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!

Continue reading «Flat modules»

Hi, I was reading in

Jet Nestruev: Smooth Manifolds and Observables, Springer, 2003

about a month ago (after I stumbled over a question on MO) and there was an exercise that resisted solution for more than a week.

Well.... now I found out that I have just misread the exercise. However, this way I basically did several exercises at once. Here comes the problem and its solution:

The problem

(inspired by page 28, chapter 3, exercise 3.17.5 in Nestruev)

Find a smooth (real) manifold $M$ such that its algebra of smooth functions $C^\infty(M,\mathbb R)$ is isomorphic to the algebra of all smooth functions $f : \mathbb R \to \mathbb R$ that have some rational period $\tau$ (i.e. there exists $\tau \in \mathbb Q$ such that $f(x)=f(x+\tau)$ for all x). Note that we don't fix a period $\tau$ here. Let's call the algebra in question (smooth functions on the real line with some rational period) $A$.

You might want to stop reading here and think for a second (or minutes) about the solution or similar problems that have easier solutions. A more vague problem would be

Find a space $M$ such that the functions $M \to \mathbb R$ correspond to functions $\mathbb R \to \mathbb R$ that are periodic with some rational period.

Continue reading «A manifold whose functions are the smooth functions on the real line with rational period»