Kernels for Noncommutative Projective Schemes

We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along lines of Rickard and Orlov.


Introduction
One of the most useful and pleasing statements in algebra is the Morita Theorem [Mor58].
Theorem 1.1. Let A and B be rings and assume we have an equivalence of module categories Then there exists an A-B bimodule which is projective as an A-module and isomorphic to B as a B-module. Consequently, P ⊗ A − is an equivalence.
For noncommutative rings, one can easily find non-isomorphic examples of Morita equivalent rings. One such easy example is the equivalence between the modules over a ring A and the modules over its matrix ring, M n (A), given by the bimodule A n . For commutative rings, two rings are Morita equivalent if and only if they are isomorphic. One can try to slacken the relationship by considering the derived version of Morita for rings [Ric89]. Even still, here two commutative rings are derived Morita equivalent if and only if they are isomorphic. To get a strictly weaker equivalence relation, we should globalize the notion of a commutative ring by passing to schemes.
Then there exists an object P ∈ D(Qcoh X × Y ) such that gives an equivalence.
−→ Y are the projections. This result, as stated, follows immediately from results of Töen and Lunts-Orlov [Toë07,LO10]. A more specific version for smooth projective schemes predates this [Orl97]. In this generality, we unlock a deep and subtle equivalence relationship. Indeed, understanding these Fourier-Mukai partnerships is a central problem in the field of derived categories in algebraic geometry. Given the richness of the globalized Morita statement for commutative rings, one is led to think about a globalized version for noncommutative rings. A framework for phrasing such a question is due to Artin and Zhang [AZ94] and goes by the name Noncommutative Projective Geometry. A graded Morita theorem for noncommutative rings is due to Zhang [Zha96].
Following the line of thought above, one wonders about derived Morita theory for noncommutative projective schemes X and Y which are associated to connected graded k-algebras A and B. As an application of the main result of this article, we have the following statement. One says that A and B form a delightful couple if they are both Ext-finite in the sense of [VdB97], both are left and right Noetherian, and both satisfy χ • (R) for R = A, A op for A and R = B, B op for B. One can think of this requirement as Serre vanishing for the twisting sheaves plus some finite-dimensionality over k.
Theorem 1.4. Let X and Y be noncommutative projective schemes associated to a delightful couple of over a field k, both of which are generated in degree one. Then for any equivalence D(Qcoh X) → D(Qcoh Y ), there exists an object P of D(Qcoh X × k Y ) whose associated integral transform is an equivalence of Fourier-Mukai type.
The interested reader can see Corollary 4.17 for a more careful statement of this result.
This result is, as the notation suggests, an application of a more general result. Work of Töen provides a good bicategorical structure for dg-categories up to quasi-equivalence [Toë07] -an internal Hom. The construction of RHom c (C, D) is abstract even if the dg-categories C and D arise geometrically (or noncommutative geometrically). To unleash the power of Töen's work, one needs to identify the internal Hom more "internally." Indeed, this is done for schemes in [Toë07], for higher derived stacks (using machinery of Lurie in place of Töen) in [BZFN10], and matrix factorizations in [Dyc11,PV12,BFK14]. The main result here is the identification of this internal Hom in Noncommutative Projective Geometry, see Theorem 4.15.
This work is part of the second author's thesis [Far18], where the homological conditions making up tastiness will be sorted out more, the calculus of kernels will be more fully developed as will conditions for fully faithfulness, and more applications to noncommutative invariants in the style of Kontsevich will be studied.
1.1. Conventions. We let k denote a field. Often, for ease of notation, C(X, Y ) will be used to refer to the morphims, Hom C (X, Y ), between objects X and Y of a category C. We shall also use an undecorated Hom again depending on the complexity of the notation. Whenever C has a natural enrichment over a category, V, we will denote by C(X, Y ) the V-object of morphisms. For example, the category of complexes of k-vector spaces, C (k), can be endowed with the the structure of a C (k)-enriched category using the hom total complex, C (k) (C, D) := C(k)(C, D) which has in degree n the k-vector space C (k) (C, D) n = m∈Z Mod k C m , D m+n and differential It should be noted that Z 0 (C (k) (C, D)) = C (k) (C, D).

Background on DG-Categories
Recall that a dg-category, A, over k is a category enriched over the category of chain 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 (F A, 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 dgcat k the 2-category of 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 It is well known that there is an isomorphism endowing dgcat k with the structure of a symmetric monoidal closed category.
