Algebraic Geometry Over C[infinity]-Rings

If $X$ is a manifold then the $mathbb R$-algebra $Cinfty (X)$ of smooth functions $c:Xrightarrow mathbb R$ is a $Cinfty $-ring. That is, for each smooth function $f:mathbb Rnrightarrow mathbb R$ there is an $n$-fold operation $Phi f:Cinfty (X)nrightarrow Cinfty (X)$ acting by $Phi f:(c1,ldots ,cn)mapsto f(c1,ldots ,cn)$, and these operations $Phi f$ satisfy many natural identities. Thus, $Cinfty (X)$ 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 $Cinfty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $Cinfty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $Cinfty $-schemes, and $Cinfty $-stacks, in particular Deligne-Mumford $Cinfty$-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: $Cinfty$-rings and $Cinfty $-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''.