{'fraction_non_alphanumeric': 0.06008059592135792, 'fraction_numerical': 0.008792282329954817, 'mean_word_length': 3.5929332585530007, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 21, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Equivalences of derived categories of sheaves on quasi-projective schemes

We extend Orlov's result on representability of equivalences to schemes projective over a field. We also investigate the quasi-projective case.


Introduction
Given two projective schemes, X and Y , and an exact functor, F : coh (Y ), one can wonder if, on the product X × Y , there exists a bounded complex coherent sheaves, E, whose associated integral transform is isomorphic to F . We can try to compare D b coh (X × Y ) with the category of exact functors between D b coh (X) and D b coh (Y ) by studying the functor that takes a complex on the product to the associated integral transform. One's first hope might be that this functor is an equivalence. A simple look at the case of the projective line shatters this hope. The space of natural endomorphisms of the identity functor of D b coh (P 1 k ) is infinitedimensional (assuming k is infinite) while D b coh (P 1 k × P 1 k ) has finite-dimensional morphisms. (The author thanks Chris Brav for pointing this out.) Moreover, any morpshim φ : E → F [2] between coherent sheaves E and F on P 1 k × P 1 k induces the zero natural transformation. Thus, the morphism sets of the two categories are very different. However, being stubborn as we are, this does not extinquish the flame of our hope; it only focuses it.
The next best scenario would be that any exact functor, F : D b coh (X) → D b coh (Y ), is isomorphic to integral transform and the kernel of the integral transform is unique up to quasi-isomorphism. This hope is more difficult to stamp out. Indeed, there are no counterexamples and there is some supporting evidence. The evidence comes from Orlov in the form of the following theorem, see [11]: Theorem 1.1. Let X and Y be smooth projective varieties and let F : coh (Y ) be a full and faithful exact functor possessing either a left or right adjoint. Then, there exists a bounded complex of coherent sheaves, E, on X × Y , whose associated integral transform, Φ E , restricted to D b coh (X), is isomorphic to F . In particular, an important case covered by Orlov's result is where F is an equivalence.
This paper is an attempt to relax the conditions of smoothness and projectivity in the hypotheses of Orlov's theorem, and, consequently, provide more evidence in favor of a bijection between exact functor on derived categories and objects on the product up to quasi-isomorphism. It contains a few results on projective schemes that are quite similar to Orlov's result. For instance, we have the following theorem: Theorem 1.2. Let X and Y be projective schemes over a field k. If F : D perf (X) → D perf (Y ) is a full and faithful functor with left and right adjoints, then F is isomorphic to the restriction of Φ E to D perf (X) for an E ∈ D b coh (X × Y ).
And a corollary of it: Corollary 1. 3. Let X and Y be projective schemes over a field k.
is an exact equivalence, then there exists an E ∈ D b coh (X×Y ) and a natural isomorphism F ∼ = Φ E | D b coh (X) . These results are natural extensions Orlov's original result. However, in the intervening decade, little knowledge about the singular case has arisen. Even when other results, such as sufficient conditions for an integral transform to be an equivalence, are extended to more singular schemes, the question of whether derived-equivalent schemes are related by a Fourier-Mukai transform is side-stepped, see [15] and especially section 4 of [14]. The results of this paper allow one to assign a kernel to any equivalence. One can then use geometric reasoning to study the kernel, a process which proves fruitful in the case of smooth and projective varieties.
If one wants to relax the projectivity assumption, we have the following: Theorem 1. 4. Let X and Y be quasi-projective schemes over a field k with either X or Y possessing an ample line bundle, large tensor powers of which have trivial higher cohomology. If D(X) and D(Y ) are equivalent, then there is an object of D(X × Y ) whose associated integral transform is an equivalence.
Note one does not necessarily have isomorphism of the two functors appearing in theorem 1.4. Indeed, the simplest case to check that such an isomorphism exists would be when both X and Y are affine. We jump wholly into the realm of commutative algebra and, surprisingly, we find no answers. The state of the knowledge remains essentially unchanged since Rickard's paper on derived Morita theory, [12].
If one wishes to extend the results in this paper, there are a few obvious cases: stacks and twisted derived categories. However, from work Dugger and Shipley, see [5], if we push too far, equivalences are no longer guaranteed to arise from "bimodules." It would be quite interesting to paint a line separating the Dugger-Shipley realm from the happy land presented here.
If one is willing to enrich the derived category by remembering more structure, i.e. the structure of a differential graded category or a stable ∞-category lying above the triangulated structure, one can show that all functors, preserving this extra structure, are integral transforms, see [4,16,3].
One can combine the results on dg-categories and dg-functors with work of Lunts and Orlov, [8]. Independently, Lunts and Orlov prove a slightly stronger result involving projective scheme; one can remove the assumption of the existence of adjoints from 1.2. They also prove a similar for bounded derived categories of coherent sheaves on projective schemes. These are applications of a central new idea: lifting structure from the triangulated category to a dg-enhancement. Whereas the methods of this paper seem to be difficult to adapt to a general exact functor, one can hope that Lunts and Orlov's results might be more amenable.
