(Well, it looks like I should stop making promises on this blog. There hasn’t been a single post about spectra yet. I hope that will change before next semester.)

So, today I am going to talk about the formal function theorem. This is more or less a statement that the properties of taking completions and taking cohomologies are isomorphic for proper schemes. As we will see, it is the basic ingredient in the proof of the baby form of Zariski’s main theorem. In fact, this is a very important point: the formal function theorem allows one to make a comparison with the cohomology of a given sheaf over the entire space and its cohomology over an “infinitesimal neighborhood” of a given closed subset. Now localization always commutes with cohomology on non-pathological schemes. However, taking such “infinitesimal neighborhoods” is generally too fine a job for localization. This is why the formal function theorem is such a big deal.

I will give the argument following EGA III here, which is more general than that of Hartshorne (who only handles the case of a projective scheme). The form that I will state today is actually rather plain and down-to-earth. In fact, one can jazz it up a little by introducing formal schemes; perhaps this is worth discussion next time. (more…)

So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field ${F}$ with an absolute value ${\left \lvert . \right \rvert}$ like the rationals with the usual absolute value.

Completions

Definition 1 The completion ${\hat{F}}$ of ${F}$ is defined as the set of equivalence classes of Cauchy sequences:  (more…)

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…)

The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.

The finite presentation trick

Our goal here is:

Theorem 1  Let ${A}$ be a Noetherian ring, and ${I}$ an ideal. If we take all completions with respect to the ${I}$-adic topology,

$\displaystyle \hat{M} = \hat{A} \otimes_A M$

for any f.g. ${A}$-module ${M}$.   (more…)

So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

Rings

The case in which we are primarily interested comes from a ring ${A}$ with a descending filtration (satisfying ${A_0 =A}$), which implies the ${A_i}$ are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal ${I}$ such that ${A_i = I^i}$, i.e. the filtration is ${I}$-adic. We have a completion functor from filtered rings to rings, sending ${A \rightarrow \hat{A}}$. Given a filtered ${A}$-module ${M}$, there is a completion ${\hat{M}}$, which is also a ${\hat{A}}$-module; this gives a functor from filtered ${A}$-modules to ${\hat{A}}$-modules. (more…)

Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

Generalities on Completions

Suppose we have a filtered abelian group ${G}$ with a descending filtration of subgroups ${\{G_i\}}$. Because of this, we can consider “Cauchy sequences” and “convergence:”

Definition 1

The sequence ${\{x_i\} \subset G}$, ${i \in \mathbb{N}}$ is Cauchy if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i,j > N \quad \mathrm{implies} \quad x_i - x_j \in G_A.$

The sequence ${\{y_i\} \subset G}$ converges to ${y}$ if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i>A \quad \mathrm{implies} \quad x_i -y \in G_A.$ (more…)