locally affine space

A τ\tau-locally affine space is a sheaf of sets on the category AffAff (affine schemes) opposite to the category of commutative unital rings in some subcanonical Grothendieck topology τ\tau. If τ\tau is the Zariski topology, then τ\tau-locally affine spaces are schemes. Other examples are sheaves in fppf, fpqc, étale or lisse (=smooth) topology. The relative version uses the category of relative affine schemes Aff/XAff/X which is, in the case of X=Spec(A)X=Spec(A) where AA is a commutative ring, simply the opposite of the category of AA-algebras.

  • Donald Knutson, Algebraic Spaces, LNM 203, Springer 1971

In noncommutative algebraic geometry, one can consider NAffNAff, that is the opposite to the category of all unital associative rings instead of AffAff. There is a noncommutative smooth topology on NAffNAff (M. Konstevich, A. L. Rosenberg, Noncommutative smooth spaces, arXiv:math/9812158), which is a genuine Grothendieck topology. On the other hand, one can not obtain a Grothendieck topology using exact affine lcoalizations: the stability axiom of Grothendieck topologies fails. However some other generalizations of topologies on NAffNAff are available for the noncommutative case (e.g. Q-categories and “quasi-topologies” of Rosenberg) or one can instead of working with sheaves on the big site NAffNAff work with noncommutative generalization of small cotopologies on rings (cf. work of van Oystaeyen on “noncommutative topologies”).

Revised on January 7, 2010 22:25:32 by Zoran Škoda (