For any dg-category, A, we denote by Z 0 (A) the category with objects those of A and morphisms Z 0 (A)(A 1 , A 2 ) := Z 0 (A(A 1 , A 2 )). By H 0 (A) we denote the category with objects those of A and morphisms Following [CS15], we say that two objects A 1 , A 2 of a dg-category, A, are dg-isomorphic (respectively, homotopy equivalent) if there is a morphism f ∈ Z 0 (A)(A 1 , A 2 ) such that f (respectively, the image of f in H 0 (A)(A 1 , A 2 )) is an isomorphism. In such a case, we say that f is a dg-isomorphism (respectively, homotopy equivalence).
2.1. The Model Structure on DG-Categories. We collect here some basic results on the model structure for dgcat k . For any dg-functor F : A → B, we say that F is (i) quasi-fully faithful if for any two objects A 1 , A 2 of A the morphism is a quasi-isomorphism of chain complexes, (ii) quasi-essentially surjective if the induced functor H 0 (F ) : H 0 (A) → H 0 (B) is essentially surjective, (iii) a quasi-equivalence if F is quasi-fully faithful and quasi-essentially surjective, (iv) a fibration if F satisfies the following two conditions: (a) for all objects A 1 , A 2 of A, the morphism F (A 1 , A 2 ) is a degree-wise surjective morphism of complexes, and (b) for any object A of A and any isomorphism η ∈ H 0 (B)(H 0 (F )A, B), there exists an isomorphism ν ∈ H 0 (C)(A, A ′ ) such that H 0 (F )(ν) = η. In [Tab05] it is shown that taking the class of fibrations defined above and the class of weak equivalences to be the quasi-equivalences, dgcat k becomes a cofibrantly generated model category. The localization of dgcat k at the class of quasi-equivalences is the homotopy category, Ho (dgcat k ). We will denote by [A, B] the morphisms of Ho (dgcat k ).
A small dg-category A is said to be h-projective if for all objects A 1 , A 2 of A and any acyclic complex, C, every morphism of complexes A(A 1 , A 2 ) → C is null-homotopic. In [CS15], it is shown that there exists an h-projective category A hp quasi-equivalent to A and, as a result, the localization of the full subcategory of dgcat k of h-projective dg-categories at the class of quasi-equivalences is equivalent to Ho (dgcat k ). In particular, when k is a field, every dg-category is h-projective and hence one can compute the derived tensor product by 2.2. DG-Modules. For any small dg-category, A, denote by dgMod (A) the dg-category of dg-functors dgcat k (A op , C (k)), where C (k) denotes the dg-category of chain complexes equipped with the internal Hom from its symmetric monoidal closed structure. The objects of dgMod (A) will be called dg A-modules. Since one may view the dg A op -modules as what should reasonably be called left dg A-modules, the terms right and left will be dropped in favor of dg A-modules and dg A op -modules, respectively. We note here that the somewhat vexing choice of terminology is such that we can view objects of A as dg A-modules by way of the enriched Yoneda embedding 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 (A op ⊗ B). We note here that the symmetric monoidal closed structure on dgcat k allows us to view bimodules as morphisms of dg-categories by the isomorphism 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 representables, h A = A(−, A) for if M is acyclic, then from the enriched Yoneda Lemma we have Noting that closure of h-proj (A) under homotopy equivalence follows immediately from the Yoneda Lemma applied to H 0 (dgMod (A)), we define A to be the full dg-subcategory of h-proj (A) consisting of the dg A-modules homotopy equivalent to representables.
We will say an h-projective dg A-B-bimodule, E, is right quasi-representable if for every object A of A the dg B-module Φ E (A) is an object of B, and we will denote by h-proj (A op ⊗ B) rqr the full subcategory of h-proj (A op ⊗ B) consisting of all right quasirepresentables.
2.2.2. The Derived Category of a DG-Category. By definition, a degree zero closed morphism ) for all objects A of A. Hence we are justified in the following definitions: is a quasi-isomorphism of chain complexes for all objects A of A, and (ii) η is a fibration if η(A) is a degree-wise surjective morphism of complexes for all objects A of A. Equipping C (k) with the standard projective model structure (see [Hov07, Section 2.3]), these definitions endow Z 0 (dgMod (A)) with the structure of a particularly nice cofibrantly generated model category (see [Toë07,Section 3]). In analogy with the definition of the derived category of modules for a ring A, the derived category of A is defined to be the model category theoretic homotopy category, obtained from localizing Z 0 (dgMod (A)) at the class, W, of quasi-isomorphisms.
It can be shown (see [Kel94,Section 3.5]) that for every dg A-module, M, there exists an h-projective, N, and a quasi-isomorphism N → M, which one calls an h-projective resolution of M. Moreover, it is not difficult to see that any quasi-isomorphism between h-projective objects is in fact a homotopy equivalence. It follows that there is an equivalence of categories between H 0 (h-proj (A)) and D(A) for any small dg-category, A.
