∨

and RQAopA(−x). It follows that RωAopπAopA(−x) and (RωAπAA(x))∨

are quasi-isomorphic h-injective complexes. Hence, they are homotopyequivalent. Therefore, dg-functor

Ξ : (QA)op → dgMod(QAop)

M 7→ Hom(−,M∨)


has image homotopy equivalent to representable modules. Consequently,Ξ is quasi-fully-faithful overall. Its image consists of h-projective set ofgenerators.

We get a induced functor

Ξ : (QA)op → h-ProjQAop.

The property that Ξ(M) is h-projective is rtj in M , as is the property thatΞ(M) is compact. Applying Proposition 4.2 shows that Ξ lands in QAop.

The property that Ξ is quasi-fully faithful on Hom(M,N) for allN is rtjin M . Thus it suffices to prove that Ξ is quasi-fully faithful on Hom(C,N)

for all N and some set of compact generators C. But since C is compactthe condition, that Ξ is quasi-fully-faithful Hom(C,N) is rtj for N . Thus,we can reduce to checking quasi-fully-faithfulness on a set of compactgenerators. But we already saw that Ξ is quasi-fully-faithful on QA.

Finally, the condition that L is homotopy equivalent to Ξ(M) for someM is rtj in L and is true for a compact set of generators. Again Propo-sition 4.2 allows us to conclude that Ξ is quasi-essentially surjective andhence a quasi-equivalence.

We have a natural map

M∨ L⊗A N → RHomA(M,RQA⊗kAop∆

L⊗A N)

φ⊗ n 7→ (m 7→ φ(m)⊗ n).

Lemma 4.13. Assume that A is delightful, the map N → RQAN is aquasi-isomorphism, and M is quasi-isomorphic to RQAM

′ for compactM ′. Then both natural maps in the diagram

M∨ L⊗A N RHomA(M,RQA⊗kAop∆

L⊗A N)

RHomA(M,N)

are quasi-isomorphisms.


Proof. The vertical map comes from the map ∆→ RQA⊗kAop∆. We havea triangle

RτA⊗kAop∆L⊗A N → ∆

L⊗A N → RQA⊗kAop∆

L⊗A N.

Using Proposition 4.6 and the assumption that N → RQAN is a quasi-isomorphisms, we have quasi-isomorphisms

RτA⊗kAop∆L⊗AN ∼= RτA⊗kAop∆

L⊗ARQAN ∼= RτAop∆

L⊗ARQAN ∼= 0.

Thus, the map

N ∼= ∆L⊗A N → RQA⊗kAop∆

L⊗A N

is a quasi-isomorphism. So the map

RHomA(M,N)→ RHomA(M,RQA⊗kAop∆L⊗A N)

is also a quasi-isomorphism.The property that the map

M∨ L⊗A N → RHomA(M,RQA⊗kAop∆

L⊗A N)

is a quasi-isomorphism is rtj. Thus, we may reduce to checking the caseM = RQAA(x) by Proposition 4.2. We have a commutative diagram

RHomA(RQAA(x),RQA⊗kAop∆)L⊗A N RHomA(RQAA(x),RQA⊗kAop∆

L⊗A N)

RHomA(A(x),RQA⊗kAop∆)L⊗A N RHomA(A(x),RQA⊗kAop∆

L⊗A N)

coming from the map A(x)→ RQAA(x). Since

RQA⊗kAop∆L⊗A N and RQA⊗Aop∆

are RτA torsion free, the vertical maps are quasi-isomorphisms. SinceA(x) is compact, RHomA(A(x),−) ∼= HomA(A(x),−) commutes withcoproducts. Consequently, HomA(A(x),−) commutes with ⊗k and hence

HomA(A(x), P )L⊗A N → HomA(A(x), P

L⊗A N)

is an isomorphism levelwise for any complex of bi-bi modules P .


4.4 Products

The following allows to re-express the tensor up to quasi-equivalence.

Proposition 4.14. If A and B are both Ext-finite, left Noetherian, rightNoetherian, and RτA and RτB have finite dimension, then the dg-functor

Υ : h-Inj(QGrA⊗k B)→ h-Proj(QA⊗k QB)

I 7→ HomC(GrA⊗kB)(RA⊗kB(−�−), ωA⊗kB(I))

is well-defined and is a quasi-equivalence.

Proof. Note here that M �k N is simply M ⊗k N as a complex of bi-bimodules.

We first reduce to the images of � and ωA⊗kB inside C(Gr(A ⊗k B)).Similar to Lemma 3.29, ωA⊗kB is quasi-fully-faithful.

