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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)
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

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Table of Contents


  • 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


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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