It should be noted that this generalizes the notion of derived categories of 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 complexes of A-modules enriched by the Hom total complex with dgMod (A), which is simply the full dg-subcategory of the functor category Fun(A, C (k)) comprised of all dg-functors. From this viewpoint it is easy to recognize the categories Z 0 (dgMod (A)), H 0 (dgMod (A)), and D(A), as the categories C (A), K(A), the usual category up to homotopy, and the derived category of Mod A, respectively. In the language of [LO10], h-proj (A) is a dg-enhancement of D(Mod A).

2.3.
Tensor Products of DG-Modules. Let M be a dg A-module, let N be a dg A opmodule, and let A, B be objects of A. For ease of notation, we drop the functor notation M(A) in favor of M A and write A A,B for the morphisms A(A, B). We have structure morphisms , which give rise to a unique morphism induced by the universal properties of the biproduct. The two collections of morphisms given by projecting onto each factor induced morphisms Ξ 1 , Ξ 2 : and we define the tensor product of M and N to be the coequalizer in C (k) It is routine to check that a morphism M → M ′ of right dg A-modules induces by the universal property for coequalizers a unique morphism . One extends this construction to bimodules as follows. Given objects E of dgMod (A ⊗ B) and F of dgMod (B op ⊗ C), we recall that we have associated to each a dg-functor Φ E : A op → dgMod (B) and Φ F : C op → dgMod (B op ) by the symmetric monoidal closed structure on dgcat k . For each pair of objects A of A and C of C, we obtain dg-modules and hence one may define the object E ⊗ B F of dgMod (A ⊗ C) by One can show that by a similar argument to the original that a morphism Remark 2.1. Denote by K the dg-category with one object, * , and morphisms given by the chain complex K( * , * ) n = k n = 0 0 n = 0 with zero differential. This category serves as the unit of the symmetric monoidal structure on dgcat k , so for small dg-categories, A and C, we can always identify A with A ⊗ K and C with K op ⊗ C. With this identification in hand, we obtain from taking B = K in the latter construction a special case: Given a dg A op -module, E, and a dg C-module, F , we have a dg A-C-bimodule defined by the tensor product

Extensions of Morphisms Associated to Bimodules. Let E be a dg A-B-bimodule.
Following [CS15, Section 3], we can extend the associated functor Φ E to a dg-functor Similarly, we have a dg-functor in the opposite direction For any dg-functor G : A → B we denote by Ind G the extension of the dg-functor and its right adjoint by Res G . By way of the enriched Yoneda Lemma we see that for any object A of A and any dg B-module, N, We record here some useful propositions regarding extensions of dg-functors.
(1) We note that for dg A-and A op -modules, M and N, part (i) implies that the dg-functors

respectively. As an immediate consequence of the enriched Yoneda Lemma
holds for any object A of A.
(2) We denote by ∆ A the dg A-A-bimodule corresponding to the Yoneda embedding, It's clear that we have a dg-functor and for any dg A-A-bimodule, E, we see that When starting with an h-projective we have a very nice extension of dg-functors: . For any h-projective dg A-B-bimodule, E, the associated functor As a direct consequence of the penultimate proposition, this means that we can view the extension of Φ E as a dg-functor Put another way, tensoring with an h-projective A-B-bimodule preserves h-projectives.
One essential result about dgcat k comes from Töen's result on the existence, and description of, the internal Hom in its homotopy category. There exists a natural bijection proving that the symmetric monoidal category Ho (dgcat k ) is closed.
To get a sense of the value of this result, let us recall one application from [Toë07, Section 8.3]. Let X and Y be quasi-compact and separated schemes over Spec k. Recall the dg-model for D(Qcoh X), L qcoh (X), is the C(k)-enriched subcategory of fibrant-cofibrant objects in the injective model structure on C(Qcoh X).
Theorem 2.7 ( [Toë07, Theorem 8.3]). Let X and Y be quasi-compact, quasi-separated schemes over k. Then there exists an isomorphism in Ho (dgcat k ) which takes a complex E ∈ L qcoh (X × k Y ) to the exact functor on the homotopy categories Proof. The first part of the statement is exactly as in [Toë07]. The second part is implicit.
3. Background on noncommutative projective schemes 3.1. Recollections and conditions. Noncommutative projective schemes were introduced by Artin and Zhang in [AZ94]. We recall the definition.
Definition 3.1. Let N be a finitely-generated abelian group. We say that a k-algebra A is N-graded if there exists a decomposition as k-modules For algebraic geometers, the most common example is the homogenous coordinate ring of a projective scheme. These are of course commutative. One has a plentitude of noncommutative examples.
Example 3.2. Let us take k = C and consider the following quotient of the free algebra for q ij ∈ C × . These give noncommutative deformations of P n .