Next we check that �k is quasi-fully-faithful. For general M,M ′ ∈C(GrA) andN,N ′ ∈ C(GrB), we have a commutative diagram of naturalmaps

Hom(M,M ′)⊗k Hom(N,N ′) Hom(M ⊗k M ′, N ⊗k N ′)

Hom(M,M ′ ⊗k Hom(N,N ′)) Hom(M,Hom(N,M ′ ⊗k N ′))

�

(2)where the right vertical map is an isomorphism coming from Hom-⊗ ad-junction. If M and N are compact objects, then the other maps are quasi-isomorphisms. Since

Hom(RQAM,RQAM′′)→ Hom(M,RQAM

′′)

Hom(RQBN,RQBN′′)→ Hom(N,RQBN

′′)

are quasi-isomorphisms and all objects in the previous diagram are Rτ -torsion free, it suffices to know that M is quasi-isomorphic to RQAM

for some compact M and similarly for N . So if we restrict attention tothe objects RQAA(x) and RQBB(y), then all the maps in Diagram 2 arequasi-isomorphisms and � is quasi-fully-faithful.


Next we check that fibrant replacement

R := RGrA⊗kB : QA⊗k QB → R(QA⊗k QB)

is a quasi-equivalence, where R(QA⊗k QB) denotes the essential image.For general complexes M,M ′ of graded A modules and N,N ′ of gradedB modules, we have the commutative diagram

Hom(M ⊗k N,M ′ ⊗k N ′)

Hom(R(M ⊗k N), R(M ′ ⊗k N ′))

Hom(M ⊗k N,R(M ′ ⊗k N ′))

R

The bottom diagonal map is a quasi-isomorphism since R(M ′⊗kN ′) is fi-brant and the natural mapM⊗kN → R(M⊗kN) is a quasi-isomorphism.The previous computation shows that, when M and N are compact, thetensor product of h-injective complexes is acyclic for Hom(M ⊗k N,−)

while R(M ′⊗k N ′) is fibrant and hence acyclic. Thus, taking M,M ′ fromQA and N,N ′ from QB, we see that the left vertical map is a quasi-isomorphism as it comes from applying a functor to a quasi-isomorphismbetween acyclic objects. Consequently, R is a quasi-fully-faithful. By def-inition of the image, it is essentially surjective and therefore is a quasi-equivalence.

Using morphisms in C(GrA⊗k B), for an object I of h-Inj(QGrA⊗kB) we have a dg module

R(QA⊗k QB)op → C(k)

M 7→ Hom(M,ω(I)).

which induces a dg-functor

h-Inj(QGrA⊗k B)→ dgModR(QA⊗k QB)


We want to check that this induces an equivalence with h-ProjR(QA ⊗kQB). If so, since Υ is a composition of this functor and R, which wasalready shown to be a quasi-equivalence, we get the desired conclusion.

Note first that Υ commutes with coproducts, up to quasi-isomorphism.Indeed, we have a sequence of quasi-isomorphisms: the map

Hom(R(RωAπAA(x)⊗k RωBπBB(y)), ωA⊗kBI)

Hom(RωAπAA(x)⊗k RωBπBB(y), ωA⊗kBI)

since P → RP is a quasi-isomorphism and ωA⊗kBI is h-injective. Usingadjunction, we have an isomorphism

HomGrA⊗kB(RωAπAA(x)⊗k RωBπBB(y), ωA⊗kBI) ∼=HomQGrA⊗kB(RQGrAπAA(x)⊗k RQGrBπBB(y), I).

Again since I is h-injective and ⊗k preserves quasi-isomorphisms since kis field, we get a quasi-isomorphism

Hom(RQGrAπAA(x)⊗k RQGrBπBB(y), I)

Hom(πAA(x)⊗k πBB(y), I).

Using adjunction again, we have an isomorphism

HomQGr A⊗kB(πAA(x)⊗k πBB(y), I) ∼=HomGrA⊗kB(A(x)⊗k B(y), ωA⊗kBI).

The end of these chains of quasi-isomorphisms commutes with coproductssince RωA⊗kB commutes with coproducts with both A and B Ext-finite byProposition 3.36.

