Related papers: On the stack of 0-dimensional coherent sheaves: st…

Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C(X) of topological O_X-modules on X, called the complete Hochschild chain complex of X. To any O_X-module M - not necessarily quasi-coherent - we…

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…

In this letter, we show that coherent structures are related to folds of horseshoes which are present in chaotic systems. We develop techniques that allow us to construct coherent structures by manipulating folds in three prototypical…

For a regular noetherian scheme $X$ with a divisor with strict normal crossings $D$ we prove that coherent sheaves satisfy descent w.r.t. the 'covering' consisting of the open parts in the various completions of $X$ along the components of…

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal…

In this paper we begin the classification of coherent systems $(E,V)$ on the projective line which are stable with respect to some value of a parameter $\alpha$. In particular we show that the moduli spaces, if non-empty, are always smooth…

In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher le vels need to respect the dynamics on the…

In this paper, we give a new constructive proof of the semi-orthogonal decomposition of the derived category of (quasi)-coherent sheaves of root stacks, through an explicit resolution of the diagonal.

We prove a general form of the statement that the cohomology of a quotient stack can be computed by the Borel construction. It also applies to the lisse extensions of generalized cohomology theories like motivic cohomology and algebraic…

We prove $v$-descent for solid quasi-coherent sheaves on perfectoid spaces as a key technical input for the development of a $6$-functor formalism with values in solid quasi-coherent sheaves on relative Fargues--Fontaine curves.

Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $\psi \colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot…

We give a homotopy theoretic characterization of stacks on a site $\cC$ as the {\it homotopy sheaves} of groupoids on $\cC$. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare…

In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather…

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent…

Given a separated and locally finitely-presented Deligne-Mumford stack $\cX$ over an algebraic space $S$, and a locally finitely-presented $\OO_{\cX}$-module $\cF$, we prove that the Quot functor $\text{Quot}(\cF/\cX/S)$ is represented by a…

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve…

Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect…

We study some properties of the characteristic cycle of a constructible complex on a smooth variety over a perfect field, push-forward and product.

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular…

A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and…