Sheaves of categories and the notion of 1-affineness

Some remarks on the derived categories of coherent sheaves on homogeneous spaces

In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.

On localizing subcategories of derived categories of categories of quasi-coherent sheaves on Noetherian algebraic spaces

Let $S$ be a scheme contained in $\mathbf{Sch}_{fppf}$. Let $\mathfrak{X}$ be a Noetherian separated algebraic space over $S$. In this paper, we interpret localizing subcategories of the derived category of $\mathfrak{X}$ by using subsets of $|\mathfrak{X}|$.

Derived Categories of Twisted Sheaves on Elliptic Threefolds

We construct an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves (modules over an Azumaya algebra) on any small resolution of its relative Jacobian.

Introduction to derived categories of coherent sheaves

In these notes, an introduction to derived categories and derived functors is given. The main focus is the bounded derived category of coherent sheaves on a smooth projective variety.

Derived categories of coherent sheaves

We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi-Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems.

An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.

A reconstruction theorem for abelian categories of twisted sheaves

We use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of alpha-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes X when alpha is in the Brauer group of X. By applying the work of Toën on derived Azumaya algebras, we give a proof of Căldăraru's conjecture.

Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces

We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.

超Kähler流形：拉格朗日纤维化、原子sheaves与范畴

Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories

本文综述了紧致超Kähler流形理论的最新发展，从拉格朗日纤维化、稳定sheaves模空间和导出范畴的角度探讨核心方法与贡献。

Nori动机sheaves的类别比较

A Comparison of Categories of Nori Motivic Sheaves

本文证明了两种Nori动机sheaves理论的自然等价性，并提供了与六操作兼容的实现实用函子。

Semiorthogonal decompositions of derived categories of equivariant coherent sheaves

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of bounded derived category of G-equivariant coherent sheaves on X into components, equivalent to derived categories of twisted representations of the group. If the group is finite or reductive over the algebraically closed field of zero characteristic, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular …

半齐次层与扭曲导出范畴

Semi-Homogeneous Sheaves and Twisted Derived Categories

本文构造了阿贝尔簇挠子与其简单半齐次层模空间间的扭曲导出等价，证明了其逆命题，并推广了Orlov与Polishchuk的导出等价准则。

tame stack上层的导出范畴的等价

Equivalences of derived categories of sheaves on tame stacks

本文将Orlov表示定理推广至光滑、proper且tame的代数stack，证明了完全忠实函子与Fourier--Mukai变换的关联，扩展了Kawamata关于Deligne--Mumford stack的先前成果。

The Pure derived category of quasi-coherent sheaves

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X.

Derived categories of sheaves on singular schemes with an application to reconstruction

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.

A Parameterization of D equivalences of coherent sheaves of symplectic resolutions of a given symplectic singularity

Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical equivalence. By refining a construction from a previous paper, we construct a local system of categories over a topological space V^0_C, where these categories are assigned to different points in V^0_C. Natural equivalence functors between these categories are parameterized by homotopy classes of paths between the corresponding points.

Tannakian Categories attached to abelian Varieties

Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G_2, F_4, as well as the groups G(m,1,n), we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces V^g/C(g), where C(g) is the centralizer subgroup of g in G. In the case of the Weyl groups the construction uses some key results about the…

Hyperplane sections and derived categories

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

Derived categories of coherent sheaves on rational homogeneous manifolds

Starting point of the present work is a conjecture of F. Catanese which says that in the derived category of coherent sheaves on any rational homogeneous manifold G/P there should exist a complete strong exceptional poset and a bijection of the elements of the poset with the Schubert varieties in G/P such that the partial order on the poset is the order induced by the Bruhat-Chevalley order. The goal of this work is to provide further evidence for Catanese's conjecture, clarify some aspects of it and supply new techniques. In particular we prove a theorem on the derived categories of quadric b…

Moduli Problems in Abelian Categories and the Reconstruction Theorem

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statement…

Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions

Let k be an algebraically closed field of characteristic p>>0. Let $X\rightarrow Y$ be a symplectic resolution. There are two questions which motivates this work. One question is a construction of an action of a group on the category $\mathcal{C}:=D^b(Coh(X))$ - The bounded derived category of coherent sheaves of the symplectic resolution X. Second question is understanding equivalence functors between derived categories of coherent sheaves for different symplectic resolutions of Y. Let G/k be a reductive group. In this paper, we construct a local system on a topological space called $V^0_{\ma…

Derived categories of coherent sheaves on some zero-dimensional schemes

Let $X_N$ be the second infinitesimal neighborhood of a closed point in $N$-dimensional affine space. In this note we study $D^b(coh\, X_N)$, the bounded derived category of coherent sheaves on $X_N$. We show that for $N\geq 2$ the lattice of triangulated subcategories in $D^b(coh\, X_N)$ has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. We also establish a relation between triangulated subcategories in $D^b(coh\, X_N)$ and universal localizations of a free graded associative algebra in $N$ variables. Our homological methods prod…

Constant families of t-structures on derived categories of coherent sheaves

We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of \cite{AJS}. As an application of the "constant" t-structures…

Kleinian singularities, derived categories and Hall algebras

We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of the category of coherent sheaves on an algebraic surface.

Euivalences of derived catgories of sheaves on smooth stacks

We extend Orlov's representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities.

