We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the categories that arise as the category of representations of an affine extension $q:G\to A$, generalizing the classical results of Tannaka Duality established for affine $\Bbbk$-group schemes (that is, when $A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant equivalence between the category of affine extensions of a given $A$ and the category of faithful commutative Hopf sheaves on $A$, generalizing in this manner the well-known op-equivalence between affine group schemes and commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$ associated to $q$, the category of representations of $q$ is equivalent to the category of $\mathcal H_q$-comodules.
We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on $n$-dimensional skew shapes…
Let G be an algebraic group over an algebraically closed field, acting on a variety X with finitely many orbits. "Staggered sheaves" are certain complexes of G-equivariant coherent sheaves on X that seem to possess many remarkable properties. In this paper, we construct "standard" and "costandard" objects in the category of staggered sheaves, and we prove that that category has enough projectives and injectives.
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 associate, in a 2-functorial way, to a 1-morphism of geometric stacks $f$, an adjunction $(f^*, f_*)$ for the corresponding categories of quasi-coherent sheaves that agrees w…
In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig …
Let $\mathbf{X}$ be an Adams geometric stack. We show that $D(A_{qc}(\mathbf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathbf{X}$.
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 of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X.
We study filtration of quasi--coherent sheaves. We prove a version of Kaplansky Theorem for quasi--coherent sheaves, by using Drinfeld's notion of almost projective module and the Hill Lemma. We also show a Lazard-like theorem for flat quasi-coherent sheaves for quasi-compact and semi-separated schemes which satisfy the resolution property.
We define the derived category of quasi--coherent modules for certain Artin stacks as the homotopy category of two Quillen monoidal model structures on the corresponding category of unbounded complexes of quasi--coherent modules.
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 prove that every at quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence K(PinfX)---> D(FlatX) of homotopy categories, whenever K(PinfX) is the homotopy category of pure injective at quasi-coherent sheaves and D(FlatX) is the pure derived category of at quasi-coherent sheaves.
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 $\mathcal E$ of $\operatorname{QCoh}(X)$, the internal hom functor into $\mathcal E$ is exact; (3) the scheme $X$ is semiseparated.
The moduli spaces of stable sheaves on projective schemes admit certain gluing data of Kapranov's NC structures, which we call quasi NC structures. The formal completion of the quasi NC structure at a closed point coincides with the pro-representable hull of the non-commutative deformation functor of the corresponding sheaf. In this paper, we show the existence of smooth non-commutative dg-resolutions of the above quasi NC structures, and call them quasi NCDG structures. When there are no higher obstruction spaces, the quasi NCDG structures define the notion of NC virtual structure sheaves, th…
We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.
For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.
A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).
We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site C. We use this to prove homotopy invariance and generalized descent statements for categories of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a generalized change of rings theorem of Hovey.
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.
We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasi-coherent sheaves on the projective line. In this model structure the cofibrant objects are the dg-locally projective complexes. We also describe the fibrations of this model structure and show that the model structure is monoidal. We point out that this model structure is necessarily…
We define quasi-coherent parabolic sheaves with real weights on a fine saturated log analytic space, and explain how to interpret them as quasi-coherent sheaves of modules on its Kato-Nakayama space. This recovers the description as sheaves on root stacks of arXiv:1001.0466 and arXiv:1410.1164 for rational weights, but also includes the case of arbitrary real weights.
Let $S$ be a scheme contained in $\mathbf{Sch}_{fppf}$. Let $\mathfrak{X}$ be a Noetherian separated algebraic space over $S$. In this paper, we interpret localizing subcategories of the derived category of $\mathfrak{X}$ by using subsets of $|\mathfrak{X}|$.
We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities and compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.
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 associated to Y, and that for every P in C, if z is an equation of C_1 in the local ring of Y at P, then (z^i) is the ideal of C_i. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e. sheaves which are locally i…
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.
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 derived category of quasicoherent sheaves on $T$ is equivalent to the derived category of usual quasicoherent sheaves on the corresponding elliptic curve. We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category $\Qcoh(η_T)$ obtained by taking the quotient of the category of quasicoherent sheaves by …
We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. We then show the homotopy category of the quasi DG category of $C$-diagrams has the structure of a triangulated category. This procedure is then applied to produce a triangulated category of mixed motives over a base variety.
We use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of alpha-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes X when alpha is in the Brauer group of X. By applying the work of Toën on derived Azumaya algebras, we give a proof of Căldăraru's conjecture.
In their article "Elementary construction of perverse sheaves", R.MacPherson and K. Vilonen show that on a Thom-Mather space X the category PervX of perverse sheaves is equivalent to the category C(F, G, T) whose objects are data of perverse sheaves on the complementary of the closed strata S, a local system on S and some gluing data. To show this equivalence of categories, they define a functor C going from the category PervX to the category C(F, G, T). This definition is based on the notion of perverse link. They do not define a quasi-inverse of this functor. moreover they have to consider f…
Raynaud--Gruson characterized flat and pure morphisms between affine schemes in terms of projective modules. We give a similar characterization for non-affine morphisms. As an application, we show that every quasi-coherent sheaf is the union of its finitely generated quasi-coherent subsheaves on any quasi-compact and quasi-separated algebraic stack.
Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove several basic results regarding these. We show that: 1.) Every quasicoherent sheaf F on MProj(A) can be constructed from a graded A--set in analogy with the construction of quasicoherent sheaves on Proj(R) from graded R--modules. 2.) High enough twists of coherent sheaves are generated by finitely many global sections, hence that every coherent sheaf is a quotien…
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 the structure of the product object of a family of locally torsion-free objects in $\mathfrak{Qcoh}(X)$, 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 quas…
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, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.
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 Toën and Vaquié. 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 "field 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 Noetheri…
We define and study the category $Coh_n(\Pone)$ of normal coherent sheaves on the monoid scheme $\Pone$ (equivalently, the $\mathfrak{M}_0$-scheme $\Pone / \fun$ in the sense of Connes-Consani-Marcolli \cite{CCM}). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of $Coh_n(\Pone)$. We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra $L {\mathfrak{gl}}_2$ and an abelian Lie algebra on infinitely many generators. This shoul…
Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
Given a smooth projective toric variety $X_Σ$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_Σ)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, Λ_Σ)$. Recently, Kuwagaki \cite{Ku2} proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent …
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 formulation of the categorical Geometric Langlands Correspondence.
We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it …
We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.
For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction 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 Durov's notion of generalized schemes over F_1.
We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the…
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent 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.
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.
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 $Y=\bigcup_{i\inΩ}Y_i$, with $X\setminus Y_i$ quasi-compact and open for all $i\inΩ$, is established. As an application, there is constructed an isomorphism of ringed spaces (X,O_X)-->(Spec(Qcoh(X)),O_{Qcoh(X)}), where $(Spec(Qcoh(X)),O_{Qcoh(X)})$ is a ringed space associated to the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories…
In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on injective resolutions for an arbitrary Grothendieck category, as has apparently also been done by Morel. In particular, this works for sheaves on a ringed space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. However, this injective model structure is not well suited to studying the derived tensor product, so we investigate other model st…
The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This i…
Algebraic structures involving both multiplications and comultiplications (such as, e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROducts and Permutations) of Adams and MacLane. To encode such structures on objects of a braided monoidal category, we need PROBs (braided analogs of PROPs). Colored PROBs correspond to multi-sorted structures. In particular, we have a colored PROB B governing non-negatively graded bialgebras in braided categories. As a category, B splits into blocks B_n according to the grading. We relate B_n with the category P_n of perverse shea…
