Related papers: Quasicoherent sheaves on complex noncommutative tw…

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…

A correspondence between quasicoherent sheaves on toric schemes and graded modules over some homogeneous coordinate ring is presented, and the behaviour of several finiteness properties under this correspondence is investigated.

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 show that in the category of analytic sheaves on a complex analytic space, the full subcategory of quasi-coherent sheaves is an abelian subcategory.

We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions,…

We build an infinite dimensional scheme parametrizing isomorphism classes of coherent quotients of a quasi-coherent sheaf on a projective scheme. The main tool to achieve the construction is a version of Grothendieck's Grassmannian…

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…

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 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…

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…

In this note we extend the main results of [E. Enochs and S. Estrada, Relative homological algebra in the category of quasi-coherent sheaves. Adv. in Math. 194(2005), 284-295] to the category of cartesian modules over a flat presheaf of…

We develop the theory of ind-coherent sheaves on schemes and stacks. The category of ind-coherent sheaves is closely related, but inequivalent, to the category of quasi-coherent sheaves, and the difference becomes crucial for the…

If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard…

A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to…

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric…

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed…

In this paper, we classify several subcategories of the category of coherent sheaves on a noetherian divisorial scheme (e.g. a quasi-projective scheme over a commutative noetherian ring). More precisely, we classify the torsionfree (resp.…

Let T be a compact complex torus, dim T>2. We show that the category of coherent sheaves on T is independent of the choice of the complex structure, if this complex structure is generic. The proof is independent of math.AG/0205210, where…

We show that the cohomology table of any coherent sheaf on projective space is a convergent--but possibly infinite--sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.

The category of coherent sheaves over a noetherian scheme is very important for studying the properties of a given scheme. For noetherian schemes it is a well-known fact that the topology can be fully recovered from the corresponding…