Algebraic Geometry over $C^\infty $-Rings

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''.