Descent in algebra, geometry, and topology

Descent in algebra, geometry, and topology

Thu, 16/10/2008
12:00 		Oscar Randal-Williams (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Descent in algebra, geometry, and topology
Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data. In algebraic geometry, faithfully flat descent says that if $ X\rightarrow Y $ is a faithfully flat morphism of schemes, then giving a sheaf on $ Y $ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $ X $, satisfying certain compatibility conditions. Translated to algebra, it says that if $ S\rightarrow R $ is a faithfully flat morphism of rings, then giving an $ S $-module is the same as giving a certain simplical module over a simplicial ring constructed from $ R $. In topology, given an etale cover $ X\rightarrow Y $ one can recover $ Y $ (at least up to homotopy equivalence) from a simplical space constructed from $ X $.