We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.
We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $λ$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \to Perf(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a s…
Using the theory infinity-categories we construct derived (dg-)categories of regular, holonomic D-modules and algebraically constructible sheaves on a complex smooth algebraic stack. We construct a natural infinity-categorical equivalence between these two categories generalising the classical Riemann-Hilbert correspondence.
Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.
For a toric Deligne-Mumford (DM) stack, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the toric DM stack. We also choose a full strong exceptional collection from the set of direct summands of the push-forward in several examples of two dimensional toric DM orbifolds.
We study matrix factorizations of a section W of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.
We study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these categories depend only on the PL homeomorphism type of the corresponding simplicial complex.
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use the general framework developed in Toen, Vezzosi, ``Homotopical Algebraic Geometry I: Topos theory'', and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of …
We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stacks and symmetric quotients.
Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations ($\ukcatiso$, $\ukcateq$, $\ukcatmor$) together with the classifying stack of abelian categories ($\ukab$), the originality of the subject being that, apart from the first one, these are higher stacks. The principal result is that, under some finiteness assumptions, these stacks are geometric in the sense of C. Simpson. In particular, one recover the Hochschild cohomology of a category $…
We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau--Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne--Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for t…
In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.
Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $Ψ_X \colon \mathsf{D}(\mathsf{QCoh}(X)) \to \mathsf{D}_{\mathrm{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.
In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic formalism for étale cohomology of Artin stacks and prove several desired properties including the base change theorem. In addition, we define perverse t-structures on Artin stacks for general perversity, extending Gabber's work on schemes. Our results generalize results of Laszlo and Olsson on adic formalism and middle perversity. We continue to work in the world of $\infty$-categories in the sense of Lurie, by enhancing all the derived categories, functors, and natural transformations to the level of $\infty$-categories…
In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale l…
We outline a proof that the stack of symplectic toric G-manifolds over a fixed orbit space W is a torsor for the stack of symplectic toric G-principal bundles over W.
We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.
We generalise recent results of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras using the formalism of algebraic stacks.
Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and the Ext groups of appropriate stacks.
We extend Orlov's representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities.
In this paper, we will provide constructions of D-module structures on the complex computing the periodic cyclic homology of a stable infinity-category defined over a scheme of characteristic zero. We give two methods. The first one is based on a canonical extension of factorization homology to the mapping stack and relation between sheaves on free loop space and D-modules. The second one uses the algebraic structure on Hochschild pairs, its moduli-theoretic interpretation, Kodaira-Spencer morphisms, and the relation between dg Lie algebras and pointed formal stacks.
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…
This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient stack which admits a stratification of the kind arising in geometric invariant theory. The use of derived algebraic geometry leads to results with pleasingly few hypotheses, even when the stack is not smooth. Using the same methods, we establish a "virtual non-abelian localization theorem" which is a K-theoretic analog of the virtual localization theorem in…
We study the category of KM fans - a "stacky" generalization of the category of fans considered in toric geometry - and its various realization functors to "geometric" categories. The "purest" such realization takes the form of a functor from KM fans to the 2-category of stacks over the category of fine fans, in the "characteristic-zero-étale" topology. In the algebraic setting, over a field of characteristic zero, we have a realization functor from KM fans to (log) Deligne-Mumford stacks. We prove that this realization functor gives rise to an equivalence of categories between (lattice) KM fa…
For an abelian tensor category a stack is constructed. As an application we show that our construction can be used to recover a quasi-compact separated scheme from the category of its quasi-coherent sheaves. In another application, we show how the "dual stack" of the classifying stack $BG$ of a finite group $G$ can be obtained by altering the tensor product on the category $\rep{G}$ of $G$-representations. Using glueing techniques we show that the dual pair of a $G$-gerbe, in the sense of [TT10], can be constructed by glueing local dual stacks.
We study the relationship between derived categories of factorizations on gauged Landau-Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and describe the complementary components. We also verify a question posed by Kawamata: we show that $D$-equivalence and $K$-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne-Mumf…
To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the triangulated category $[T]$ associated to $T$. The main result of this work states that under certain finiteness conditions on $T$ (e.g. if it is saturated) the $D^{-}$-stack $\mathcal{M}_{T}$ is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of a saturated dg-cate…
We generalise the techniques of arXiv:0908.1963 to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms). We also give a general approach for studying deformations of diagrams.
We calculate explicit formulas for the general equivariant Bondal-Orlov functors on the localized K-theory groups for a crepant birational transformation of toric DM stacks. We recall some facts that the Bondal-Orlov functors give equivalences on the bounded derived categories. Applying twice of these functors we get the Seidel-Thomas spherical twists for the derived category.
