On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let $\mathsf{X}$ be an Adams geometric stack. We show that $\mathsf{D}({\mathsf{A}}_{\mathsf{qc}}(\mathsf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in [13]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathsf{X}$.     

On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme

In this paper, we prove that for a noetherian formal scheme \mathfrak X\mathfrak X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is generated by a single compact object. In an Appendix we prove that the category of compact objects in D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is skeletally small.     

Totally acyclic complexes over noetherian schemes

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves on a semi-separated noetherian scheme, and study these complexes using the pure derived category of flat quasi-coherent sheaves. We prove that a scheme is Gorenstein if and only if every acyclic complex of flat quasi-coherent sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jørgensen's proof of the existence of Gorenstein projective precovers.     

Locally torsion-free quasi-coherent sheaves

Let X   be an arbitrary scheme. It is known that the category Qcoh(X)$\mathfrak{Qcoh}(X)$ of quasi-coherent sheaves admits arbitrary products. However its structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in Qcoh(X)$\mathfrak{Qcoh}(X)$, for X an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi-coherent sheaves on a Dedekind scheme X   is closed under arbitrary direct products, and that the class of all locally torsion-free quasi-coherent sheaves induces a hereditary torsion theory on Qcoh(X)$\mathfrak{Qcoh}(X)$. Finally torsion-free covers are shown to exist in Qcoh(X)$\mathfrak{Qcoh}(X)$.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Relative homological algebra in the category of quasi-coherent sheaves

In this paper, we prove the existence of a flat cover and of a cotorsion envelope for any quasi-coherent sheaf over a scheme (X,O_X). Indeed we prove something more general. We define what it is understood by the category of quasi-coherent R-modules, where R is a representation by rings of a quiver Q, and we prove the existence of a flat cover and a cotorsion envelope for quasi-coherent R-modules. Then we use the fact that the category of quasi-coherent sheaves on (X,O_X) is equivalent to the category of quasi-coherent R-modules for some Q and R to get our result.     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Oriented cohomology and motivic decompositions of relative cellular spaces

For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL.     

Auslander–Reiten Sequences,Brown–Comenetz Duality,and the <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-local Generating Hypothesis

In this paper, we construct a version of Auslander–Reiten sequences for the K(n)-local stable homotopy category. In particular, the role of the Auslander–Reiten translation is played by the local Brown–Comenetz duality functor. As an application, we produce counterexamples to the K(n)-local generating hypothesis for all heights n > 0 and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

Configurations in abelian categories—I: Basic properties and moduli stacks

This is the first in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or π(J,K):σ(J)→σ(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects, and are useful for studying stability conditions on A.We define and motivate the idea of configurations, and explain some natural operations upon them—subconfigurations, quotient configurations, substitution, refinements and improvements. Then we study moduli spaces of (I,?)-configurations in A, and natural morphisms between them, using the theory of Artin stacks. We prove well-behaved moduli stacks exist when A is the abelian category of coherent sheaves on a projective scheme P, or of representations of a quiver Q.In the sequels, given a stability condition (τ,T,?) on A, we will show the moduli spaces of τ-(semi)stable objects or configurations are constructible subsets in the moduli stacks of all objects or configurations. We associate infinite-dimensional algebras of constructible functions to a quiver Q using the method of Ringel-Hall algebras, and define systems of invariants of P that ‘count’ τ-(semi)stable coherent sheaves on P and satisfy interesting identities.     

A Gabriel Theorem for coherent twisted sheaves

The aim of this work is to give a generalization of Gabriel’s Theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes. We start by showing that we can recover a noetherian scheme X from the category Coh(X, α) of coherent α-twisted sheaves over X, where α lies in the cohomological Brauer group of X. This follows from the bijective correspondence between closed subsets of X and Serre subcategories of finite type of Coh(X, α). Moreover, any equivalence between Coh(X, α) and Coh(Y, β), where X and Y are noetherian schemes, and , β Br ′(Y) induces an isomorphism between X and Y.     

Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A_∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A_∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.     

Moduli stacks and invariants of semistable objects on K3 surfaces

For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D. Joyce's work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.     

Model category structures on chain complexes of sheaves

The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.

     

Relative uniform completeness and order-convex representations of archimedean ?-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ?-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ?-group, G, we show: if G admits any representation G?D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ?-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A?D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G?D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).