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Algebraic Geometry over $C^\infty$-rings
About this Title
Dominic Joyce, The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 260, Number 1256
ISBNs: 978-1-4704-3645-2 (print); 978-1-4704-5336-7 (online)
DOI: https://doi.org/10.1090/memo/1256
Published electronically: July 17, 2019
Keywords: $C^\infty$-ring,
$C^\infty$-scheme,
$C^\infty$-stack,
quasicoherent sheaf,
manifold,
orbifold
MSC: Primary 58A40; Secondary 14A20, 46E25, 51K10
Table of Contents
Chapters
- 1. Introduction
- 2. $C^\infty$-rings
- 3. The $C^\infty$-ring $C^\infty (X)$ of a manifold $X$
- 4. $C^\infty$-ringed spaces and $C^\infty$-schemes
- 5. Modules over $C^\infty$-rings and $C^\infty$-schemes
- 6. $C^\infty$-stacks
- 7. Deligne–Mumford $C^\infty$-stacks
- 8. Sheaves on Deligne–Mumford $C^\infty$-stacks
- 9. Orbifold strata of $C^\infty$-stacks
- A. Background material on stacks
Abstract
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.
We explain 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. We also study 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 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).
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