Here is a outline of the paper. In section 2, we recall the results of [1] which serve as the main new ingredient. In section 3, we study integral transforms generally. We focus on the interplay between the existence of adjoints and the preservation of certain subcategories of the unbounded derived category of quasi-coherent sheaves. Section 4 recalls Orlov's notion of convolution of a complex over a triangulated category. We mention how one can extend Orlov's ideas to totalize unbounded complexes via homotopy colimits of convolutions. In section 5, we recall another of Orlov's useful definitions: ample sequences in derived categories. We focus on ample sequences consisting of perfect objects. We talk about resolutions of the diagonal in section 6. In the final two sections, we use the ideas and results of the previous sections to prove new results. In section 7, we discuss quasi-projective schemes, in particular proving the theorem 1.4, In section 8, we focus on the projective case and prove theorem 1.2 amongst other results.
This work was a portion of the author's thesis at the University of Washington. The author would like to thank his advisor, Charles Doran, for his patience, energy, and dedication. While preparing this paper, the author was supported by NSF Research Training Group Grant, DMS 0636606.

Preliminaries
Some notional preliminaries: given a category, C, the morphism set from an object, A, to an object, B, is denoted as [A, B]. If a category is endowed with shift functor, the shift is denoted by [1].
Given a scheme X, the category of perfect complexes, D perf (X), is the full subcategory of the unbounded derived category of quasi-coherent sheaves on X, D(X), consisting of complexes locally quasi-isomorphic to bounded complexes of finite rank locally-free sheaves. If X is quasi-compact and separated, the objects of D perf (X) are the compact objects of D(X), meaning the natural map, is an isomorphism for any perfect A and any collection B i , [9]. Any nonzero object of D(X) admits a nonzero morphism from a compact object. Because of this, we call D(X) compactly-generated. Brown's theorem on representability of cohomological functors on the category of spectra can be extended to compactly-generated triangulated categories, see [10]. It provides a useful tool for studying D(X).
If we restrict to the situation where X is quasi-projective over a field k, a complex is perfect if and only if it is globally quasi-isomorphic to a bounded complex of finite rank locally-free sheaves. While the notion of a perfection of an object is manifestly geometric, it is often not as useful as the more natural notion of compactness. Consequently, the identification of D perf (X) as the subcategory of compact objects is helpful. Many other subcategories of D(X) are defined geometrically; one can ask for a more intrinsic characterization of these subcategories. One such category is the bounded derived category of coherent sheaves with proper support, D b coh,c (X). The desired characterization comes from viewing any object, B ∈ D b coh,c (X), as a functor, [−, B] : D perf (X) • → vect k , that takes triangles to long exact sequences and satsifies the following finiteness condition: for any perfect A. Any functor, φ : D perf (X) • → vect k , taking triangles to long exact sequences and satisfying the finiteness condition is called a locally-finite cohomological functor.
Theorem 2.1. For any quasi-projective scheme over a field, the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite cohomological functors. Moreover, the equivalence is the functor given above.
For a proof, see [1]. This result succeeds in providing the requested intrinsic characterization of D b coh,c (X). It is in some sense dual to the category of compact objects. The duality is rather strong. It is easy to check that any object A satisfying coh,c (X) must be perfect. Moreover, an impressive representability result of Rouquier, see [13], immediately implies the following proposition.
Proposition 2.2. Let X be a projective scheme over a perfect field k. The category of locally-finite homological functors on D b coh (X) is equivalent to D perf (X) via the Yoneda embedding.
This duality at the level of categories, including morphisms, allows us access to a wider range of tools than just an observation concerning objects would allow. For instance, given two quasi-projective schemes, X and Y , and a functor, F : coh,c (Y ). The proof of the existence is a simple application of the previous theorem. Uniqueness is clear. We call F ∨ a right pseudo-adjoint to F . We also call F left pseudo-adjoint to F ∨ . For a given functor , the existence of a left pseudo-adjoint, ∨ G, is guaranteed in the case of the proposition 2.2 above. There are other cases where the existence of a left pseudo-adjoint is known. For instance, if G : coh (X) already possesses a left adjoint, ∨ G, ∨ G must take perfect objects to perfect objects. Specializing further, if G is an equivalence, then its inverse is its left pseudo-adjoint.
With this knowledge fresh in our memory, we begin the investigation in earnest by studying integral transforms homologically.

Integral transforms
Let X and Y be quasi-compact, separated schemes and let f : X → Y be a morphism.
Proof. f * (E ⊗ −) is left adjoint to Hom(E, f ! −) so the result follows from the next lemma. Lemma 3.3. Let T be a compactly-generated triangulated category and S a triangulated category. Let F : T → S be a functor which commutes with coproducts and let G : S → T be the right adjoint. G commutes with coproducts if and only if F takes a generating set of compact objects to compact objects.