Six operations on dg enhancements of derived categories of sheaves

We lift Grothendieck-Verdier-Spaltenstein's six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Our main tools come from enriched model category theory.

Crossingless sheaves and their classes in equivariant K-theory

We introduce crossingless sheaves in certain equivariant derived categories which are analogous to the Bezrukavnikov-Mirkovic exotic sheaves for two-block nilpotents. We calculate the classes of crossingless sheaves in equivariant K-theory of Cautis-Kamnitzer varieties.

Twisted Fourier-Mukai functors

Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor. As an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties.

External tensor product of categories of perverse sheaves

Under some assumptions we prove that the Deligne tensor product of categories of constructible perverse sheaves on pseudomanifolds X and Y is the category of constructible perverse sheaves on X\times Y. The Deligne external tensor product functor is identified with the geometrical external tensor product.

Explicit models for perverse sheaves

We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing $t$-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular, we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict ourselves to the two-strata case, but our results extend to the general case.

Knot homology via derived categories of coherent sheaves I, sl(2) case

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology.

Knot homology via derived categories of coherent sheaves II, sl(m) case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.

Derived categories of coherent sheaves

We show how derived categories build bridges across the current mathematical mainstream, linking geometric and algebraic, commutative and noncommutative, local and global banks. Arches in these bridges are pieces of semiorthogonal decompositions of triangulated categories. To appear in the Proceedings of the ICM 2002 in Beijing.

Quotient categories, stability conditions, and birational geometry

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.

Ind-abelian categories and quasi-coherent sheaves

We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

Points in algebraic geometry

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, étale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show direct image along a closed immersion is exact for all these topologies except qfh. The methods …

二次束的导出范畴和旋量sheaf的模堆

Derived categories of quadric bundles and moduli stacks of spinor sheaves

本文通过旋量sheaf的模堆，证明了二次束族的Kuznetsov组件与扭曲导出范畴的等价性，并提供了Brauer类的几何解释。

Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves

The purpose of this paper is to develop an efficient computational model for Abelian categories of coherent sheaves over certain classes of varieties. These categories are naturally described as Serre quotient categories. Hence, our approach relies on describing general Serre quotient categories in a constructive way which leads to an efficient computer implementation.

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a fam…

Derived categories of Thaddeus pair moduli spaces via d-critical flips

We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence of fully-faithful functors between derived categories of coherent sheaves on these moduli spaces. Our result gives an evidence of a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also a categorification of wall-crossing formula of Donaldson-Thomas type invariants on ADHM sheaves introduced by Diaconescu.

阿贝尔簇上凝聚层的导出范畴及其间的等价

Derived categories of coherent sheaves on abelian varieties and equivalences between them

本文研究阿贝尔簇上凝聚层的导出范畴，给出两个此类范畴等价的判据，并刻画相应的自等价群。

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.

Derived categories of toric varieties II

We prove two theorems on the derived categories of toric varieties, the existence of an exceptional collection consisting of sheaves for a divisorial extraction and the finiteness of Fourier-Mukai partners.

New enhancements of derived categories of coherent sheaves and applications

We introduce new enhancements for the bounded derived category $D^b(Coh(X))$ of coherent sheaves on a suitable scheme $X$ and for its subcategory $Perf(X)$ of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of $Perf(X)$ to geometric smoothness of $X,$ and for proving homological smoothness of $D^b(Coh(X)).$ Moreover, we characterize properness of $Perf(X)$ and $D^b(Coh(X))$ geometrically.

Knot homology via derived categories of coherent sheaves IV, coloured links

We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a spectral sequence starting with the old homology and converging to this deformed homology.

Derived categories of sheaves: a skimming

These lecture notes were prepared for the workshop ``Algebraic Geometry: Presentations by Young Researchers'' in Snowbird, Utah, July 2004, and for the autumn school in Lukecin, Poland, September 2004. In six lectures I attempted to present a non-technical introduction to derived categories of sheaves on varieties, including some important results like: -- the description of the derived category of P^n; -- Serre duality; -- the Bondal-Orlov result about derived categories of varieties with ample or antiample canonical class; -- the Mukai-Bondal-Orlov-Bridgeland criterion for equivalence; -- eq…

Perverse sheaves on semiabelian varieties

We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves on abelian varieties. As an application we explain how the arising Tannaka groups on abelian varieties can be studied in terms of semiabelian degenerations via the functor of nearby cycles.

Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities

In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W …

MW-homotopy sheaves and Morel generalized transfers

We explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with Morel generalized transfers and MW-homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets MW-transfers and use theses results to discuss the conservativity conjecture in A^1-homotopy due to Bachmann and Yakerson.

Equivalences of derived factorization categories of gauged Landau-Ginzburg models

For a given Fourier-Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier-Mukai equivalences of derived factorization categories of gauged Landau-Ginzburg (LG) models. As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier-Mukai equivalence is linearizable wi…

Derived Categories of Quadric Fibrations and Intersections of Quadrics

We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration, generalizing the Kapranov's description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety $(\PP(S^2W^*),\CB_0)$, where $\CB_0$ is the universal sheaf of even parts of Clifford alg…

Equivariant Coherent Sheaves, Soergel Bimodules, and Categorification of Affine Hecke Algebras

We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.

Pure injective and absolutely pure sheaves

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally we introduce the class of locally absolutely pure (quasi--coherent) sheaves, with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.