Next, we note that

Υ(R(πAA(x)⊗ πBB(y))) = Hom(−,RωA⊗kBπAA(x)⊗k πBB(y))


and RωA⊗kBπAA(x)⊗k πBB(y)) and R(RωAπAA(x)⊗RωBπBB(y) arequasi-isomorphic. Since both objects are h-injective, they are homotopyequivalent. Thus, Υ(R(πAA(x) ⊗ πBB(y))) is homotopy equivalent to arepresentable functor. Consequently, Υ is quasi-fully-faithful on the fullsubcategory consisting of the R(πAA(x) ⊗ πBB(y). Moreover, the im-ages of these complexes are h-projective dg modules since h-projectivityis preserved under homotopy equivalence and representable modules areh-projective.

The property that Υ(I) is h-projective is RTJ in I since Υ commuteswith coproducts. Thus, by Proposition 4.2, we see that Υ has image inh-ProjR(QA⊗QB).

Similarly, the property that Υ is quasi-fully-faithful on Hom(M,N) forall N is RTJ. By Proposition 4.2 we just need to check that

RA⊗kB(πAA(x)⊗ πBB(y))

have this property. But, the property that Υ is quasi-fully-faithful on

Hom(RA⊗kB(πAA(x)⊗ πBB(y)),M)

is RTJ in M . This is due to the quasi-isomorphism

HomQGrA⊗kB(RπAA(x)⊗ πBB(y), I)

HomGrA⊗kB(A(x)⊗k B(y), ωA⊗kBI)

plus the facts that ωA⊗kB commutes with coproducts and A(x)⊗kB(y) arecompact.

Thus, we just need to check that Υ is quasi-fully-faithful on the fullsubcategory consisting of the RA⊗kB(πAA(x) ⊗ πBB(y)) which we haveseen. We can conclude that Υ is quasi-fully-faithful overall.

The property that J ∼= Υ(I) in the homotopy category is RTJ in J andis satisfied by a compact set of generators. Thus, Υ is quasi-essentiallysurjective. Hence, Υ is a quasi-equivalence.


4.5 The quasi-equivalence

Now we turn to the main result.

Theorem 4.15. Let k be a field. Let A and B be connected graded k-algebras. IfA andB form a delightful couple, then there is a natural quasi-equivalence

F : h-Inj (QGr(Aop ⊗k B))→ RHomc (h-Inj (QGrA) , h-Inj (QGrB))

such that for P ∈ D(QGr(Aop ⊗k B)), the exact functor H0(F (P )) isisomorphic to

ΦP (M) := πB

(RωAop⊗kBP

L⊗A RωAM

).

Proof. Appealing to Proposition 4.14 we have quasi-equivalences

h-Inj(QGrA) ∼= h-Proj(QA)

h-Inj(QGrB) ∼= h-Proj(QB).

As RHomc (−,−) preserves quasi-equivalences, it suffices to construct aquasi-equivalence between RHomc (h-ProjQA, h-ProjQB) andh-Inj(QGrA⊗k B).

Applying Corollary 2.6, it then suffices to provide a quasi-equivalence

h-Inj (QGr(Aop ⊗k B)) ∼= h-Proj ((QA)op ⊗QB)

In general, the inclusion C → C induces a quasi-equivalence

h-Proj (C ⊗ D) ∼= h-Proj(C ⊗ D

)for two small dg-categories C and D.

Using Corollary 4.12, we have a quasi-equivalence

h-Proj(

(QA)op ⊗QB)∼= h-Proj

(QAop ⊗QB

)and thus a quasi-equivalence

h-Proj ((QA)op ⊗QB) ∼= h-Proj (QAop ⊗QB) .


From Proposition 4.14 we have the quasi-equivalence

Υ : h-Inj(QGrAop ⊗k B)→ h-Proj(QAop ⊗k QB).

Combining these gives the desired quasi-equivalence.Tracing out the quasi-equivalences, one just needs to note that

Hom(RQAA(x)∨ ⊗k RQBB(y), P ) ∼=Hom(RQBB(y),Hom(RQAA(x)∨,RωAop⊗kBP )) ∼=

Hom(RQBB(y),RωAop⊗kBPL⊗A RQAA(x))

using Proposition 4.6 and Lemma 4.13. This says that the induced contin-uous functor is

M 7→ πB

(RωAop⊗kBP

L⊗A RωAM

).

The following statement is now a simple application of Theorem 4.15and results of [26].

Corollary 4.16. Let k be a field. Let A and B be a delightful couple ofconnected graded k-algebras. Assume that there exists an equivalence

f : D(QGrA)→ D(QGrB).

Then there exists an object P ∈ D(QGr(Aop ⊗k B)) such that

ΦP : D(QGrA)→ D(QGrB)

is an equivalence.

