Geometry working seminar

Talk 1. Mohamed El Alami - 1:00pm

Pure Mathematics Department, University of Waterloo

“Introduction to schemes, part 2”

The theory of sheaves that was presented in the first talk will be used to introduce ringed spaces, locally ringed spaces and their morphisms. The structure sheaf on Spec R will be revisited to reveal some of its properties that will motivate the definition of schemes. The last part of the talk will cover some examples and properties of schemes.

Talk 2. Ehsaan Hossain - 2:30pm

Pure Mathematics Department, University of Waterloo

“Projective modules, part 2”

Last time we learned what it means for a module to be projective. We also established this fact: a finitely-presented R-module P is projective if and only if the localized module Pm = Rm ⊗R P is a free latexRm-module for every maximal ideal m. Thus projective modules are “locally free” in a specific sense. In this talk we’ll see how this connects to local freeness for sheaves of modules on (X,OX), or local triviality for vector bundles.