Example 3.3. Building off of Example 3.2, we recall the following class of noncommutative algebras of Kanazawa [Kan15]. Pick φ ∈ C and q ij according to [Kan15, Theorem 2.1] with n i=1 q ij = 1 for all j. And set This is the noncommutative version of the homogeneous coordinate rings of the Hesse (or Dwork) pencil of Calabi-Yau hypersurfaces in P n .
We define left limited grading analogously.
For a connected graded k-algebra, A, one has the two-sided ideal Definition 3.5. 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 such that We let τ denote the functor that takes a module, M, to its torsion submodule. The module M is torsion if τ M = M.
Proposition 3.6. Let A be a connected graded k-algebra. Denote by Tors A, the full subcategory of Gr A consisting of all torsion modules. If A is finitely generated in positive degree, then Tors A is a Serre subcategory.
be a set of generators for A as a k-algebra and let d i = deg(x i ). Consider a short exact sequence It's clear that if M is an object of Tors A, then so are M ′ and M ′′ . Hence it suffices to show that if M ′ and M ′′ are both objects of Tors A, then so is M.
First assume that there exists some N such that for any (X 1 , X 2 , . . . , ) and take any a ∈ A ≥dN . By assumption we It follows that N ≤ s i and hence am = 0. Thus it suffices to find such an N. Fix an element m ∈ M. Since M ′′ is an object of Tors A, there exists some n such that A ≥n p(m) = 0 and hence A ≥n m ∈ M ′ . In particular, if we let T = n i=1 S, then for any element t = (X 1 , X 2 , . . . , X n ) ∈ T we have an element a t = X 1 X 2 · · · X n ∈ A ≥n and so a t m ∈ M ′ . Let n t be such that A ≥nt (a t m) = 0 and take N = 2 max({n t } t∈T ∪ {n}) + 1. If we take any element (X 1 , X 2 , . . . , As such, we can form the quotient.
Definition 3.7. Let A be connected graded and finitely generated as a k-algebra. Then denote the quotient of the category of graded A-modules by torsion as π : Gr A → QGr A denote the quotient functor. By Proposition 3.6 and [Gab62, Cor. 1, III.3], π admits a fully faithful right adjoint which we denote by Finally, we denote the composition Q := ωπ.
The category QGr A is defined to be the quasi-coherent sheaves on the noncommutative projective scheme X.
Remark 3.8. Note that, traditionally speaking, X is not a space, in general. In the case A is commutative and finitely-generated by elements of degree 1, then a famous result of Serre says that X is Proj A.
One can give more explicit descriptions of Q and τ .
Proposition 3.9. Let A be a finitely generated connected graded k-algebra and let M be a graded A-module. Then Proof. This is standard localization theory, see [Ste75].
In studying questions of kernels and bimodules, we will have to move outside 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 Z 2 -gradings of the following form.
Definition 3.10. Let A and B be connected graded k-algebras. The tensor product A ⊗ k B will be equipped with its natural bi-grading Remark 3.11. As noted in the remarks above [VdB97, Lemma 4.1], the notion of A-torsion and B-torsion bi-bi modules is well-defined provided that A and B are finitely generated as k-algebras. From this point on, all of our k-algebras will be assumed to be finitely generated.
There are a couple notions of torsion for a bi-bi module that one can dream up. We use the following.
Definition 3.12. 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 it lies in the smallest Serre subcategory containing A-torsion bi-bi modules and B-torsion bi-bi modules.
In the case that A = B, there is a particular bi-bi module of interest.
Definition 3.13. For A a finitely generated, connected graded k-algebra, we define ∆ A to be the A-A bi-bi module with (∆ A ) i,j = A i+j and the natural left and right A actions. If the context is clear, we will often simply write ∆.
Lemma 3.14. Let A and B be finitely generated, connected graded k-algebras. One can form the quotient category Lemma 3.15. The quotient functor has a fully faithful right adjoint Proof. This is just an application of [Gab62, Cor. 1, III.3].
Corollary 3.16. We have an isomorphism Proof. This follows from Lemma 3.15 using tensor-Hom adjunction.
We also have the following standard triangles of derived functors.
and exactness of Hom(−, I) plus Lemma 3.14 to get exactness.
In general, good behavior of QGr A occurs with some homological assumptions on the ring A. We recall two common such ones. Corollary 3.24. Let A and B be finitely generated, connected graded k-algebras, and let P be a chain complex of bi-bi A ⊗ k B modules. Assume RQ A commutes with coproducts. Then, RQ A P is naturally also a chain complex of bi-bi modules. In particular, if A is Ext-finite, RQ A P has a natural bi-bi structure.