Proof. Assume that G commutes with coproducts. Let X be a compact object of T . Then, The resulting map agrees with the natural map Recall that a generating set is a set of objects, {X i }, for which [X i , A] = 0 for all i implies that A is isomorphic to the zero object. Assume that we have a set of compact objects, {X i }, which is a generating set. Assume that F (X i ) is compact for all i. Then, for each i, This morphism coincides with applying [X i , −] to the natural map = 0 for all i by the above calculation. Thus, Z is isomorphic to 0 and G commutes with coproducts.
Proof. If E is perfect, then, for any perfect object F , E ⊗ F is perfect. If E is id X -perfect, then Hom(E, −) commutes with coproducts. Taking global sections, we see that [E, −] commutes with coproducts.

Lemma 3.5.
If E is f -perfect, then the natural map, is an isomorphism for any G and H in D(Y ).
Proof. First, let us describe the origin of the aforementioned natural map. By adjunction, it is sufficient to provide a map Using the projection formula, see Proposition 5.3 of [10], Using the counit of adjunction, we get a map The composition of these two maps is our desired natural map, which we call ν G,H .

Since both
Hom(E, f ! −) ⊗ f * − and Hom(E, f ! (− ⊗ −)) commute with coproducts, it suffices to show that ν F,F ′ is an isomorphism for any compact F and F ′ . Moreover, we only need to show that [D, ν F,F ′ ] is an isomorphism for any D in D(Y ). We have a sequence of natural isomorphisms: It is straightforward, but tedious, to check the end result of this sequence coincides with the precomposition by ν F,F ′ . Lemma 3. 6. If E is f -perfect, then we have a natural isomorphism, Proof. From the counit, f * f ! G → G, we get a natural map f * (Hom(E, f ! G)) → Hom(f * E, G). To show that is it an isomorphism of sheaves, we need to know that it induces an isomorphism of the derived global sections over any open affine subset, U , of Y . Let j : U → Y and j ′ : f −1 U → X be the inclusions and Taking adjoints, we get We use the unit, id → j * j * , of adjunction and reduce to checking that the resulting natural map, is an isomorphism. The cone over the unit, id → j * j * , is the functor of local cohomology on Z = Y − U . Let us denote this by Γ Z . It is now sufficient to check Thus, after we apply (j ′ ) * , we get zero.
for any F . Since Hom(−, O X ) is a involution on D perf (X), [F, E] = 0 for any perfect F . Thus, E is zero.
is an isomorphism. Therefore, Hom(−, f ! O Y ) is an involution on the collection of f -perfect objects.
Proof. By lemma 3.8, it suffices to show that f * (ν ⊗ F ) is an isomorphism for any perfect object F . By lemma 3.6, It is straightforward to check that the resulting map coincides with the double Proof. We compute: Corollary 3.11. Let X and Y be defined over a field k. If E is f -perfect, E ⊗ f * − takes locally-finite objects to locally-finite objects. In particular, if X and Y are quasi-projective over k, The right adjoint must therefore take locally-finite objects to locally-finite objects. If X and Y are quasiprojective over k, then locally-finite objects are exactly the objects of D b coh,c (X). The following is a slight extension of proposition 1.6 from [15]. Proposition 3.12. Let f : X → Y be a morphism of quasi-projective schemes over k and E ∈ D(X). The following are equivalent: • coh (X) and f is proper, then the above are equivalent to: Proof. Condition one implies condition two because Hom(E, f ! −) is right adjoint to a functor that takes perfect objects to perfect objects. Similarly, condition two implies condition one since f * (E⊗−) is left adjoint to a functor that sends coh,c (X). Corollary 3.11 says that condition one implies condition three. If condition three holds for E, it also holds for E ⊗ F for any perfect F . Using the projection formula, . The proof of this proposition is finished by the following lemma.
Replacing E p by its minimal free resolution, we see that E p is quasi-isomorphic to bounded complex of finite rank free modules. This must be true on a neighborhood of p. So E is locally quasi-isomorphic to a bounded complex of locally-free sheaves.
We can also test for perfection by using Hom instead of ⊗.
. We can apply the argument from the previous lemma except we consider Hom(E p , k(p)). If E p is replaced by its minimal free resolution, then Hom(E p , k(p)) is bounded if and only if E p quasi-isomorphic to a bounded complex of free modules. E must then be perfect.
coh (X). Now we can begin applying the above homological algebra to the case of integral transforms. Let X and Y be be quasi-projective schemes. We have the projections From an object, E ∈ D(X × Y ), we can create a functor We shall often drop the superscript if the context is clear.
We can immediately apply the results about relative perfection to the case of an integral transform between quasi-projective schemes over a field.
Using the projection formula, we get . First, we prove a lemma about boundness. Lemma 3.19. Let X be a quasi-projective scheme over k and G ∈ D(X). There exists an Using the Beilinson resolution of the diagonal on N -dimensional projective space and pulling back to X, we obtain a quasi-isomorphism and we can apply proposition 3.12 to conclude that E is Proof. This is a simple application of standard adjunctions: From lemma 3.5 and proposition 3.12, we get the following statement.
coh,c (X). The following tells us when an integral transform cannot have a left adjoint.
