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Homotopical Algebraic Geometry II: Geometric Stacks and Applications

Bertrand Toën Université Paul Sabatier, Toulouse, France
Gabriele Vezzosi Università di Firenze, Firenze, Italy
Available Formats:
Electronic ISBN: 978-1-4704-0508-3
Product Code: MEMO/193/902.E
224 pp
List Price: $86.00 MAA Member Price:$77.40
AMS Member Price: $51.60 Click above image for expanded view Homotopical Algebraic Geometry II: Geometric Stacks and Applications Bertrand Toën Université Paul Sabatier, Toulouse, France Gabriele Vezzosi Università di Firenze, Firenze, Italy Available Formats:  Electronic ISBN: 978-1-4704-0508-3 Product Code: MEMO/193/902.E 224 pp  List Price:$86.00 MAA Member Price: $77.40 AMS Member Price:$51.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1932008
MSC: Primary 14; 18; 55;

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.

• Chapters
• Introduction
• Part 1. General theory of geometric stacks
• Introduction to Part 1
• 1.1. Homotopical algebraic context
• 1.2. Preliminaries on linear and commutative algebra in an HA context
• 1.3. Geometric stacks: Basic theory
• 1.4. Geometric stacks: Infinitesimal theory
• Part 2. Applications
• Introduction to Part 2
• 2.1. Geometric $n$-stacks in algebraic geometry (after C. Simpson)
• 2.2. Derived algebraic geometry
• 2.3. Complicial algebraic geometry
• 2.4. Brave new algebraic geometry
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Volume: 1932008
MSC: Primary 14; 18; 55;

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.

• Chapters
• Introduction
• Part 1. General theory of geometric stacks
• Introduction to Part 1
• 1.1. Homotopical algebraic context
• 1.2. Preliminaries on linear and commutative algebra in an HA context
• 1.3. Geometric stacks: Basic theory
• 1.4. Geometric stacks: Infinitesimal theory
• Part 2. Applications
• Introduction to Part 2
• 2.1. Geometric $n$-stacks in algebraic geometry (after C. Simpson)
• 2.2. Derived algebraic geometry
• 2.3. Complicial algebraic geometry
• 2.4. Brave new algebraic geometry
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