Algebraic Geometry over C-Infinity Rings (Memoirs of the American Mathematical Society) - Softcover

Joyce, Dominic

Softcover

ISBN 10: 1470436450 ISBN 13: 9781470436452

Publisher: Amer Mathematical Society, 2019

Synopsis

If $X$ is a manifold then the $mathbb R$-algebra $Cinfty (X)$ of smooth functions $ mathbb R$ is a $Cinfty $-ring. That is, for each smooth function $ Rnrightarrow mathbb R$ there is an $n$-fold operation $Phi (X)nrightarrow Cinfty (X)$ acting by $Phi ,cn)mapsto f(c1,ldots ,cn)$, and these operations $Phi f$ satisfy many natural identities. Thus, $Cinfty (X)$ 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 $Cinfty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $Cinfty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $Cinfty $-schemes, and $Cinfty $-stacks, in particular Deligne-Mumford $Cinfty$-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not $Cinfty$-rings and $Cinfty $-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''.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2019
• Language: English
• ISBN 10: 1470436450
• ISBN 13: 9781470436452
• Binding: Paperback
• Number of pages: 139