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Homotopical Algebraic Geometry II: Geometric Stacks and Applications
Bertrand Toën, Université Paul Sabatier, Toulouse, France, and Gabriele Vezzosi, Università di Firenze, Italy
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Memoirs of the American Mathematical Society
2008; 224 pp; softcover
Volume: 193
ISBN-10: 0-8218-4099-1
ISBN-13: 978-0-8218-4099-3
List Price: US$81
Individual Members: US$48.60
Institutional Members: US$64.80
Order Code: MEMO/193/902
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This is the second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category \(C\), and prove that this notion satisfies the expected properties.

Table of Contents

  • Introduction
Part 1. General theory of geometric stacks
  • Introduction to Part 1
  • Homotopical algebraic context
  • Preliminaries on linear and commutative algebra in an HA context
  • Geometric stacks: Basic theory
  • Geometric stacks: Infinitesimal theory
Part 2. Applications
  • Introduction to Part 2
  • Geometric \(n\)-stacks in algebraic geometry (after C. Simpson)
  • Derived algebraic geometry
  • Complicial algebraic geometry
  • Brave new algebraic geometry
  • Appendix A. Classifying spaces of model categories
  • Appendix B. Strictification
  • Appendix C. Representability criterion (after J. Lurie)
  • Bibliography
  • Index
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