Valery A. Lunts

Uniqueness of enhancement for triangulated categories

The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded de- rived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de- rived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Grothendieck ring of pretriangulated categories

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated categories. We introduce an operation of “multiplication” • on the collection of DG categories, which makes this abelian group into a commutative ring. A few applications are considered: representability of “standard” functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

Categorical Resolutions of Irrational Singularities

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.

Rationality criteria for motivic zeta functions

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is rational if and only if its Kodaira dimension is negative.

Localization for derived categories of (,)-modules

0.1. This paper arose from an attempt to solve the following problem. Let (g, K) be a Harish-Chandra pair, i.e. g is a complex reductive Lie algebra, and K is an algebraic group with an action K -Aut(g) and an embedding k = Lie K c g, satisfying some standard conditions (see 1.1 below). Let Z be the center of the enveloping algebra U(g) . Fix a regular character 0: Z -C. Let ((g, K) be the category of (g, K)-modules and t,(g, K) c l((g, K) the subcategory consisting of modules annihilated by Ker 0 . Then by the localization theorem this category X (g, K) can be described geometrically. Namely, fix a Borel subalgebra b c g and a dominant weight A corresponding to 0. Consider the algebra D, of twisted differential operators on the flag space X of g. Then X4/g, K) = J((DZ, K), the K-equivariant D,-modules on X. This result allows us to study many properties of Harish-Chandra modules geometrically. But it does not give a geometric interpretation of Ext-groups of modules in t, (g, K). Namely, let M, N E 4t (gi, K). From the point of view of representation theory the interesting objects are Ext.,, (, K) (M, N), the Ext-groups in the category of all (g, K)-modules. But these Exts do not admit localization since arbitrary (g, K)-modules do not localize. Our main result is a geometric interpretation of these Ext-groups and, more precisely, of the corresponding derived category. Let us describe it. Let to(g, K) c l((g, K) be the subcategory of 0-finite modules. That is, each element m of M E Ito(g, K) is annihilated by some power of Ker 0. Recall the localization for the category to(g, K) (precise definitions will be given later). Let G be the algebraic group of automorphisms of g, H c G a maximal torus, [ = Lie H. The flag variety X has a natural H-monodromic structure X -X. Let At(Dk) denote the category of weakly H-equivariant Dk-modules. Elements of 1#(Dk) are called monodromic D-modules on X.

Deformation theory of objects in homotopy and derived categories III: abelian categories

Abstract This is the third paper in a series. In Part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces.

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor☆

Abstract This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.

Matrix factorizations and semi-orthogonal decompositions for blowing-ups

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct and inverse image functors and dg enhancements. In the second part we prove that the category of matrix factorizations on the blowing-up of a suitable regular scheme X along a regular closed subscheme Y has a semi-orthogonal decomposition into admissible subcategories in terms of matrix factorizations on Y and X. This is the analog of a well-known theorem for bounded derived categories of coherent sheaves, and is an essential step in our forthcoming article which defines a Landau-Ginzburg motivic measure using categories of matrix factorizations. Finally we explain some applications.

Smoothness of equivariant derived categories

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant derived category of X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if all orbits O satisfy H^*(O; R)=R. On the way we prove several results concerning smoothness of dg categories over a graded commutative dg ring.

Geometricity for derived categories of algebraic stacks

We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.