One of the standard facts in algebraic geometry is that a projective scheme is proper. In the language of varieties, one says that the image of a projective variety is closed. The precise statement one proves is that:

Theorem 1 Let ${Y}$ be any variety over the algebraically closed field ${k}$. Let ${Z \subset \mathbb{P}^n_k \times Y}$ be a closed subset. Then the projection of ${Z}$ to ${Y}$ is closed.

This statement is sometimes phrased as saying that ${\mathbb{P}_k^n}$ is “complete.” In many ways, it is a compactness statement. Recall that a compact space ${X}$ has the property that the map ${X \times Y \rightarrow Y}$ for any ${Y}$ is a closed map. The converse is also true if there are reasonable assumptions (I think locally compact Hausdorff on ${X}$ will do it). Of course, these reasonable assumptions don’t apply to a variety with the Zariski topology, but they do if we are working with a variety (say, a quasiprojective variety) over ${\mathbb{C}}$ and we can define the complex topology. And in fact, it turns out that projective varieties are indeed compact in the complex topology. The scheme-theoretic version is a little different. First, the theorem for varieties said that the object ${\mathbb{P}_k^n}$ was “complete” in a certain sense. But the philosophy of Grothendieck is to consider not so much schemes but morphisms of schemes, and to do everything in a relative context. The idea is:

Definition 2 A morphism ${f: X \rightarrow Y}$ is proper if it is separated, of finite type, and universally closed (i.e. any base change is a closed morphism).

I don’t really want to define separated in this post. On the other hand, I should explain what’s going on for quasiprojective varieties over an algebraically closed fields. In this case, the conditions of finite type and separated are redundant. The key condition is universal closedness, which won’t be satisfied for a general morphism. For instance, to say that ${\mathbb{P}^n_k \rightarrow \mathrm{Spec} k}$ is universally closed implies the first lemma about closedness of the projection.
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I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. But this tidbit from EGA 1 that I just learned today was too much fun to resist. Besides, it puts the material on completions in more context, so I think the digression is justified.

Lifting Idempotents

The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:

Theorem 1 Let ${A}$ be a ring complete with respect to the ${I}$-adic filtration. Then if ${\bar{e} \in A/I}$ is idempotent (i.e. ${\bar{e}^2=\bar{e}}$) then there is an idempotent ${ e \in A}$ such that ${e}$ reduces to ${\bar{e}}$  (more…)

There is a useful fact in algebraic geometry that if you have a coherent sheaf over a Noetherian integral scheme, then it is locally free on some dense open subset. That is the content of today’s post, although I will use the language of commutative algebra than that of schemes (except at the end), to keep the presentation as elementary as possible. The goal is to get the generic freeness in a restricted case. Later, I’ll discuss the full “generic freeness” lemma of Grothendieck.

Noetherian Rings and Modules

All rings are assumed commutative in this post.

As I have already mentioned, a ring is Noetherian if each ideal of ${A}$ is finitely generated. Similarly, a module is Noetherian if every submodule is finitely generated. I will summarize the basic facts below briefly.