Algebraic Geometry over C-infinity rings

Details Algebraic Geometry over C-infinity rings Algebraic Geometry over C-infinity rings

2009-12-31

arxiv.org/abs/1001.0023v2

Mathematics

Algebraic Geometry

(v2) 93 pages. New material added

Scientific paper

If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation \Phi_f : C^\infty(X)^n --> C^\infty(X) acting by \Phi_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations \Phi_f satisfy many natural identities. Thus, C^\infty(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-infinity rings. As schemes are the basic objects in algebraic geometry, the new basic objects are "C-infinity schemes", a category of geometric objects which generalize smooth manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent and coherent sheaves on C-infinity schemes, and "C-infinity stacks", in particular Deligne-Mumford C-infinity stacks, a 2-category of geometric objects generalizing orbifolds. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems. It is part of a project to develop a theory of "derived differential geometry", with applications to areas of symplectic geometry involving moduli spaces of J-holomorphic curves. Many of these ideas are not new: C-infinity rings and C-infinity schemes have long been part of synthetic differential geometry. But we develop them in new directions. This paper is surveyed in arXiv:1104.4951.

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