In math.AG/0207028 we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of S-site and of model site, and the associated categories of stacks on them. This led us to study a notion of \textit{model topos} (orginally due to C. Rezk), a model category version of the notion of Grothendieck topos. In this paper we treat the analogous theory starting from (1-)Segal categories in place of S-categories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over t…
We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.
This paper has been withdrawn, because I have merged it with paper I of the series, math.AG/0312190. The main results of this paper now appear in sections 7-9 of the revised version of math.AG/0312190, with shortened and improved proofs.
Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups.
For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\infty}-ring spectra. This first versi…
Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. The classification is actually a purely category-theoretic result about groupoid fibrations over fibered categories, though most of the known applications occur in the setting of log geometry, where our categorical framework encompasses many notions of "minimality" previously extant in the literature.
We show that the obstruction to the existence of a strict symmetric monoidal structure on a monoidal stack $\cal C$ is determined by a commutator biextension associated to $\cal C$, and that this biextension is alternating under an additional hypothesis. The analogous construction for monoidal 2-stacks is described, together with the corresponding generalizations of the concept of a biextension.
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 rings and on an arbitrary small category. This provides with new applications of that paper to the categories of quasi-coherent sheaves on an Artin stack or on a Deligne-Mumford stack.
Let Y be a smooth algebraic stack exhausted by quotient stacks. Given a Kirwan-Ness stratification of the cotangent stack T^*Y, we establish a recollement package for twisted D-modules on Y, gluing the category from subquotients described via modules microsupported on the Kirwan-Ness strata of T^*Y. The package includes unusual existence and "preservation-of-finiteness'' properties for functors of the full category of twisted D-modules, extending the standard functorialities for holonomic modules. In the case that Y = X/G is a quotient stack, our results provide a higher categorical analogue o…
It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $Λ$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,Λ)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by thi…
We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We show that the inertia operator is locally finite and diagonalizable. This is proved for the Grothendieck group of Deligne-Mumford stacks over a base scheme and the category of quasi-split Artin stacks defined over a field of characteristic zero. Motivated by the quasi-splitness condition, in [2] we consider the inertia operator of the Hall algebra of algebroids, and applications of it in generalized Donaldson-Thomas theory.
Using Quillen-Lurie deformation theory formalism we develop an obstruction theory for studying the stable $\infty$-category of modules over a given geometric $\infty$-stack. The obstruction theory studies the problem of lifting compact objects to the stable $\infty$-category of quasi-coherent modules over a derived geometric stack from the category of modules over its underlying classical stack. The obstructions live in Andre-Quillen cohomology. An explicit description of the space of realizations of a given module over X as a colimit of perfect modules can be given in terms of the k-invariant…
We prove that the moduli stack of Bridgeland semistable holomorphic triples over a curve of $g(C)\geq 1$ with a fixed numerical class and phase is an algebraic stack of finite type over $\mathbb{C}$ and admits a proper good moduli space. We prove that this also holds for a class of Bridgeland stability conditions on the category of holomorphic chains $\mathcal{T}_{C,n}$. In the process, we construct an explicit geometric realisation of $\mathcal{T}_{C,n}$ and prove the open heart property for noetherian hearts in admissible categories of $D^b(X)$, where $X$ is a smooth projective variety over …
Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in this case the category of its finite dimensional representations. Lurie's result that the pseudofunctor which sends a geometric stack to its category of quasi-coherent sheaves is fully faithful provides a conceptual explanation for why this works. In this paper we prove a generalized Tannakian recognition theorem, in order to characterize a part of the imag…
The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the model structure defined by Quillen in 1969 for 2-reduced coalgebras. In our case, the notion of weak equivalence is structly stronger than that of quasi-isomorphism. A pair of adjoint functors connecting the category of coalgebras with the category of dg Lie algebras, induces an equivalence of the corresponding homotopy categories. The model category structure allows one to consider dg coalgebras as very genera…
We associate to a braided 2-stack ${\cal C}$ a torsor, endowed with a symmetric cube structure (or $Σ$-structure), whose triviality is equivalent to the existence on ${\cal C}$ of a fully symmetric monoidal structure. In order to perform the analogous construction for braided 4-stacks, it is necessary to introduce the notion of a $Γ_3$-torsor pair, which is in the same relation to cubic forms as torsors with $Σ$-structure are to quadratic ones.
Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. In this article we give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed in arXiv:1211.1877. As an application, we prove a Lefschetz-Verdie…
We define a closed model category containing the $n$-nerves defined by Tamsamani, and admitting internal $Hom$. This allows us to construct the $n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré $n$-groupoid of a topological space. We give a still-speculative discussion of $n$-stacks, and similarly of comparison with other possible definitions of $n$-category.
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373). In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of \infty-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on s…
