Related papers: Sur la cat\'egorie d\'eriv\'ee des faisceaux tordu…

Given a smooth morphism of schemes $X\rightarrow T$, denote by $\mathcal D_{X/T}^{\mathsf{cr}}$ the sheaf of rings of fiberwise crystalline differential operators on $X$ relative to $T$ and by $\Omega^\bullet_{X/T}$ the de Rham sheaf of…

We show that for a quasicompact quasiseparated scheme $X$, the following assertions are equivalent: (1) the category $\operatorname{QCoh}(X)$ of all quasicoherent sheaves on $X$ has a flat generator; (2) for every injective object $\mathcal…

Let X be e quasi-compact and semi-separated scheme. If every at quasi- coherent sheaf has finite cotorsion dimension, we prove that X is n-perfect for some n > 0. If X is coherent and n-perfect(not necessarily of finite krull dimension), we…

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $X\setminus Y_i$…

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably…

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of To\"en and Vaqui\'{e}. In this paper we show that when $X$ is…

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe…

In this paper, we study the category of twisted sheaves over a scheme $X$. Let $\mathcal{M}$ be a quasi-coherent sheaf on $X$, and $\alpha$ in $\operatorname{Br}(X)$. We show that the functor $ - \otimes_{\mathcal{O}_X} \mathcal{M} :…

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…

We expand \v{C}ech cohomology of a topological space $X$ with values in a presheaf on $X$ to \v{C}ech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open…

For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open subset U of X, C(U) is quasi-isomorphic to the Hochschild complex of the scheme U. Since C is moreover acyclic for taking sections on…

Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[\eta] \in H^0(X, \Omega^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In…

Let ${\mathcal X}$ be a category fibered in groupoids over a finite field $\mathbb{F}_q$, and let $k$ be an algebraically closed field containing $\mathbb{F}_q$. Denote by $\phi_k\colon {\mathcal X}_k\to {\mathcal X}_k$ the arithmetic…

The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$, we construct a sheaf $\Omega^{ch}_X$, called the {\bf chiral de Rham complex} of $X$. It…

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u\colon \mathcal{Z} \hookrightarrow \mathfrak{X}$ be a closed immersion of smooth, quasi-compact, separated formal schemes over…

Let $X$ be an unbounded metric space, $B(x,r) = \{y\in X: d(x,y) \leqslant r\}$ for all $x\in X$ and $r\geqslant 0$. We endow $X$ with the discrete topology and identify the Stone-\v{C}ech compactification $\beta X$ of $X$ with the set of…

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus $T$ as an ind-object in the category of holomorphic vector bundles on $T$. Extending the results of math.QA/0211262 and math.QA/0308136 we prove that the…

We prove that the tensor category of quasi-coherent modules $\mathsf{Qcoh}(X \times_S Y)$ on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ over…

We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck…

We prove that for $X$ a regular quasi-affine scheme of dimension $d$, $\mathcal{O}_X$ is a $d$-step generator of $D^b_{coh}(X)$, establishing Orlov's conjecture in this case. We prove something weaker in the projective case. The main…