Proof. Note we already have an A-module structure so we only need to provide a Z 2 grading and a B-action. If we write P = v∈Z P * ,v as a direct sum of left graded A-modules, then we set The B module structure is precomposition with the B-action on P . The only non-obvious condition of the bi-bi structure is that which is equivalent to pulling the coproduct outside of RQ A . We can do this for Ext-finite A thanks to Proposition 3.23.
Corollary 3.25. Assume that A and B are left Noetherian, and that Rτ A and Rτ B both commute with coproducts. There exist natural morphisms of bimodules β l P : RQ A P → RQ A⊗ k B P β r P : RQ B P → RQ A⊗ k B P. We first record a more explicit version of [AZ94, Prop 7.1 (5)], which states that every injective object of Gr A is of the form I 1 ⊕ I 2 , with I 1 a torsion-free injective and I 2 an injective torsion module.
Lemma 3.26. Every injective I of Gr A is isomorphic to τ A I ⊕ Q A I.
Proof. By [AZ94, Prop 2.2] any essential extension of a torsion module is torsion, so if τ A I → E is an injective envelope, then we have a monic extension over the inclusion of the torsion submodule 0 τ A I E I ∃ since the injective envelope is an essential monomorphism. By maximality of τ A I amongst torsion subobjects of I, it follows that τ A I = E is injective. Denoting by ε A the unit of the adjunction π A ⊣ ω A : Gr A QGr A the exact sequence splits, as desired.
Proof of 3.25. Thanks to Corollary 3.24, we see that the question is well-posed. We handle the case of β l P and note that case of β r P is the same argument, mutatis mutandis. First we make some observations about objects of Gr (A ⊗ k B). If we regard such an object, E, as an A-module, the A-action is a · e = (a ⊗ 1) · e and we can view τ A E as the elements e of E for which a · e = (a ⊗ 1) · e = 0 whenever a ∈ A ≥m for some m ∈ Z. As such, τ A E inherits a bimodule structure from E and Thanks to Lemma 3.14, we can view τ A⊗ k B E as the elements e of E for which there exists integers m and n such that a ⊗ b · e = 0 for all a ∈ A ≥m and b ∈ B ≥n . From this viewpoint it's clear that We equip C (Gr A) with the injective model structure and use the methods of model categories to compute the derived functors (see [Hov01] for more details). Since we can always replace P by a quasi-isomorphic fibrant object, we can assume that each P n is an injective graded A ⊗ k B-module. Moreover, the fact that the canonical morphisms A → A ⊗ k B is flat implies that the associated adjunction is Quillen, and hence P is fibrant when regarded as an object of C (Gr A). Since Q A preserves injectives, it follows that each Q A P n is an injective object of Gr A. It's clear from the fact that τ A P n is an A ⊗ k B-module that 0 → τ A P n → P n → P n /τ A P n → 0 is an exact sequence of Gr(A⊗ k B) for each n. Moreover, by Lemma 3.26 we have P n /τ A P n ∼ = Q A P n . We thus define (β l P ) n to be the epimorphism induced by the unversal property for cokerenels as in the commutative diagram We observe here that by the Snake Lemma, (β l P ) n is an isomorphism if and only if τ A⊗ k B P n ∼ = τ A P n , which is equivalent by the remarks above to the condition that τ B τ A P n = τ A P n .
To see that β actually defines a morphism of complexes, we have by naturality of ε A , ε A⊗ k B , and the commutative diagram defining (β l P ) n above For naturality, we note that as the fibrant replacement is functorial if we have a morphism of bi-bi modules, then there is an induced morphism of complexes ϕ : P 1 → P 2 between the replacements and for each n a commutative diagram The left square commutes by naturality of ε A and the right square commutes because ) and ε A (P n 1 ) is epic. Proposition 3.27. Assume that A and B are left Noetherian and Ext-finite. Then, we have natural quasi-isomorphisms Consequently, β l P (respectively β r P ) is an isomorphism if and only if RQ A P (respectively RQ B P ) is Q B (respectively Q A ) torsion-free.
Proof. As above, we can replace P with a quasi-isomorphic fibrant object, so it suffices to assume that P is fibrant. We see from Corollary 3.16 that The result now follows from the natural isomorphism (see, e.g., [Hov07, Theorem 1.3.7]) Using the standard homological assumptions above, one has better statements for P = ∆.
Proposition 3.28. Let A be left (respectively, right) Noetherian and assume that the condition χ • (A) holds (respectively, as an A op -module). Then the morphism β l ∆ (respectively, β r ∆ ) of Corollary 3.25 is a quasi-isomorphism.

