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Author Joyce, Dominic D., author.

Title Algebraic geometry over C[infinity]-rings / Dominic Joyce
Published Providence, RI, USA : American Mathematical Society, [2019]
©2019

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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
Other Titles Algebraic geometry over C infinity-rings