Homotopical algebraic geometry II : geometric stacks and applications / Bertrand Toën, Gabriele Vezzosi.

Format:

Book

Author/Creator:

Toën, Bertrand, 1973- author.

Vezzosi, Gabriele, 1966- author.

Series:

Memoirs of the American Mathematical Society ; no. 902.

Memoirs of the American Mathematical Society, 0065-9266 ; number 902

Language:

English

Subjects (All):

Algebraic stacks.

Algebra, Homological.

Geometry, Algebraic.

Categories (Mathematics).

Physical Description:

1 online resource (242 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category. 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.

Contents:

""Contents""; ""Abstract""; ""Introduction""; ""Reminders on abstract algebraic geometry""; ""The setting""; ""Linear and commutative algebra in a symmetric monoidal model category""; ""Geometric stacks""; ""Infinitesimal theory""; ""Higher Artin stacks (after C. Simpson)""; ""Derived algebraic geometry: D[sup(�)]-stacks""; ""Complicial algebraic geometry: D-stacks""; ""Brave new algebraic geometry: S-stacks""; ""Relations with other works""; ""Acknowledgments""; ""Notations and conventions""; ""Part 1. General theory of geometric stacks""; ""Introduction to Part 1""

""Chapter 1.1. Homotopical algebraic context""""Chapter 1.2. Preliminaries on linear and commutative algebra in an HA context""; ""1.2.1. Derivations and the cotangent complex""; ""1.2.2. Hochschild homology""; ""1.2.3. Finiteness conditions""; ""1.2.4. Some properties of modules""; ""1.2.5. Formal coverings""; ""1.2.6. Some properties of morphisms""; ""1.2.7. Smoothness""; ""1.2.8. Infinitesimal lifting properties""; ""1.2.9. Standard localizations and Zariski open immersions""; ""1.2.10. Zariski open immersions and perfect modules""; ""1.2.11. Stable modules""

""1.2.12. Descent for modules and stable modules""""1.2.13. Comparison with the usual notions""; ""Chapter 1.3. Geometric stacks: Basic theory""; ""1.3.1. Reminders on model topoi""; ""1.3.2. Homotopical algebraic geometry context""; ""1.3.3. Main definitions and standard properties""; ""1.3.4. Quotient stacks""; ""1.3.5. Quotient stacks and torsors""; ""1.3.6. Properties of morphisms""; ""1.3.7. Quasi-coherent modules, perfect modules and vector bundles""; ""Chapter 1.4. Geometric stacks: Infinitesimal theory""; ""1.4.1. Tangent stacks and cotangent complexes""; ""1.4.2. Obstruction theory""

""2.2.6.1. Local systems""""2.2.6.2. Algebras over an operad""; ""2.2.6.3. Mapping D[sup(�)]-stacks""; ""Chapter 2.3. Complicial algebraic geometry""; ""2.3.1. Two HA contexts""; ""2.3.2. Weakly geometric D-stacks""; ""2.3.3. Examples of weakly geometric D-stacks""; ""2.3.3.1. Perfect modules""; ""2.3.3.2. The D-stacks of dg-algebras and dg-categories""; ""2.3.4. Geometric D-stacks""; ""2.3.5. Examples of geometric D-stacks""; ""2.3.5.1. D[sup(�)]-stacks and D-stacks""; ""2.3.5.2. CW-perfect modules""; ""2.3.5.3. CW-dg-algebras""; ""2.3.5.4. The D-stack of negative CW-dg-categories""

""Chapter 2.4. Brave new algebraic geometry""

Notes:

"May 2008, volume 193, number 902 (second of 5 numbers)."

Includes bibliographical references (pages 219-221) and index.

Description based on print version record.

ISBN:

1-4704-0508-3