By Proposition 3.27, RQ
we obtain the triangle and so we are reduced to showing that which then implies that Rτ op A (RQ A ∆) = 0, as desired. First we note that for any bi-bi module, P , the natural morphism

is a quasi-isomorphism if and only if the natural morphism
x, * has right limited grading for each x and j. Now, by [AZ94, Cor. 3.6 (3)], for each j for fixed x and sufficiently large y. This implies that the natural morphism is a quasi-isomorphism, as desired.
Similar hypotheses of Proposition 3.28 will appear often so we attach a name.
Definition 3.29. Let A and B be connected graded k-algebras. If A is Ext-finite, left and right Noetherian, and satisfies χ • (A) and χ • (A op ) then we say that A is delightful. If A and B are both delightful, then we say that A and B form a delightful couple.

Segre Products.
Definition 3.30. Let A and B be connected graded k-algebras. The Segre product of A and B is the graded k-algebra Proposition 3.31. If A and B are connected graded k-algebras that are finitely generated in degree one, then A × k B is finitely generated in degree one.
As a nice corollary, we can relax the conditions on [VR96, Theorem 2.4] to avoid the Noetherian conditions on the Segre and tensor products.
Proof. As noted in Van Rompay's comments preceding the Theorem, the hypothesis is necessary only to ensure that QGr S and QGr T are well-defined. Thanks to Proposition 3.6 and Lemma 3.14, the equivalence follows by running the same argument.

A Comparison with the Commutative Situation.
To provide a touchstone for the reader, we interpret the definitions and results when A and B are commutative and finitelygenerated by elements of weight 1. Then, A = k[x 1 , . . . , x n ]/I A and B = k[y 1 , . . . , y m ]/I B for some homogenous ideals I A , I B . So Spec A is a closed G m -stable subscheme of affine space A n and similarly for Spec B. Let X and Y be the associated projective schemes. Then, The category Gr A ⊗ k B is equivalent to the G 2 m -equivariant quasi-coherent sheaves on Spec A ⊗ k B with Tors A ⊗ k B being those modules supported on the subscheme corresponding to (x 1 , . . . , x n )(y 1 , . . . , y m ). Descent then gives that 3.4. Graded Morita Theory. This section demonstrates how the tools of dg-categories yield a nice perspective on derived graded Morita. Compare with the well-known graded Morita statement in [Zha96].
In order to utilize the machinery of dg-categories, we must first translate chain complexes of graded modules into dg-categories. While one can naïvely regard this category as a dg-category by way of an enriched Hom entirely analogous to the ungraded situation, the relevant statements of [Toë07] are better suited to the perspective of functor categories. As such, we adapt the association of a ringoid with one object to a ring from Section 2.2 to the graded situation, considering instead a ringoid with multiple objects.
Throughout this section, we will let G = (G, +) be an abelian group, and let A and B be not necessarily commutative G-graded algebras over k. We will generally be concerned with the groups Z and Z 2 . In the sequel, there will be many instances where there are two simultaneous gradings on an object: homological degree and homogenous degree. We avoid the latter term, preferring weight, and use degree solely when referring to homological degree.
For clarity, consider the example of a complex of G-graded left A-modules, M. The degree n piece of M is the G-graded left A-module M n . The weight g piece of the graded module M n is the A 0 -module of homogenous elements of (graded) degree g, M n g . Note that in this terminology, the usual morphisms of graded modules are the weight zero morphisms.
Definition 3.33. Denote by C (Gr A) the dg-category with objects chain complexes of Ggraded left A-modules and morphisms defined as follows.
We say that a morphism f : M → N of degree p is a collection of morphisms f n : M n → N n+p of weight zero. We denote by C (Gr A) (M, N) p the collection of all such morphisms, which we equip with the differential N) to be the resulting chain complex. Composition is the usual composition of graded morphisms.
We denote by C (Gr (A op )) the same construction with G-graded right A-modules, which are equivalently left modules over the opposite ring, A op .
Remark 3.34. One should note that the closed morphisms are precisely the morphisms of complexes M → N[p] and, in particular, the closed degree zero morphisms are precisely the usual morphisms of complexes.
Definition 3.35. To each G-graded k-algebra, A, associate the category A with objects the group G and morphisms given by A(g 1 , g 2 ) = A g 2 −g 1 and composition defined by the multiplication A g 2 −g 1 A g 3 −g 2 ⊆ A g 3 −g 1 .
We regard A as a dg-category by considering the k-module of morphisms, A(g 1 , g 2 ), as the complex with A g 2 −g 1 in degree 0 and zero differential.
Lemma 3.36. Let G be an abelian group. If A is a G-graded algebra over k and A the associated dg-category, then there is an isomorphism of dg-categories Proof. We first construct a dg-functor F : C (Gr A) → dgMod (A). For each g ∈ G, denote by A(g)[0] the complex with A(g) in degree zero and consider the full subcategory of C (Gr A) of all such complexes. We see that a morphism indexed by G.