, then it does not possess a left adjoint.
Proof. Assume such a left adjoint exists. It would have to take perfect objects to perfect objects since Φ E commutes with coproducts. By duality, Φ E would then Lemma 3.23. Assume that X and Y are projective and E lies in Proof. From lemma 3.18, we know that E is p 1 -perfect. Take F from D(Y ) and G from D(X). We apply lemma 3.10 in the following sequence of natural isomorphisms:

Totalizing complexes in triangulated categories
The majority of the ideas and results of this section are due to Orlov [11], see also [7]. Let T be a triangulated category. commutative.
where faces denoted by are commutative and faces denoted by △ are exact triangles in T . A left convolution of a complex (A, d) over T is an object, Y ∈ T , and a map, l : Y → A s , such that there exists a diagram where faces denoted by are commutative and faces denoted by △ are exact triangles in T .
] is zero for all l < m and j < 0. Then, there exists a right convolution X of (A, d).
Proof. We proceed by induction on the length of the complex. Assume that the lemma is true for any complex that satisfies the hypotheses of the lemma and has length less than −s. Form the triangle and consider the long exact sequence commute. Now consider the long exact sequence is a complex over T satisfying the hypotheses of the lemma and having length less than −s. It possesses a right convolution, which is also a right convolution of the original complex. ] is zero for l < m and j < 0. Then, for any right convolutions (X, r) of (A, d) and (X ′ , r ′ ) of (A ′ , d ′ ) and any map h : X ′ → Z ′ there exists a (non-canonical) morphism g : X → Z ′ making the diagram ] is zero for all l, j < 0, then this morphism is unique.
Proof. We again proceed by induction. Assume the result is true for morphisms satisfying the hypotheses of the lemma and having length less than −s. Form the triangles and let e : Y → A s+2 and e ′ : Y ′ → A ′ s+2 be the maps constructed as in the previous lemma. There is a y : with all but the right parallelogram commutative. Consider the long exact sequence and note that e ′ • y and f s+2 • e both map to d ′ s+1 • f s+1 = f s+2 • d s+1 and that [A s [1], A ′ s+2 ] is zero by assumption. Thus, e ′ • y = f s+2 • e and we reduce to a map of complexes satisfying the hypotheses of the lemma and of length less than −s.
Let (X, r) and (X ′ , r ′ ) be convolutions of (A, d) and (A ′ , d ′ ), respectively. And, let h : X ′ → Z ′ be a morphism. We use the notation from the definition of a convolution. Let C be the cone over the map, h • r ′ : X ′ → Z ′ . We have a morphism, B −1 → C[−1], which makes the diagram is zero for all l < m and j < 0. All right convolutions are (non-canonically) isomorphic. If we assume, in addition, that [A t , X[j]] is zero for all l, j ≤ 0, and some right convolution X, then all right convolutions of (A, d) are canonically isomorphic.
Proof. Apply lemma 4.4 to the identity map between two complexes and note that, if all f i are isomorphisms, the resulting morphism between the convolution is an isomorphism.
We also have duals of these results.
] is zero for all l, j < 0, then this morphism is unique.
Example 4.8. If T = D(X) and each A i is a quasi-coherent sheaf placed in degree zero, then we can convolve and the convolution is simply the complex itself as an object of D(X).
We can also use these results to totalize unbounded complexes over T . This is done by taking a homotopy colimit of the convolutions of the brutal truncations. Assume we are given a bounded above complex over a triangulated category T possessing small coproducts. The brutal sth truncation of (A, d) is the complex ] is zero for l < m and j < 0. Then, we can convolve (σ ≥s A, d). Denote the convolution by X s . Using the obvious morphism σ ≥s A → σ ≥s−1 A, we get a morphism X s → X s−1 . Then, we set Tot(A, d) = hocolim X s . Tot(A, d) is determined up to (a non-canonical) isomorphism by (A, d).
be a complex over D(X) with each A i a quasi-coherent sheaf in degree zero. Then, from the construction of the convolution, the convolution of σ ≥s A is just the complex itself as an object in D(X). We shall make no distinction between the two in notation. Let A denote the complex as an object of D(X). Note that A is the colimit of the σ ≥s A in the category of chain complexes. Therefore, there is a short exact sequence of chain complexes. This induces an exact triangle in D(X) and the map σ ≥s A → σ ≥s A is the same as in the definition of the homotopy colimit. Thus, Tot(A, d) ∼ = A.
If Tot(A, d) has bounded and coherent cohomology, then there are no phantom maps Tot(A, d) → Tot(A, d). Thus, the homotopy colimit is unique up to a unique isomorphism, given the uniqueness of the convolutions of the σ ≥s A.

Ample sequences
Let A be a k-linear abelian category. The following definition is due to Orlov.
