Related papers: Algebraic Geometry over Non-Algebraically Closed F…

We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher)…

Let (X,D) be a D-scheme in the sense of Beilinson and Bernstein, given by an algebraic variety X and a morphism O_X -> D of sheaves of rings on X. We consider noncommutative deformations of quasi-coherent sheaves of left D-modules on X, and…

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable…

We develop a theory of quasimaps to a moduli space of sheaves $M$ on a surface $S$. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic…

Let G be an algebraic group over an algebraically closed field, acting on a variety X with finitely many orbits. "Staggered sheaves" are certain complexes of G-equivariant coherent sheaves on X that seem to possess many remarkable…

This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a…

We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category $X$. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when $X$ is the category of…

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…

G-structures and Cartan geometries are two major approaches to the description of geometric structures (in the sense of differential geometry) on manifolds of some fixed dimension $n$. We show that both descriptions naturally extend to the…

Let $X$ be either a quasi-compact semi-separated scheme, or a Noetherian scheme of finite Krull dimension. We show that the Grothendieck abelian category $X{-}\mathsf{Qcoh}$ of quasi-coherent sheaves on $X$ satisfies the Roos axiom…

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution…

We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $\omega\_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of…

In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of…

A general framework is described which associates geometrical structures to any set of $D$ finite-dimensional hermitian matrices $X^a, \ a=1,...,D$. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and…

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…

Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent…

Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincar\'e duality theories are obtained. We assume given a (twisted) cohomology theory $H^*$ having a cup product structure and we let consider…

We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed…

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are…

In this work we study a kind of coherence condition on FI_G-modules, which generalizes the usual notion of finite generation. We prove that a module is coherent, in the appropriate sense, if and only if its generators, as well as its…