Today I’ll continue the series on graded rings and filtrations by discussing the resulting topologies and the Artin-Rees lemma.
All filtrations henceforth are descending.
Recall that a topological group is a topological space with a group structure in which the group operations of composition and inversion are continuous—in other words, a group object in the category of topological spaces.
So, let’s now consider a filtered module over some filtered ring with a descending filtration. Recall that this means that we have descending subgroups such that for all , .
We can actually generalize this as follows:
Definition 1 A filtered abelian group is an abelian group with a sequence of subgroups such that , for .
(Alternatively this is a filtered module over , with the filtration for .)
Given a filtered additive commutative group , we can give it a topology by taking a base at to be . It can directly be checked that addition and inversion are continuous operations since the are subgroups: for instance, the inverse image of under the addition map contains the open set .
In general, however, the topology will be Hausdorff if and only if
if this is true, then given , we can find some with so is Hausdorff, and the argument can be reversed.
Now return to the case of a filtered module over a filtered ring . As above, and have topologies. Moreover, the condition implies for instance that the map
is continuous as well.
The result I am aiming for today is the Artin-Rees lemma, which states under suitable conditions, if you pick an ideal in a ring , -modules , then the -adic topology on (from the filtration ) is induced by the -adic topology on . But first, we need to talk more about a variant of .
The Artin-Rees Lemma
The variant of I was talking about is the blowup algebra or Rees algebra defined as
where multiplication of is . This is now a graded algebra. Similarly, given a filtered module , we can make into a graded module over the blowup algebra. Then is functorial, from the category of filtered rings to the category of graded rings (resp. from the category of filtered modules to the category of graded modules over ).
The case we care about most is with the filtration or . In this case we denote by , following Eisenbud.
Proposition 2 If is Noetherian and an ideal, the blowup algebra is Noetherian.
Indeed, the algebra is ; then if generate as an ideal, it follows that generate as an -algebra, so the blowup algebra is Noetherian by Hilbert’s basis theorem.
Now let be a filtered -module, where is given the -adic filtration.
Proposition 3 Let be a Noetherian ring. Give the -adic filtration, and let be a finitely generated and filtered -module. The -module is finitely generated if and only if the filtration satisfies for sufficiently large (this means the filtration is -stable; recall that we already have the inclusion ).
If the filtration is -stable, then let be large enough that implies
Then choose generating sets such that each generates as an -module. Then consider the generating set consisting of all the in the -th homogeneous slot. Then generates as an -module, hence as an -module. By (1), generates as an -module.
Conversely, if is finitely generated, then some subset
generates as an -module; by the definition of , it follows that , and similarly it follows more generally that for .
Now we move to proving the Artin-Rees lemma.
Theorem 4 (Artin-Rees) Let be a Noetherian ring, an ideal, a finitely generated -module, and a submodule. If is an -stable filtration on , then is -stable too.
Indeed, considering as filtered -modules (where has the -adic filtration), we find is finitely generated over ; thus is too, and the filtration on (that is ) is thus -stable.
Corollary 5 The -adic topology on is that induced by the -adic topology on .
The topology on induced by the -adic topology on can be described by intersecting a basis at zero for with : that is, we take as a basis at . We have to show this basis is equivalent to .
By the previous theorem, the filtration is -stable. Thus for large , we have
Thus the basis is finer than . Conversely, the basis is indexwise smaller so finer than .
There is a nice consequence of this:
Corollary 6 (Krull Intersection Theorem)Let be a Noetherian local ring with maximal ideal . Then if is a finitely generated -module
Let be the submodule given by the intersection. Then the -adic topology on is given by restriction of the -adic topology on . But each element of the basis at zero in contains , so the basis at zero for is just itself, i.e. has the indiscrete -adic topology. This means just , so by Nakayama we have .
Next I would like to talk about completions. Artin-Rees plays a key role in showing that completion is an exact functor.