Category Of Quasi-coherent Sheaves Research Articles

A categorification of the quantum Lefschetz principle

The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus 0, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the ∞-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.

Locally finitely presented and coherent hearts

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal{G}$, we study the local finite presentability and local coherence of the heart $\mathcal{H}{\mathbf{t}}$ of the associated Happel–Reiten–Smalø $t$-structure in the derived category $\mathbf{D}(\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf{t}$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf{t}$ for which $\mathcal{H}{\mathbf{t}}$ is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category $\mathcal{G}$, which are general enough to include all locally coherent Grothendieck categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction, we characterize those torsion pairs $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in a locally finitely presented $\mathcal{G}$ for which $\mathcal{H}{\mathbf{t}}$ is locally coherent in two cases: when the tilted $t$-structure in $\mathcal{H}{\mathbf{t}}$ is assumed to restrict to finitely presented objects, and when $\mathcal{F}$ is cogenerating. In the last part of the paper, we concentrate on the case when $\mathcal{G}$ is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations in this more classical setting, underlying connections with the notion of an elementary cogenerator.

The Poincaré line bundle and autoduality of Hitchin fibers

In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves.

Reconstruction theorem for monoid schemes

We aim to reconstruct a monoid scheme X from the category of quasi-coherent sheaves over it. This is much in the vein of Gabriel's original reconstruction theorem. Under some finiteness condition on a monoid schemes X , we show that the localisations of the topos Qc ( X ) of quasi-coherent sheaves on X are in a one-to-one correspondence with open subsets of X , while the elements of X correspond to the topos points of Qc ( X ) . This allows us to reconstruct X from Qc ( X ) .

On Noetherian schemes over (\mathcal{C},\otimes,1)$ and the category of quasi-coherent sheaves

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 Toen and Vaquie. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of objects in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a familiy of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.

Tensor generators on schemes and stacks

We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group, or equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space. This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks. Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.

Tensor generators on schemes and stacks

We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group, or equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space. This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks. Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.

Absolute purity in the category of quasi coherent sheaves

In this paper we consider the class of absolutely pure and Fp-injective quasi-coherent sheaves. We show that these two classes of quasi-coherent sheaves over a locally coherent scheme are equivalent. As a corollary we will show that the class of absolutely pure quasi-coherent sheaves over such a scheme is an enveloping and a covering class. It is proved that over a locally coherent scheme, the pair (?(Abs(X),Abs(X)) is a cotorsion theory. The existence of a duality between absolutely pure envelopes and flat covers is proved as expected.

A functorial formalism for quasi-coherent sheaves on a geometric stack

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 a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category.We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f:X→Y, an adjunction f∗⊣f∗ for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid.

Kaplansky Classes and Cotorsion Theories of Complexes

In this article we provide arguments for constructing Kaplansky classes in the category of complexes out of a Kaplansky class of modules. This leads to several complete cotorsion theories in such categories. Our method gives a unified proof for most of the known cotorsion theories in the category of complexes and can be applied to the category of quasi-coherent sheaves over a scheme as well as the category of the representations of a quiver.

A lifting functor for toric sheaves

For a variety $X$ which admits a Cox ring, we introduce a functor from the category of quasi-coherent sheaves on $X$ to the category of graded modules over the homogeneous coordinate ring of $X$. We show that this functor is right adjoint to the sheafification functor and therefore left-exact. Moreover, we show that this functor preserves torsion-freeness and reflexivity. For the case of toric sheaves, we give a combinatorial characterization of its right derived functors in terms of certain right derived limit functors.

Tensor functors between categories of quasi-coherent sheaves

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction f↦f⁎ implements an equivalence between the discrete category of morphisms Y→X and the category of cocontinuous tensor functors Qcoh(X)→Qcoh(Y). This is an improvement of a result by Lurie and may be interpreted as the statement that algebraic geometry is 2-affine. Moreover, we prove the analogous version of this result for Durovʼs notion of generalized schemes.

Invariant rings through categories

We formulate a notion of “geometric reductivity” in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result applies to the category of modules over a bialgebra, the category of comodules over a bialgebra, and the category of quasi-coherent sheaves on an algebraic stack of finite type over an affine base.

A Structure Theorem for $\displaystyle\mathbb{P}^{{1}}$ — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.

The Eilenberg–Watts theorem over schemes

We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg–Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002) [6] are noncommutative P 1 -bundles in the sense of M. Van den Bergh [10].

D-modules on the affine flag variety and representations of affine Kac-Moody algebras

The present paper studies the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme G ( ( t ) ) / I G((t))/I , where I I is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of the authors in 2006.

Algebroid prestacks and deformations of ringed spaces

For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules (Lowen and Van den Bergh, 2006) are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation equivalence between Mod(O) and Qch(O).

Auslander–Reiten sequences on schemes

Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories. This note considers X, a smooth projective scheme of dimension at least 1 over the field k, and $\mathcal{C}$, an indecomposable coherent sheaf on X. It is proved that in the category of quasi-coherent sheaves on X, there is an Auslander–Reiten sequence ending in $\mathcal{C}$.

Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line#

In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line , where R is any commutative ring. We first prove a general result that guarantees the existence of ℱ-covers and ℱ⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class ℱ satisfies some “standard” conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., García Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. J. Algebra 243:615–630].