Algebraic Geometry over 𝐶^{∞}-rings

Abstract

If X is a manifold then the R-algebra C ∞ (X) of smooth functions c : X → R is a C ∞ -ring. That is, for each smooth function f : R n → R there is an n-fold operation Φ f : C ∞ (X) n → C ∞ (X) acting by Φ f : (c1, . . . , cn) → f (c1, . . . , cn), and these operations Φ f satisfy many natural identities. Thus, C ∞ (X) actually has a far richer structure than the obvious R-algebra structure. We explain the foundations of a version of algebraic geometry in which rings or algebras are replaced by C ∞ -rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C ∞ -schemes, a category of geometric objects which generalize manifolds, and whose mor-phisms generalize smooth maps. We also study quasicoherent sheaves on C ∞ -schemes, and C ∞ -stacks, in particular Deligne–Mumford C ∞ -stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C ∞ -rings and C ∞ -schemes have long been part of synthetic differential geometry. But we develop them in new directions. In [36–38], the author uses these tools to define d-manifolds and d-orbifolds, 'derived' versions of manifolds and orbifolds related to Spivak's 'derived manifolds' [64].