Related papers: Horrocks theory and the Bernstein-Gelfand-Gelfand …

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for…

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated $K$-theory groups are "stable algebraic…

A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for…

In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the "Hecke family"; on the other, we find…

To an abelian category A of homological dimension 1 satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth…

We define functors on the derived category of the moduli space M of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain relations. These relations allow us to prove…

In the appendix of the famous book "Commutative Algebra with a View Towards Algebraic Geometry" one can find an infinite family of complexes indexed by integers. This family includes Eagon-Northcott and Buschsbaum-Rim complexes. The…

If V is a bundle of Tate vector spaces over a base B, its determinantal gerbe has a class C_1(V) in the second cohomology group of the sheaf of invertible functions which can be seen as the Deligne cohomology H^3(B, Z(2)). An example of…

Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective…

We set up a homological algebra for N-complexes, which are graded modules together with a degree -1 endomorphism d satisfying d^N=0. We define Tor- and Ext-groups for N-complexes and we compute them in terms of their classical counterparts…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

We define and study a certain category of vector bundles on a p-adic curve to which we can associate in a functorial way finite dimensional p-adic representations of the geometric fundamental group. Among other things we investigate two…

We offer here a more direct approach to twisted K-theory, based on the notion of twisted vector bundles (of finite or infinite dimension) and of twisted principal bundles. This is closeely related to the classical notion ot torsors and…

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack ${\mathscr B}un_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of…

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular…

Using the theory infinity-categories we construct derived (dg-)categories of regular, holonomic D-modules and algebraically constructible sheaves on a complex smooth algebraic stack. We construct a natural infinity-categorical equivalence…

The current article is a short survey on the theory of Hecke algebras, and in particular Kazhdan-Lusztig theory, and on the theory of symplectic reflection algebras, and in particular rational Cherednik algebras. The emphasis is on the…

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we…

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting…

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between…