# Derived L-Adic Categories for Algebraic Stacks

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American Mathematical Society
Algebraic geometry, Algebraic topology, Algebra - Linear, Mathematics, Algebra, Homological, Algebraic stacks, Moduli theory, Science/Mathem
The Physical Object ID Numbers Series Memoirs of the American Mathematical Society, No. 774 Format Mass Market Paperback Open Library OL11420020M ISBN 10 0821829297 ISBN 13 9780821829295

Derived category. (The equivariant case is the case of quotient stacks.) The simi-larity can be seen from their Appendix B, where a description of their category is given, which is formally close to our category. We construct an ℓ-adic derived category for stacks, called D+ c (X,Qℓ), where cCited by: Get this from a library.

Derived l-adic categories for algebraic stacks. [K Behrend]. Get this from a library. Derived l-adic categories for algebraic stacks. [K Behrend] -- Introduction The $\ell$-adic formalism Stratifications Topoi Algebraic stacks Convergent complexes Bibliography.

We construct an ℓ-adic formalism of derived categories for algebraic stacks. Over finite fields we generalize the theory of mixed complexes to a theory of so called convergent complexes.

Geometricity for dg enhancements of algebraic stacks 25 Appendix A. Bounded derived category of coherent modules 28 References 29 1. Introduction The derived category of a variety or, more generally, of an algebraic stack, is traditionally studied in the context of triangulated categories. Although triangu.

Behrend, Kai A., Derived -adic categories for algebraic stacks Behrend, Kai and Conrad, Brian and Edidin, Dan and Fantechi, Barbara and Fulton, William and Göttsche, Lothar and Kresch, Andrew, Algebraic stacks Behrend, Kai and Fantechi, Barbara, The intrinsic normal cone.

Preprints: Algebraic Stacks • Derived l-adic categories for algebraic stacks This is work I did while I was a postdoc at MIT. I construct l-adic derived categories for algebraic stacks defined over a finite field. I define pushforward and pullback functors between these categories.

The key notion is that of convergent complex. EXAMPLES OF ALGEBRAIC STACKS 3 8. GIT-stacks. De nition Let k be a eld. A GIT-stack1 Xover k is a quotient stack of the form X= [Xss=G] where G is a reductive linear algebraic group over k, and Xss is the semi-stable locus for an action of G on a is a projective scheme (X;O.

The book discusses fibered categories, stacks, torsors and gerbes over general: sites but does not discuss algebraic stacks. For instance, if $G$ is a sheaf: of abelian groups on $X$, then in the same way $H^ 1 (X, G)$ can be identified: with $G$-torsors, $H^ 2 (X, G)$ can be identified with an appropriately defined: set of $G$-gerbes.

### Details Derived L-Adic Categories for Algebraic Stacks PDF

$\begingroup$ Stacks generalize sheaves, fibered categories (equivalently pseudofunctors) generalize presheaves (contravariant functors [into Sets]). The key key idea behind stacks is not only the generalization of functors, but also a generalization of glueing. I disagree with the idea that algebraic spaces should be learned first for the following reason: all of the 2-categorical.

Approximately ten years ago C. Simpson introduced in [S3] a notion of algebraic n-stack, and more recently notions of derived scheme and of derived algebraic n-stacks have been introduced in [To-Ve3, HAGII, Lu1].

The purpose of this text is to give an overview on the recent works on the theories of higher algebraic stacks and of higher derived. We also include a detailed study of the intersection cohomology of algebraic stacks and their associated moduli spaces. Smooth group scheme actions on singular varieties and the associated derived category turn up as special cases of the more general results on algebraic stacks.

Then the category of lisse ℓ-adic sheaves on X is equivalent to the category of continuous representations of (,) on finite free -modules. This is an analog of the correspondence between local systems and continuous representations of the fundament group in algebraic topology (because of this, a lisse ℓ-adic sheaf is sometimes also called a.

K. Behrend, Derived l-Adic Categories for Algebraic Stacks, Mem. Amer. Math. Soc., vol.no.Amer. Math. Soc., Providence, RI, 2. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I (Luminy, ), Astérisque, vol.pp. 5–, Soc.

Math.

