Saturday, October 30th, 2010 | Author: Konrad Voelkel
What is a flat module? How should I think of it?
To answer that question, I will provide some background, then define what a flat module is, clarify the definition by means of example and counter-example and finally show some nice and useful properties which you can memorize later by doing exercises. If you're lost, take a look at the references below.
Some far-fetched motivation to understand flat modules:
Flat modules are the "local" model for flat morphisms of schemes. Flatness is an essential part of the definition of étale morphisms. Etale morphisms are used in the definition of étale cohomology, which was used by Deligne 1974 to prove an analogue of the Riemann hypothesis over finite fields. The proof of the Riemann hypothesis over finite fields finished the proof of the Weil conjectures, some of the most influential conjectures in algebraic geometry.
You think you already know flat modules are? Then look if you can do all the exercises!