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B. Toen

Publications and source records attributed to B. Toen.

At least 19 recordsLinked to original sources
arXiv2015

Shifted Poisson Structures and Deformation Quantization

This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the way for shifted deformation quantization of many interesting derived moduli spaces, like those studied in [PTVV] and probably many others.

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arXiv2011

Shifted Symplectic Structures

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks. We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures. Our main existence theorem states that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic structure, the derived mapping stack $\textbf{Map}(X,F)$ is equipped with a canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a Calabi-Yau condition in dimension $d$. These two results imply the existence of many examples of derived moduli stacks equipped with $n$-shifted symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We also show that Lagrangian intersections carry canonical (-1)-shifted symplectic structures.

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arXiv2010

Derived Azumaya algebras and generators for twisted derived categories

We introduce a notion of derived Azumaya's algebras over rings and schemes. We prove that any such algebra $B$ on a scheme $X$ provides a class $\phi(B)$ in $H^{1}_{et}(X,\mathbb{Z})\times H^{2}_{et}(X,\mathbb{G}_{m})$. We prove that for $X$ a quasi-compact and quasi-separated scheme $\phi$ defines a bijective correspondence, and in particular that any class in $H^{2}_{et}(X,\mathbb{G}_{m})$, torsion or not, can be represented by a derived Azumaya's algebra on $X$. Our result is a consequence of a more general theorem about the existence of compact generators in \emph{twisted derived categories, with coefficients in any local system of reasonable dg-categories}, generalizing the well known existence of compact generators in derived categories of quasi-coherent sheaves of \cite{bv} (corresponding to the trivial local system of dg-categories). A huge part of this paper concerns the correct treatment of these twisted derived categories, as well as the proof that the existence of compact generators locally for the fppf topology implies the existence of a global compact generator.

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arXiv2010

Flat descent for Artin n-stacks

We prove two flat descent statements for Artin n-stacks. We first show that an n-stack for the etale topology which is an Artin n-stack in the sense of HAGII, is also an n-stack for the fppf topology. Moreover, an n-stack for the fppf topology which possess a fppf n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these results some comparison statements between fppf and etale (non-ablelian) cohomolgies. This paper is written in the setting of derived algebraic geometry and its results are also valid for derived Artin n-stacks.

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arXiv2009

Grothendieck rings of Artin n-stacks

We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite field". We also introduce the notion of "special Artin stacks", which by definition have affine homotopy groups π_{i}, and furthermore unipotent for i>1. Our principal theorem states that the natural inclusion morphism from the Grothendieck ring of varieties to the Grothendieck ring of special Artin stacks is an isomorphism after inverting the class of the affine line L and the classes of L^{i}-1 for all i>0. We deduce from this that several numerical invariants defined for varieties (e.g. Hodge numbers, l-adic Euler characteristic ...) extend uniquely to invariants defined for special Artin stacks. In particular we obtain a trace formula for special Artin stacks of finite type over a finite field, identifying the number of rational points as the trace of the Frobenius acting on the l-adic Euler characteristic.

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arXiv2009

S^1-Equivariant simplicial algebras and de Rham theory

This is a companion paper our previous submission "\infty-categories monoidales rigides et caracteres de Chern", in which we give a comparison between functions on the derived loop space of a smooth scheme of caracteristic zero, and its algebraic de Rham cohomology. As a consequence we obtain functorial and multiplicative versions of the HKR decomposition theorem relating Hochschild homology and Hodge cohomology.