Definition 5.1. Let {L i } i∈Z be a sequence of objects in A. We say that {L i } is an ample sequence if, for any object A ∈ A, there exists an N ∈ Z so that for i < N the following conditions hold: • The canonical map Lemma 5.2. Let X be a quasi-projective scheme over k. Assume that, on X, we have an ample line bundle, L, with L ⊗l = O X for all l = 0 and H k (X, L ⊗l ) is zero for k > 0 and l ≫ 0. Then, {L ⊗i } i∈Z form an ample sequence for Coh(X).
Proof. The first condition is a classical result of Serre. The second condition can be restated as the vanishing of [L ⊗i , A[j]] = H j (X, A ⊗ L ⊗−i ) for i ≪ 0. This follows from a standard argument as in [6].
If H 0 (X, L ⊗i−j ) is nonzero for i − j < 0, then L ⊗j−i is trivial as it is effective and its inverse has a section. Thus, [L ⊗j , L ⊗i ] is zero for j < i. We can find a surjective map [L ⊗j , A]⊗L ⊗j → A and apply [−, L ⊗i ] to get an injection [A, Proof. Since the set of all locally-free coherent sheaves generates D perf (X) as a triangulated category, it is sufficient to show we can get any locally-free coherent sheaf from the {P i } by finite iteration of the operations of forming cones, forming finite direct sums, and forming direct summands. Let V be any locally-free coherent sheaf. Using the first property of an ample sequence, we see that can find a resolution P ⊕mi If X is of dimension d, then the only map between ker(P ⊕mi ni → P ⊕mi−1 ni−1 )[n + 1] and V if n ≥ d is the zero map. Thus, V is a direct summand of Lemma 5.7. If Q is a perfect object such that Hom(Q, P i [j]) is zero for all i ≪ 0 and all j. Then, Q is isomorphic to zero.
Consequently, Q is also isomorphic to zero.
are isomorphisms for all i, j, k. Then F is full and faithful.
Proof. Let ∨ F denote the left adjoint and F ∨ the right adjoint. Take the unit of adjunction applied to S i , f i : S i → F ∨ F S i , and form a triangle

R R R R R R R
For all j and k, Hom(S j , C i [k]) = 0. From the definition of a spanning class, we conclude that C i ∼ = 0. Consider the counit of adjunction, g Q : ∨ F F Q → Q, and form a triangle The isomorphisms imply that Hom(C Q , S i [k]) = 0 for all i and all k. We have C Q ∼ = 0. g Q is an isomorphism and F is full and faithful. We have the subsequent relevant corollary.
Corollary 5.11. Let X be a quasi-projective scheme over k with a perfect ample sequence {P i } and F : D perf (X) → D perf (Y ) an exact functor possessing left and right adjoints. If the maps are isomorphisms for i < j and for all k. Then F is full and faithful on D perf (X).
The following is a very useful result due to Orlov, see [11]. We shall not recall the proof as we require no modification.

Resolutions of the diagonal
Let X and Y be projective schemes over a field k. Let O X (1) and O Y (1) denote choices of very ample sheaves on X and Y , respectively. Then, O X (1) ⊠ O Y (1) is a very ample sheaf for X × Y since we can use O X (1) and O Y (1) to embed X × Y into P N k × P M k and then apply the Segre embedding, Lemma 6.1. Given any coherent sheaf C on X × Y , there exists a j and m and a surjection and immediately deduce the following lemma. Lemma 6.2. Any coherent sheaf C on X × Y has a bounded above resolution where E i are invertible sheaves on X and F i are locally-free coherent sheaves on Y . Corollary 6.3. On X × X, there is a resolution of the structure sheaf of the diagonal ∆X where A i and B i are locally-free coherent sheaves. Corollary 6.4. Assume that X is quasi-projective. Then, there is a resolution of the structure sheaf of the diagonal ∆X where A i and B i are locally-free coherent sheaves.
Proof. LetX be a choice of projective closure of X. By corollary 6.3, we have a resolution of the structure sheaf of the diagonal ∆X. A i and B i are locally-free coherent sheaves onX. Since X is an open subset ofX, the restriction to X is exact. So We now record and prove a useful lemma found in [7].
Lemma 6.5. Let f : X → Y be a quasi-projective morphism with L the corresponding ample sheaf. Assume that large tensor powers of L have trivial higher cohomology. If D is a bounded above complex of coherent sheaves on X with H m0 (D) nonzero, then there exists an integer N so that, for all k > N , H m0 (f * (D ⊗ L k )) is nonzero.
Proof. Take N large enough so that, for k > N , H p (f * (H q (D) ⊗ L k )) = 0 for p > 0 and q > m 0 − dim X and H 0 (f * (H m0 (D) ⊗ L k )) = 0. We have a spectral sequence ) and we see that, thanks to our choices, H m0 (f * (D ⊗ L k )) = 0.
