C^infty - Schemes

DUBUC, EDUARDO J.

AMERICAN JOURNAL OF MATHEMATICS

The Johns Hopkins University Press

Año: 1981 vol. 104 p. 683 - 690

0002-9327

<!-- @page { size: 8.5in 11in; margin: 0.79in } PRE.cjk { font-size: 12pt } P { margin-bottom: 0.08in } --> We are concerned here with the development of techniques, like those of algebraic geometry, but directly applicable to the study of differential geometry. A. Grothendieck in 1957 introduced Affine Schemes, which are a concrete specification of the dual of the category of commutative rings. In this way he was able to glue them together to con- struct geometric objects. This came after the realization that the most general object of study in algebraic geometry was something locally iden- tifiable with an object equal to the formal categorical dual of a commuta- tive ring. Schemes made possible to apply in algebraic geometry the con- cepts of differential geometry connected with infinitely small changes of points on an algebraic variety. For example, the Spec of the ring of dual numbers is an infinitesimal subscheme D of the algebraic line L, and the tangent space at a point x of an algebraic variety X is the set of maps t from D to X such that (0) = x. In particular, LD = L X L. F. W. Lawvere proposed in 1967 an axiomatic approach to the category of schemes, abstracted from these developments, but intended to be applicable to differential geometry. He also set basic guidelines on how to construct models of his axioms. In the mid 70´s, A. Kock, G. Reyes and G. Wraith started the development of this theory, to be called Synthetic Differential Geometry. E. Dubuc in 1978 explicitly demonstrated its applicability to the study of ordinary smooth manifolds by introducing the notion of (fully) well adapted model, and constructing one. The purpose of this note is twofold: First, to incorporate differential geometry into the field of applicability of the techniques and tools of alge- braic geometry; second, to construct a model of Synthetic Differential Geometry sufficiently general and with enough good properties to suggest stronger axioms.