eBook

Algebraic geometry over C∞-rings

Joyce, Dominic D.

2019

Overview

If 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\".

• Publisher: American Mathematical Society
• Subject: Differentiable manifolds / Geometry, Algebraic / Rings (Algebra) / Smooth affine curves
• ISBN: 1470436450, 9781470436452