There is one concrete and important resolution of the diagonal that we should discuss further: the Beilinson resolution of the diagonal for P n k , [2]. For the polynomial algebra, S = k[x 0 , · · · , x n ] we have a map of graded modules S(−1) ⊕n+1 → S given by sending a basis vector e i to x i . Let M be the kernel of this map. The corresponding maps of coherent sheaves on P n k , is exact. By localising to the standard open subsets, one can check thatM ∼ = Ω P n k /k . (On the affine subset where x i is nonzero, d(x j /x i ) gets mapped to 1/x i (e j − x j /x i e i ), and these maps glue). If we take duals and twist by −1, we get 1)). This shows that the vector fields ∂ ∂x i are a basis of the global sections of T P n k . Define a global section, are coordinates on the first factor and y i are coordinates on the second. One can then check, by localizing to the affine charts, that the divisor corresponding to s is exactly the diagonal. Taking the Koszul resolution associated to the section we get the resolution

A derived Morita theorem for some quasi-projective schemes
In this section, we assume that X and Y are quasi-projective over a field k. We also assume that X possesses a line bundle L which is ample and satisfies the following condition: there exists an N so that, for l > N , H i (X, L l ) = 0 for i > 0. We denote L by O X (1). Any scheme that is projective over a finitely generated k-algebra satisfies this condition. Any scheme that is affine over a projective scheme satisfies this condition.
Example 7.1. One might speculate that we just need to pullback the twisting sheaf from P n k , but this does not always work. Consider A 2 k − {0, 0}. Its structure sheaf is very ample, but it has higher cohomology. Since its Picard group is Z, we see that, on A 2 k − {0, 0}, there is no ample sheaf whose large tensor powers possess trivial higher cohomology.
Also, in this section, F : D perf (X) → D perf (Y ) is a full and faithful exact functor. If X is not projective, we assume that F admits an extension, F : D(X) → D(Y ), which commutes with coproducts.
Lemma 7.2. If F has a left adjoint, then there exists m 1 and m 2 so that F (E) is concentrated in [m 1 , m 2 ] for any finite rank locally-free sheaf, E.
Proof. Let us denote the left adjoint by F * . Choose some embedding Y ֒→ P n k . On P n k × P n k , we can resolve the pushforward of Applying p 1 * , we see that O P n k (j) can be obtained from {O P n k (−n), · · · , O P n k } by taking shifts [−j] with 0 ≤ j ≤ n, direct sums, and then n cones of objects within the subcategory formed by those direct sums of shifts of {O P n k (−n), . . . , O P n k }. This statement pulls back to Y .
We choose k 1 and k 2 so that Hom(F * (O Y (j)), O X ) is quasi-isomorphic to a complex of locally-free coherent sheaves which is zero outside [k 1 , k 2 ] for all 0 ≤ j ≤ n. Take any locally-free coherent sheaf, E, on Y . Via adjunction,  coh (X)) concentrated in degree zero are uniformly bounded, we say that F is bounded.
Remark 7.4. Note that, if X is projective, F has a left adjoint if and only if F extends to a functorF : , then this argument actually shows thatF is bounded. We shall shift F , if necessary, and assume that the cohomology of F (A), for any locally-free coherent sheaf, A, is concentrated in [k 0 , 0]. Consider the following complex over D(X × Y ): Here A i , B i come from our choice of the resolution of the diagonal on X × X.
Lemma 7.5. For any objects C 1 , C 2 in D perf (X) and D 1 , D 2 in D perf (Y ), Proof. We manipulate some adjunctions: Remark 7.6. As long as there are no morphisms between D 1 and D 2 [l] for |l| large, we do not need perfection.
For r > 0 and p < q, we have Thus a right convolution of this complex exists. Denote it by E ′ m . Lemma 7.7. Assume that F (G) is concentrated in [k 0 , 0] for any locally-free co- From our assumption on X, for any fixed bounded complex D of coherent sheaves, there is an N so that tensoring D by p * 1 O X (l), for k ≥ N , and pushing forward by p 2 yields an exact sequence. Thus, for l large, we have an exact sequence which must be zero if we choose m > dim X. Thus, the complex, viewed as lying over D(X), is convolvable. The convolution is just the complex itself as an object of D(X). Hence, the convolution is quasi-isomorphic to S l,m−1 [m] ⊕ O X (l). Apply F to and consider it as a complex over D(Y ). It is convolvable as F is fully-faithful. Since exact functors map convolutions to convolutions and convolutions are unique by corollary 4.5, the convolution of ). Now, note that the complex also results from applying for l large(r than a fixed constant depending on dim X and A i , B i for 0 ≤ i ≤ m). Thus, the cohomology of Φ E ′ m (O X (l)) is concentrated in [k 0 +m, m]∪[k 0 , 0]. Lemma 6.5 says that E ′ m has cohomology concentrated in [k 0 + m, m] ∪ [k 0 , 0]. We record a corollary of the proof.