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arXiv2009

Caract\`eres de Chern, traces \'equivariantes et g\'eom\'etrie alg\'ebrique d\'eriv\'ee

The purpose of this work is to provide details about the construction of the Chern character for categorical sheaves mentioned in our previous work "Chern character, loop spaces and derived algebraic geometry". For this, we introduce and study the notion of rigid symmetric monoidal \infty-category. We show how trace maps can be constructed in this higher categorical setting, and using a recent work of Hopkins-Lurie we prove the existence of a "cyclic trace", which is the main ingredient in the construction of the Chern character. Our Chern character is then constructed for any pair (T,A), consisting of a \infty-topos T and a stack of rigid symmetric monoidal \infty-categories A on T. We propose two main applications of this construction. First of all, we show how to recover the Chern character of perfect complexes on schemes, with values in cyclic homology, and show how it can be extended to an interesting new Chern character for Artin stacks. Our second application provides invariants of famillies of dg-categories. A consequence of the existence of these invariants is the construction of a Gauss-Manin connexion on the cyclic homology complex of such a familly, generalizing previous constructions by Getzler and of Dolgushev-Tamarkin-Tsygan. Another consequence is the construction of the "character sheaf" associated to a representation of an algebraic group into a dg-category, which is a categorification of the character function of a linear representation. Finally, for a familly of saturated dg-categories we construct a "secondary Chern character", taking values in a new cohomology theory called "secondary cyclic homology".

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arXiv2007

Homotopical finiteness of smooth and proper dg-algebras

We show that any smooth and proper dg-algebra (over some base ring k) is determined, up to quasi-isomorphism, by its underlying A_n-algebra, for a certain integer n. Similarly, any morphism between two smooth and proper dg-algebras is determined, up to homotopy, by the morphism induced on the underlying A_n-algebras, for a certain integer n. When the base ring k is local, we show that the integer n can be chosen uniformally for all smooth and proper dg-algebras for which two numerical invariants (the "type" and the "cohomogical dimension") are bounded.

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arXiv2007

On derived equivalence classes of algebraic varieties

Let $X \to S$ be a miniversal family of smooth and projective varieties and D be a fixed triangulated category. We show that the set of points s in S such that the derived category of the fiber X_s at s is equivalent to D is at most countable. We deduce from this that the derived equivalence classes of smooth and projective complex varieties is at most countable.

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arXiv2007

Moduli of objects in dg-categories

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-category. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.

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arXiv2006

Rings of definition of smooth and proper dg-algebras

This is a companion paper to math.AT/0609762. For a filtered colimit of commutative rings k=colim k_i, we prove that the homotopy theory of smooth and proper dg-algebras over k is the colimit of the homotopy theories of smooth and proper dg-algebras over k_i. As a consequence, we deduce that any smooth and proper dg-algebra can be defined over a commutative Z-algebra of finite type.

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arXiv2006

The homotopy theory of dg-categories and derived Morita theory

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.

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arXiv2006

Higher and derived stacks: a global overview

These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.

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arXiv2006

Affine stacks (Champs affines)

This is an expended and revised version of the preprint "Schematization of homotopy types". The purpose of this work is to introduce a notion of \emph{affine stacks}, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and $p$-adic homotopy theory. This gives a first solution to A. Grothendieck's \emph{schematization problem} described in \cite{gr}. We also use affine stacks in order to introduce a notion of \emph{schematic homotopy types}. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyound rational and $p$-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various cohomology theories (Betti, de Rham, $l$-adic, ...), extending the well known constructions of the various fundamental groups. Finally, as algebraic stacks are obtained by gluing affine schemes we define \emph{$\infty$-geometric stacks} as a certain gluing of affine stacks. Example of $\infty$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.

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arXiv2006

Derived Hall Algebras

The purpose of this work is to define a derived Hall algebra $\mathcal{DH}(T)$, associated to any dg-category $T$ (under some finiteness conditions). Our main theorem states that $\mathcal{DH}(T)$ is associative and unital. It is shown that $\mathcal{DH}(T)$ contains the usual Hall algebra $\mathcal{H}(T)$ when $T$ is an abelian category. We will also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to $T$. As an example, we describe the derived Hall algebra of an hereditary abelian category.

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arXiv2005

Algebraic and topological aspects of the schematization functor

We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types: co-simplicial Hopf alegbras and equivariant co-simplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor helpful for the description of the schematization of smooth projective complex varieties. In a companion paper these results are used in the construction of a non-abelian Hodge structure on the schematic homotopy type of a smooth projective variety.

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