### Description Derived L-Adic Categories for Algebraic Stacks EPUB

France, Paris, A Study in Derived Algebraic Geometry Volume I: Correspondences and Duality Dennis Gaitsgory Now, even though, as was mentioned above, for a usual Lie algebra g,the categories QCoh(B(exp(g))) which we consider quasi-coherent sheaves, the latter being schemes (or algebraic stacks).

Anaturalclassofalgebro. smooth and proper algebra, the e´tale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories. Key Words Commutative ring spectra, derived algebraic geometry, moduli spaces, Azumaya algebras, and Brauer groups.

This appears as (deJong, lemmatheorem ).Properties. Tannaka duality for geometric stacks.; Examples. Orbifolds are an example of an Artin orbifolds the stabilizer groups are finite groups, while for Artin stacks in general they are algebraic groups.

Generalizations Noncommutative spaces. What is (derived) algebraic geometry about.

1 What is done in this Chapter beyond the de nition. 3 1. The category of quasi-coherent sheaves 4 Setting up the theory of quasi-coherent sheaves 4 Basic properties of QCoh 5 Descent 6 Quasi-coherent sheaves on Artin stacks 7 The t-structure 9 2.

Direct image for. The Intersection Cohomology and Derived Category of Algebraic Stacks. equivariantly perfect and from it deduce that each of the l-adic cohomology groups of. give an introduction to algebraic stacks with a particular emphasis on moduli stacks of vector bundles on algebraic curves.

The main goal was to present recent joint work with Ulrich Stuhler on the ac-tions of the various Frobenius morphisms on the l-adic cohomology of the moduli stack of vector bundles of xed rank and degree on an. Lectures on algebraic stacks Alberto Canonaco Abstract.

These lectures give a detailed and almost self-contained introduction to algebraic stacks. A great part of the paper is devoted to preliminary technical topics, both from category theory (like Grothendieck topologies, bred categories and stacks) and algebraic geometry (like faithfully at.

Note that the converse is not true: some random Fourier-Mukai transform (i.e. some random choice of the sheaf $\mathcal{P}$) is probably not a derived equivalence. I think Huybrechts' book "Fourier-Mukai transforms in algebraic geometry" is a good book to look at.

We develop the notion of strati ability in the context of derived categories and the six operations for stacks in [26, 27]. Then we reprove Behrend’s Lefschetz trace formula for stacks, and give the meromorphic continuation of the L-series of F q-stacks. We give an upper bound for the weights of the cohomology groups of stacks, and as an.

Papers in the literature. Below is a list of research papers which contain fundamental results on stacks and algebraic spaces.

The intention of the summaries is to indicate only the results of the paper which contribute toward stack theory; in many cases these results are subsidiary to the main goals of the paper.

Behrend, Kai (), "Derived l-adic categories for algebraic stacks", Memoirs of the American Mathematical Society, Laumon, Gérard; Moret-Bailly, Laurent (), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. On l-adic Cohomology of Artin stacks: L-functions, Weights, and the Decomposition theorem We develop the notion of stratifiability in the context of derived categories and the six operations for stacks in the work of Laszlo and Olsson.

and give a comparison between the lisse-etale topos of a complex algebraic stack and the lisse. Behrend, K. Derived l-adic categories for algebraic stacks. Memoirs of the American Mathematical Society Vol.

Laumon, Gérard; Moret-Bailly, Laurent (), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. Folge. Introduction to derived algebraic geometry Bertrand To en Our main goal throughout these lectures will be the explicate the notion of a derived Artin stack.

We will avoid homotopy theory wherever possible. Throughout, we will keep the following conventions: Everything will be over a base eld kof characteristic 0.

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent.

This book is an introduction to the theory of algebraic spaces and stacks intended for graduate students and researchers familiar with algebraic geometry at the level of a first-year graduate course.

The first several chapters are devoted to background material including chapters on Grothendieck topologies, descent, and fibered s: 1.

We also study quasicoherent sheaves on -schemes, and -stacks, in particular DeligneâĂŞMumford -stacks, a 2-category of geometric objects generalizing of these ideas are not new: rings and -schemes have long been part of synthetic differential geometry. But we develop them in new directions.an open source textbook and reference work on algebraic geometry.

Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays. This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the Reviews: 1.