Conversely, we note that the data of a functor M : A op → C (k) is a collection of chain complexes, M g := M(g), indexed by G and morphisms of complexes The non-zero arrow factors through Z 0 (C (k) (M g , M h )), so the structure morphism is equivalent to giving a morphism and thus M determines a complex of graded A-modules A morphism η : M → N is simply a collection of natural transformations η p such that for each g ∈ G we have η p (g) ∈ C (k) (M g , N g ) p and the naturality implies that η p (g) is A-linear. The natural transformation η p thus determines a morphism Remark 3.37. It is worth noting that it is natural from the ringoid perspective to reverse the weighting on the opposite ring in that, formally, With this convention, when considering right modules, one can dispense with the formality of the opposite ring by constructing from a complex, M, the dg-functor A → C (k) mapping g to M g := C (Gr (A op )) (A(−g)[0], M).
When G = Z 2 , and A, B are Z-graded algebras over k, we denote the dg-category of chain complexes of G-graded B-A-bimodules by C (Gr A op ⊗ k B). We associate to the Z 2 -graded k-algebra A op ⊗ k B the tensor product of the associated dg-categories, A op ⊗ B. Note that in the identification the weighting coming from the A-module structure is reversed, as in the remark above. From this construction, we have a dg-enhancement, h-proj (A), of the derived category of graded modules, D(Gr A). Passing through the machinery of Corollary 2.6, we have an isomorphism in Ho (dgcat k )

Derived Morita Theory for Noncommutative Projective Schemes
Let A and B left Noetherian connected graded k-algebras. We want to extend the ideas from Section 3.4 to cover dg-enhancements of D(QGr A).

4.1.
Vanishing of a tensor product. We recall a particularly nice type of property of objects in the setting of compactly generated triangulated categories. In the sequel, many of our properties will be of this type, so we give this little gem a name.
Definition 4.1. Let T be a compactly generated triangulated category. Let P be a property of objects of T . We say that P is RTJ if it satisifies 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 set and A i be objects of T for each i ∈ I. If P holds for each A i , then P holds for i∈I A i .
Proposition 4.2. Let P be an RTJ property that holds for a set of compact generators of T . Then P holds for all objects of 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 coproducts. Thus, P is all of T .  There are various types of projection formulas. We record here two which will be useful in the sequel.
Proposition 4.5. Let A be a finitely generated, connected graded k-algebra. Let P be a complex of bi-bi A-modules and let M be a complex of left graded A-modules. Assume Rτ A commutes with coproducts. There is natural quasi-isomorphism Assume RQ A commutes with coproducts. There is natural quasi-isomorphism Proof. We treat the τ projection formula. The Q projection formula is analogous. By Corollary 3.24, we see that the tensor product is well-defined. It suffices to exhibit a natural transformation for the underived functors applied to modules to generate the desired natural transformation. Given Taking the colimit gives the natural transformation. Let us look at the natural transformation in the case that P = A(u) ⊗ k A(v) and M = A(w). Recall that which are compatible with the natural transformation. The property that the natural transformation is a quasi-isomorphism is RTJ in each entry. Thus, it holds for all P and M by Proposition 4.2.
For the hypothesis, recall Definition 3.29.
Proposition 4.6. Assume A is delightful. Then ⋆ holds for A.
Proof. By Proposition 4.4, it suffices to check ⋆(M, A(v)) for each v. This is equivalent to ⋆(M, v A(v)). Equipping the sum with a bi-bi structure as ∆, we reduce to checking ⋆(M, ∆). Using Proposition 3.28 and Lemma 3.17 for A and A op , we have a natural quasiisomorphism Using Proposition 4.5, we have a natural quasi-isomorphism 4.2. Duality. One can regard the bimodule RQ A⊗ k A op ∆ as a sum of A-modules and define for any object, M, of C (Gr A) the object of C (Gr (A op )). Consider the functor Lemma 4.7. Assume A is delightful. Then the natural map given by evaluation is a quasi-isomorphism for RQ A A(x), for all x. Furthermore, there are quasi-isomorphisms Proof. We first exhibit the latter quasi-isomorphisms. We have Applying this twice, we get We just need to check that the natural map ν : 1 → (−) ∨∨ induces the identity after this quasi-isomorphism. Note that we found a map . One can identify the image of 1 as a map , 0)) which, after applying the quasi-isomorphism, is the natural inclusion. Evaluating this at a ∈ RQ A A(x) gives a back. Thus, we see that ν is quasi-fully faithful on RQ A A(x) for all x.  Proof. From Lemma 4.7, we see that (−) ∨ is quasi-fully faithful on QA and has quasi-essential image Q(A op ).