Proof. Applying [26, Theorem 1] we know there is a quasi-equivalencebetween the unique enhancements, i.e. there is an

F ∈ [h-Inj (QGrA) , h-Inj (QGrB)]


giving an equivalence

H0(h-Inj (QGrA)) D(QGrA)

H0(h-Inj (QGrB)) D(QGrB)

H0(F )

Then, by Theorem 4.15, there exists a P ∈ D(QGr(Aop ⊗k B)) such thatΦP = H0(F ).

Remark 4.17. In particular, one can ask what the kernel associated to theidentity functor on D(QGrA). In this case, it is easy to see that

ΦπA⊗kAop∆∼= IdD(QGrA)

justifying the notation.

One also has a statement for bounded and finitely generated categorywhich is analogous to Orlov’s theorem for equivalences between boundedderived categories of coherent sheaves on smooth and projective varieties.

Corollary 4.18. Let A and B be a delightful couple of connected gradedk-algebras with k a field. Assume that the enhancements of Db(qgrA) andDb(qgrB) are both smooth and proper as dg-categories. If there exists anequivalence

f : Db(qgrA)→ Db(qgrB),

then there exists an object P ∈ Db(qgrAop ⊗k B) such that

ΦP : Db(qgrA)→ Db(qgrB)

is an equivalence.

Proof. Since any generator of a smooth and proper dg-category must bea strong generator, both QGrA and QGrB necessarily have finite coho-mological dimension. From [12, Lemma 3.4.2], we know that the compact


objects in D(QGrA) are exactly Db(qgrA). Using Corollary 4.12, we seethat the enhancement of Db(qgrAop) is smooth and proper and that

D(QGrAop)c ∼= Db(qgrAop).

Applying [26, Theorem 2.8] we know that there is a quasi-equivalencebetween the unique enhancements, i.e. there is an

F ∈ [h-Inj (QGrA)c , h-Inj (QGrB)c]

giving an equivalence

H0(h-Inj (QGrA)c) Db(qgrA)

H0(h-Inj (QGrB)c) Db(qgrB)

H0(F )

There is then an induced quasi-equivalence between the two big cate-gories, h-Inj(QGrA) and h-Inj(QGrB), and hence corresponds to an ob-ject in D(QGrAop ⊗k B) by Theorem 4.15. Since the induced functortakes compact objects to compact objects and h-Inj (QGrA)c (an enhance-ment of Db(qgrA)) is smooth as a dg-category, [39, Lemma 2.8] saysthe corresponding object of D(QGrAop ⊗k B) is a compact object. AsDb(qgrAop ⊗k B) is also smooth and proper, the same arguments of [12]give that

D(QGrAop ⊗k B)c ∼= Db(qgrAop ⊗k B).

We wish to identify the kernels as objects of the derived category ofan honest noncommutative projective scheme. In general, one can onlyhope that kernels obtained as above are objects of the derived category ofa noncommutative (bi)projective scheme. However, we have the followingspecial case in which we can collapse the Z2-grading to a Z-grading.


Corollary 4.19. Let A and B be a delightful couple of connected gradedk-algebras with k a field that are both generated in degree one. Assumethat there exists an equivalence

f : D(QGrA)→ D(QGrB).

Then there exists an object P ∈ D(QGr(Aop×kB)) that induces an equiv-alence

D(QGrA) D(QGrB)

M πB(V(P )⊗L RωAM

)Proof. The equivalence V of Theorem 3.46 extends naturally to a quasi-equivalence

V : h-Inj (QGrS)→ h-Inj (QGrT ) .

Now choose P such that V(P ) is homotopy equivalent to the kernel ob-tained by an application of Corollary 4.16, so the desired equivalence isΦV(P ).

Coming back to Example 3.3. We ask the following question.

Question 4.20. Fix qij ∈ C. Then two noncommutative projective schemesAφq and Aφ

′q are isomorphic if and only if they are derived equivalent.

In the commutative case, this is a derived Torelli statement which onecan understand via matrix factorizations [30] and the Mather-Yau theorem[27].

Acknowledgments. Both authors were partially supported by a NSF Standard Grant DMS-1501813. The first author would like to thank the Institute for Advanced Study for pro-viding an enriching and focused environment during his membership. The paper benefitedfrom valuable comments by Pieter Belmans and Paolo Stellari; we thank them. Both au-thors are incredibly honored to have known Ragnar-Olaf Buchweitz.

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Kernels for noncommutative projective schemes › farmanb › files › 2019 › 12 › KNCP.pdf · mutative graded rings has proven itself fruitful as well. Indeed, moduli spaces