Corollary 7.8. There is an M (depending on m) so that, for any l > M , Denote the truncation by E. Lemma 7.9. There is an N so that, for l > N , there are isomorphisms ) has no higher cohomology on any element of E ′ m for l > N . Then, ). Corollary 7.10. There exists an N so that, for l > N , Lemma 7.11. There exists a natural isomorphism of Φ E and F on the full subcategory of D perf (X) whose objects are {O X (l)} l>N Proof. We have just seen that the for each l > N we have the isomorphisms of objects. The need is to make them functorial. Consider projection ǫ l : F (S l,m−1 [m] ⊕ O X (l)) → F (O X (l)). Since Hom(H 0 (X, A p (l))⊗F (B p )[r], F (O X (l))) ∼ = Hom(H 0 (X, A p (l))⊗B p [r], O X (l)) ∼ = 0 for r > 0 and any p (both are sheaves), ǫ l is the only morphism which makes Here r is the map coming from the convolution of commute. Here r ′ is the map coming from the convolution of From above, we have isomorphisms f l : commute. It follows from the previous observations that any map fitting in the slot of f l must be unique. If there were two such maps f 1 and f 2 , then (f 1 − f 2 )ǫ l = 0 by uniqueness. But, ǫ l is just a projection and we can precompose with the splitting be a morphism for l 1 , l 2 > N . Then there is uniquely determined g so that We can now state a derived Morita theorem (in the sense of Rickard) for certain quasi-projective schemes. For the original statement, see [12].
Theorem 7.13. Let X and Y be quasi-projective schemes over a field k. Assume that X possesses an ample line bundle, sufficiently high powers of which have trivial higher cohomology. The following are equivalent: Moreover, if X is projective, then the following are also equivalent: In the quasi-projective case, the work of this section showed the equivalence of the first and third conditions. Tracing out the arguments, we see that we can restrict ourselves to bounded complexes in the arguments and still conclude that the third condition holds.
In the projective case, we can restrict our attention to perfect objects to conclude that the old third condition holds. The fourth and fifth conditions are equivalent by locally-finite duality.
Orlov's original theorem says something a bit stronger. It says that F and Φ E are isomorphic on all of D b coh (X) if X and Y are smooth and projective. In the next section, we push a little harder and provide an extension of Orlov's result.

Equivalences and Fourier-Mukai transforms
As in last section, F is a full and faithful exact functor D perf (X) → D perf (Y ). However, in this section, we take X and Y to be projective schemes over a field.
We are almost ready to deduce an extension of Orlov's result. We first need to know that Φ E takes bounded complexes of coherent sheaves to bounded complexes of coherent sheaves. Proof. Instead of fixing m, we shall now let m run starting from m < k 0 − 2. From lemma 4.4, there is a morphism φ m : ). Applying lemma 4.4 again, we see that this corresponds to a morphism F (S l,m−1 [−m] ⊕ O X (l)) → F (S l,m−2 [−m + 1] ⊕ O X (l)). It is easy to check that the morphism, F (S l,m−1 [−m] ⊕ O X (l)) → F (S l,m−2 [−m + 1] ⊕ O X (l)), comes from applying F to the morphism λ m ⊕ id OX (l) : Since this is true for all l ≫ 0, we see that Φ φm (O X (l)) is a quasi-isomorphism in degrees > m + 1. We truncate E ′ m above k 0 − 1. Denote the result by E m . We have a diagram which can be completed to a morphism of triangles The map fitting into the slot of Φ ψm (O X (l)) is unique, and the composition of the inclusion of F (O X (l)), Φ φm (O X (l)), and projection onto F (O X (l)) also fits there. Thus, Φ ψm (O X (l)) is a quasi-isomorphism for l ≫ 0. By lemma 6.5, ψ m is an quasi-isomorphism for each m.
and that F is bounded (with [k 0 , 0] being the region with nonvanishing cohomology).
Proof. Consider the complex Since A i , F (B i ) are perfect, this complex can be convolved, and the convolution is quasi-isomorphic to E ′ m ⊗ p * 1 C. Let F m denote the convolution of the complex Then, F m ⊗ p * 1 C is the convolution of the complex Choose N large enough so that A i ⊗ C(l) has no higher cohomology for l > N and i ≥ m and p 2 * R l,m−1 = p 2 * (p * 1 O X (l) ⊗ R m−1 ) is concentrated in degree zero. We have an triangle The convolution of the complex is quasi-isomorphic to Φ E ′ m (C(l)) by uniqueness of convolutions. Applying F to the previous triangle then gives the triangle Applying Hom(O X , −) gives us a long exact sequence of cohomology sheaves. Since , 0] and the claim follows.
to which we apply Φ − (C): is independent of the choice of m for m large. By considering the long exact sequence of cohomology sheaves as m grows large, we see Φ E (C) must be concentrated in [k 0 , dim X]. Consequently, Φ E must take an bounded complex of coherent sheaves to another bounded complex of coherent sheaves.
We next extend the natural isomorphism found in lemma 7.11.
Lemma 8.4. There exists a natural isomorphism between Φ E | Ω and F | Ω where Ω is the full subcategory formed by O X (l) for l ∈ Z.