Lemma 4.10. Assume that A is delightful. There is a natural map Then η is a quasi-isomorphism for any M and any N ∼ = RQ A N.
Proof. First, note that we have the natural map For M = A(x), we see this map is a quasi-isomorphism using the fact that A satisfies ⋆ from Proposition 4.6. Since A satisfies ⋆, the map is also a quasi-isomorphism. Combining the two gives the desired quasi-isomorphism for M = A(x). But the condition η is a quasi-isomorphism is RTJ in M so is true for all M by Proposition 4.2 4.3. Products.
Definition 4.11. For a finitely generated, connected graded k-algebra, A, let h-inj (Gr A) be the full dg-subcategory of C(Gr A) with objects the K-injective complexes of Spaltenstein [Spa88]. Similarly, we let h-inj (QGr A) be the full dg-subcategory of C(QGr A) with objects the K-injective complexes.
Lemma 4.12. The functor is well-defined. Moreover, H 0 (ω) is an equivalence with its essential image.
Proof. For the first statement, we just need to check that ω takes K-injective complexes to K-injective complexes. This is clear from the fact that ω is right adjoint to π, which is exact.
To see this is fully faithful, we recall that πω ∼ = Id so Remark 4.13. Using Lemma 4.12, we can either use h-inj (QGr A) or its image under ω in h-inj (Gr A) as an enhancement of D(QGr A).
Consider the full dg-subcategory of h-inj (QGr A ⊗ k B) consisting of the objects for all u, v. Denote this subcategory by E.
Lemma 4.14. If A and B are both Ext-finite, left Noetherian, and right Noetherian, then the dg-category E is naturally quasi-equivalent to QA ⊗ k QB.
Proof. Recall that QA is the full dg-subcategory of C (Gr A) consisting of Q A applied to injective resolutions of A(u), loosely denoted by RQ A A(u), and similarly for QB. We have the exact functor ⊗ k : C (Gr A) ⊗ k C (Gr B) → C (Gr A ⊗ k B) which tensors a pair of modules over k to yield a bimodule. First consider the triangle By Proposition 3.27, we have Since Rτ B commutes with coproducts, we have a natural quasi-isomorphism ) is quasi-isomorphism for all u, v with τ A⊗ k B torsion cone. The same consideration shows that the map with τ A⊗ k B torsion kernel. Now we check that these morphisms induce quasi-isomorphisms on the morphism spaces giving our desired quasi-equivalence. We have a commutative diagram and we want to know first that a and b are quasi-isomorphisms. We know that b is a quasiisomorphism since Rτ A⊗ k B is left orthogonal to RQ A⊗ k B so we only need to check a. Since A(u) ⊗ k B(v) is free and is a quasi-isomorphism, d is a quasi-isomorphism. Since RQ A and RQ B commute with coproducts, using tensor-Hom adjunction shows that c is a quasi-isomorphism. Finally, since the cone over the map is annihilated by τ A⊗ k B , we see that e is also a quasi-isomorphism. This implies that a is a quasi-isomorphism. By an analogous argument, the endomorphisms of RQ A⊗B (A(u)⊗ k B(v)) and RQ A⊗B (RQ A A(u) ⊗ k RQ B B(v)) are quasi-isomorphic.
4.4. 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. If A and B form a delightful couple, then there is a natural quasi-equivalence F : h-inj (QGr A op ⊗ k B) → RHom c (h-inj (QGr A) , h-inj (QGr B)) such that for an object P of D(QGr A op ⊗ k B), the exact functor H 0 (F (P )) is isomorphic to Proof. Applying Corollary 2.6, it suffices to provide a quasi-equivalence From Lemma 4.14 we have a quasi-fully faithful functor using Propostion 4.6 and Lemma 4.10. This says that the induced continuous functor is The following statement is now a simple application of Theorem 4.15 and results of [LO10]. Then, by Theorem 4.15, there exists a P ∈ D(QGr A op ⊗ k B) such that Φ P = H 0 (F ).
We wish to identify the kernels as objects of the derived category of an honest noncommutative projective scheme. In general, one can only hope that kernels obtained as above are objects of the derived category of a noncommutative (bi)projective scheme. However, we have the following special case in which we can collapse the Z 2 -grading to a Z-grading. Now choose P such that V(P ) is homotopy equivalent to the kernel obtained by an application of Corollary 4.16, so the desired equivalence is Φ V(P ) .
Coming back to Example 3.3. We ask the following question.
Question 4.18. Fix q ij ∈ C. Then two noncommutative projective schemes A φ q and A φ ′ q are isomorphic if and only if they are derived equivalent.
In the commutative case, this is a derived Torelli statement which one can understand via matrix factorizations [Orl09] and the Mather-Yau theorem [MY82].