Homotopical Algebraic Geometry II: Geometric Stacks and Applications
Title | Homotopical Algebraic Geometry II: Geometric Stacks and Applications PDF eBook |
Author | Bertrand Toën |
Publisher | American Mathematical Soc. |
Pages | 242 |
Release | 2008 |
Genre | Mathematics |
ISBN | 0821840991 |
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.
Homotopical Algebraic Geometry II
Title | Homotopical Algebraic Geometry II PDF eBook |
Author | Bertrand Toën |
Publisher | |
Pages | 242 |
Release | 2014-09-11 |
Genre | Algebra, Homological |
ISBN | 9781470405083 |
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.
Algebraic Geometry over C∞-Rings
Title | Algebraic Geometry over C∞-Rings PDF eBook |
Author | Dominic Joyce |
Publisher | American Mathematical Soc. |
Pages | 139 |
Release | 2019-09-05 |
Genre | |
ISBN | 1470436450 |
If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,…,cn), and these operations Φf satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞ -schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.
Simplicial Methods for Operads and Algebraic Geometry
Title | Simplicial Methods for Operads and Algebraic Geometry PDF eBook |
Author | Ieke Moerdijk |
Publisher | Springer Science & Business Media |
Pages | 186 |
Release | 2010-12-01 |
Genre | Mathematics |
ISBN | 3034800525 |
"This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures delivered at the Centre de Recerca Matemàtica in February 2008, as part of a special year on Homotopy Theory and Higher Categories"--Foreword
Higher Categories and Homotopical Algebra
Title | Higher Categories and Homotopical Algebra PDF eBook |
Author | Denis-Charles Cisinski |
Publisher | Cambridge University Press |
Pages | 449 |
Release | 2019-05-02 |
Genre | Mathematics |
ISBN | 1108473202 |
At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.
Derived Algebraic Geometry
Title | Derived Algebraic Geometry PDF eBook |
Author | Renaud Gauthier |
Publisher | Walter de Gruyter GmbH & Co KG |
Pages | 489 |
Release | 2024-01-29 |
Genre | Mathematics |
ISBN | 311133421X |
C?-Algebraic Geometry with Corners
Title | C?-Algebraic Geometry with Corners PDF eBook |
Author | Kelli Francis-Staite |
Publisher | Cambridge University Press |
Pages | 223 |
Release | 2023-12-31 |
Genre | Mathematics |
ISBN | 1009400169 |
Crossing the boundary between differential and algebraic geometry in order to study singular spaces, this book introduces 'C∞-schemes with corners'.