Proof. We proceed by downward induction. Choose an embedding of X in P n k . Then, we have a exact sequence ). This comes from the Beilinson resolution of the diagonal. Twisting we have an exact sequence Note that O X (k) is the left convolution of the complex V n ⊗ O X (k + 1) → · · · → V 0 ⊗ O X (k + n + 1) We already know that we have a natural isomorphism Φ E → F on the subcategory formed by O X (l) for l > N . If we set k = N above and use the natural transformation, we get a morphism of complexes (N + n + 1) IdV n−1 ⊗fN+2 IdV 0 ⊗fN+n+1 which gives a unique morphism f N : Φ E (O X (N )) → F (O X (N )) by lemma 4.7.
Working downward, we get all f l for l ∈ Z. The morphism of the complexes is an isomorphism as each f l is an isomorphism.
To check that these are natural, we take a morphism α : O X (l) → O X (l ′ ). It induces a morphism of complexes which corresponds to a unique morphism Φ E (O X (l)) → F (O X (l ′ )). Both F (α)f l and f l ′ Φ E (α) fit into the diagram given in lemma 4.7. Thus, they are equal and the isomorphism between Φ E and F on the subcategory consisting of {O X (l)} l∈Z is natural. We now need a useful lemma: Assume G preserves perfection, and there is an isomorphism of functors f Ω : F | Ω → G| Ω . Then there exists an isomorphism of functors f : F | D perf (X) → G| D perf (Y ) which extends f Ω . Moreover, if F is an equivalence, then F and G are naturally isomorphic.
Proof. Since F posseses a left adjoint, the pseudo-right adjoint of F is an extension of F to the bounded derived categories. We will denote this extension by F also. Recall that a perfect ample sequence must split generate D perf (X). Thus, G must take perfect objects to perfect objects. The restriction of G to D perf (X) has a right pseudo-adjoint which we will simply denote G ∨ .
We have a natural morphism of functors, Id D perf (X) → F ∨ F , which is an isomorphism as F is full and faithful. Since G is isomorphic to F when restricted to Ω, it also full and faithful on D perf (X) and the natural morphism of functors, ∨ GG| D perf (X) → Id D perf (X) , is an isomorphism.
∨ GF is the left adjoint to F ∨ G| D perf (X) on D perf (X). They are both isomorphic to the identity when restricted to Ω so they are both fully faithful. Since they are adjoints to each other, they are quasi-inverses. F ∨ G is right pseudo-adjoint (on D b coh (X)) to ∨ GF . Thus, F ∨ G is an autoequivalence of D b coh (X). By proposition 5.12, there is an natural isomorphism Id D b coh (X) → F ∨ G. Restricting to D perf (X) and using adjunction, we obtain a morphism of functors f : F | D perf (X) → G| D perf (X) .
Take any object a of D perf (X). Let Assume that F is an auto-equivalence. Note that by uniqueness ∨ F ∼ = F −1 and ∨ G ∼ = G| −1 D perf (X) . We have an natural isomorphism g : which corresponds to f : F | D perf (X) → G| D perf (Y ) . Taking the right dual of this natural transformation we get a natural isomorphism g ∨ : ( ∨ F ) ∨ → ( ∨ G) ∨ which gives the isomorphism between F and G.
If X and Y are projective, we have seen that Φ E must have left and right pseudoadjoints because it preserves perfection and bounded coherence.
Theorem 8.6. Let X and Y be projective schemes over a field k. If F : D perf (X) → D perf (Y ) is a full and faithful functor with a left and a right adjoint, then F is isomorphic to the restriction of Φ E to D perf (X) for E ∈ D b coh (X × Y ). A particular case of this is: Corollary 8.7. Let X and Y be projective schemes over a field k. Let F : D perf (X) → D perf (Y ) be an exact equivalence. There exists an E ∈ D b coh (X × Y ) and a natural isomorphism F ∼ = Φ E | D perf (X) .
We can give another corollary that is a consequence of locally-finite duality.
Corollary 8.8. Let X and Y be projective schemes over a field k. If F : D b coh (X) → D b coh (Y ) is an exact equivalence, then there exists an E ∈ D b coh (X×Y ) and a natural isomorphism F ∼ = Φ E | D b coh (X) . Remark 8.9. The complex, E, appearing in the previous results is unique.
Proof. Note that the complex, E, given in these results is unique. Suppose we have two complexes, E andẼ, and natural isomorphisms between F and both Φ E and ΦẼ. The complexes and A m ⊠ ΦẼ(B m ) → · · · → A 0 ⊠ ΦẼ(B 0 ) are isomorphic. We can take A 0 and B 0 to be O X . Using the counit p * 2 p 2 * → Id, we get a morphism A 0 ⊠ ΦẼ(B 0 ) →Ẽ. Using the isomorphism of Φ E and ΦẼ, we can view this as a map from the complex A i ⊠Φ E (B i ) toẼ. Consequently, we get a map between their convolutions E ′ m →Ẽ. The induced map Φ Em (O X (l)) → ΦẼ(O X (l)) is a quasi-isomorphism in degrees > m + dim X + dim Y . Consequently, the map