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Memoirs of the American Mathematical Society Ser.: Algebraic Geometry over $C^ Infty $-Rings by Dominic Joyce (2019, Trade Paperback)

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About this product

Product Identifiers

PublisherAmerican Mathematical Society

ISBN-101470436450

ISBN-139781470436452

eBay Product ID (ePID)7038676993

Product Key Features

Number of Pages140 Pages

Publication NameAlgebraic Geometry over $C^ Infty $-Rings

LanguageEnglish

Publication Year2019

SubjectGeometry / Algebraic

TypeTextbook

Subject AreaMathematics

AuthorDominic Joyce

SeriesMemoirs of the American Mathematical Society Ser.

FormatTrade Paperback

Dimensions

Item Height0.8 in

Item Weight8 Oz

Item Length10 in

Item Width7 in

Additional Product Features

Intended AudienceScholarly & Professional

LCCN2019-033051

Series Volume Number260

IllustratedYes

Table Of ContentIntroduction $C^\infty$-rings The $C^\infty$-ring $C^\infty (X)$ of a manifold $X$ $C^\infty $-ringed spaces and $C^\infty $-schemes Modules over $C^\infty$-rings and $C^\infty $-schemes $C^\infty $-stacks Deligne-Mumford $C^\infty $-stacks Sheaves on Deligne-Mumford $C^\infty $-stacks Orbifold strata of $C^\infty $-stacks Appendix A. Background material on stacks Bibliography Glossary of Notation Index.

SynopsisIf $X$ is a manifold then the $\mathbb R$-algebra $C^\infty (X)$ of smooth functions $c:X\rightarrow \mathbb R$ is a $C^\infty $-ring. That is, for each smooth function $f:\mathbb R^n\rightarrow \mathbb R$ there is an $n$-fold operation $\Phi _f:C^\infty (X)^n\rightarrow C^\infty (X)$ acting by $\Phi _f:(c_1,\ldots ,c_n)\mapsto f(c_1,\ldots ,c_n)$, and these operations $\Phi _f$ satisfy many natural identities. Thus, $C^\infty (X)$ actually has a far richer structure than the obvious $\mathbb R$-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^\infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^\infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $C^\infty $-schemes, and $C^\infty $-stacks, in particular Deligne-Mumford $C^\infty$-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: $C^\infty$-rings and $C^\infty $-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''., Explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^\infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^\infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps.

LC Classification NumberQA614.3.J69 2019