Description |
1 online resource (v, 139 pages) : illustrations |
Series |
Memoirs of the American Mathematical Society, 0065-9266 ; volume 260, number 1256 |
|
Memoirs of the American Mathematical Society ; no. 1256. 0065-9266
|
Contents |
C[infinity]-rings -- The C[infinity]-ring C[infinity](X) of a manifold X -- C[infinity]-ringed spaces and C[infinity]-schemes -- Modules over C[infinity]-rings and C[infinity]-schemes -- C[infinity]-stacks -- Deligne-Mumford C[infinity]-stacks -- Sheaves on Deligne-Mumford C[infinity]-stacks -- Orbifold strata of C[infinity]-stacks |
Summary |
"If X is a manifold then the R-algebra C[infinity](X) of smooth functions C : X [right arrow] R is a C[infinity]-ring. That is, for each smooth function f : Rn [right arrow] R there is an n-fold operation]Phi]f : C[infinity](X)n [right arrow] C[infinity](X) acting by [Phi]f : (c1 ..., cn) [right arrow] f(c1 ..., cn), and these operations [Phi]f satisfy many natural identities. Thus, C[infinity](X) actually has a far richer structure than the obvious R-algebra structure. We explain the foundations of a version of algebraic geometry in which rings or algebras are replaced by C[infinity]-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C[infinity]-schemes, a category of geometric objects which generalize manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent sheaves on C[infinity]-schemes, and C[infinity]-stacks, in particular Deligne- Mumford C[infinity]-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C[infinity]-rings and C[infinity]-schemes have long been part of synthetic differential geometry. But we develop them in new directions. In Joyce (2014, 2012, 2012 preprint), the author uses these tools to define d-manifolds and d-orbifolds, 'derived' versions of manifolds and orbifolds related to Spivak's 'derived manifolds' (2010)"-- Provided by publisher |
Notes |
"July 2019, Volume 260, Number 1256 (fifth of 5 numbers)." |
|
Title page displays an infinity sign rather than the word "infinity." |
Bibliography |
Includes bibliographical references (pages 131-133) and index |
Notes |
Online resource, title from digital title page (viewed on September 16, 2020) |
Subject |
Differentiable functions.
|
|
Smooth affine curves.
|
|
Rings (Algebra)
|
|
Geometry, Algebraic.
|
|
Geometría algebraica
|
|
Anillos (Álgebra)
|
|
Funciones diferenciables
|
|
Smooth affine curves
|
|
Geometry, Algebraic
|
|
Differentiable functions
|
|
Rings (Algebra)
|
|
Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory, see 49Q15} -- General theory of differentiable manifolds [See also 32Cxx]
|
|
Algebraic geometry -- Foundations -- Generalizations (algebraic spaces, stacks)
|
|
Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx} -- Linear function spaces and their duals [See also 30H05, 32A38, 46F05] {For function algebras, see 46J10}
|
|
Geometry {For algebraic geometry, see 14-XX} -- Distance geometry -- Synthetic differential geometry.
|
Form |
Electronic book
|
ISBN |
1470453363 |
|
9781470453367 |
|
1470436450 |
|
9781470436452 |
|