Fix a noetherian local ring .
Let (for the derived category of , or preferably its higher-categorical analog). Let us define the local cohomology functor
We can think of this at a number of levels: for instance, it is the (derived) functor of the ordinary functor on -modules which sends an -module to its submodule
of -power torsion elements. From this point of view, we can think of the cohomology groups
as defining “cohomology with supports” for the pair with coefficients in the sheaf . I’ll try to elaborate more on this point of view later.
Notation: The derived categories in this post will use cohomological grading conventions, for simplicity.
Our first goal here is to describe the calculation (which is now quite formal) of in the regular case, in terms of dualizing objects. So, let’s suppose is regular local on, of dimension . In this case, each lives in the smaller perfect derived category , and we will use the duality in that category.
Namely, recall that we have a functor given by , which induces a duality on the perfect derived category of , as we saw yesterday.
Let . We saw in the previous post that is cohomologically concentrated in the degree , and it is a shift of the module : we saw that was the injective envelope of . The next result will reduce the computation of to an computation.
Theorem 5 (Local duality) If and is regular, then we have a canonical isomorphism in ,
Proof: In fact, we can do this using purely formal manipulations. We have:
In particular, since is an injective module, we can now calculate the cohomology groups ,
In the previous post, we saw that the functor defined a duality on the category of finite length -modules. In general, the functor can be defined for any -module, though it is no longer a duality. We will denote it by ; it is an exact functor. We can thus state the special case of the above calculation for a single module.
Corollary 6 (Local duality) For a finitely generated -module over the regular local ring , there is a canonical isomorphism for all ,
2. Definitions of local cohomology
This is a powerful result. Although we have not said much about the functors , they have a number of interpretations:
- is a special case of a sheaf-theoretic construction of “cohomology with supports:” that is, given a space and a closed subspace , one can consider the functor
sending a sheaf to its group of sections supported on . The derived functors of are the local cohomology functors, and a special case of them (for ) is given by the above. From this point of view, the are analogous to relative cohomology groups.
- can be computed using a Koszul-type complex tensored with , and is related to questions of depth. Namely, choose generators of (or, more generally, of a power of ). For , let be the complex
Then one has an isomorphism
These complexes are colimits of Koszul complexes, so the depth of can be used to show vanishing of the groups . This definition has, intriguingly, been generalized to spectra, though I don’t know much about it.
- In the case of a regular local ring, we have (as we just saw) a third interpretation as the dual to an group. The case of a general local ring can often be reduced to this case: passage to completion does not affect local cohomology, and any complete local ring is a quotient of a regular local ring.This interpretation has the disadvantage of involving the somewhat inexplicit duality functor . Nonetheless, in particular examples duality functors can often be written down explicitly. For instance, for , we can take as the dualizing module.
Playing these various ideas off each other leads to interesting results; I’ll try to describe some in the next few posts. It’s not really my intention to spend time on this blog proving the above equivalences: there are a number of references that do this very well, and I think blogging tends to work better when the key ideas are highlighted. So, we’ll freely use either of the above definitions.
3. Vanishing and non-vanishing
Let be a local ring (not necessarily regular) of dimension . Let be any -module, non-zero. We have the following fundamental vanishing and non-vanishing theorem.
Theorem 7 (Grothendieck) Let be a finitely generated module over . The groups are zero for , and are nonzero at the two endpoints.
In particular, a Cohen-Macaulay module (e.g., a regular local ring over itself) has only one local cohomology group that does not vanish.
Let’s start with the bound
The rest of the bounds will be sketched in later posts.
By replacing by (which does not affect the local cohomology groups), we can assume has support on all of , and thus reduce to the vanishing assertion
There are several ways we could prove this:
A classical vanishing theorem of Grothendieck asserts that on an -dimensional noetherian space , we have for and any sheaf of abelian groups on . The local cohomology groups can be reconstructed from sheaf cohomology: that is, we can use the interpretation of “cohomology with supports.”
Here is the functor which assigns to a sheaf the group of sections supported on . Using this exact sequence, we find that if has dimension , then for . Since local cohomology can be described in this way (as the for ), the vanishing result (1) follows.
The ring has dimension , so there exist which generate an ideal whose radical is : that is, they form a system of parameters for . Then, the Koszul-type complex described earlier can be used to compute ; since this complex has length , we get the result.
Finally, we can also prove vanishing using local duality. First, we can replace with its completion. This will change nothing, since is flat over , and the local cohomology modules are all -power torsion anyway (so tensoring up to has no effect). In particular,
Thus, we can assume complete at the outset. It follows that is a quotient of a complete regular local ring (by Cohen’s structure theorem). We may as well compute the over instead of . In fact, we can just throw out the old and assume that it is a complete regular local ring to begin with, though we then lose the hypothesis and have to prove the stronger statement. In other words, we have to show that if is regular local (even complete) and a finitely generated -module, then
We can prove (2) by induction on . When , it suffices to show that
for of finite length over . We can reduce by exact sequences to the case where , in which case we can use the Koszul resolution of by choosing a regular system of parameters for . Then self-duality of the Koszul complex, together with the fact that is regular, gives the result.
Now, suppose . Suppose is regular on (see below); then we have a short exact sequence
and the dimension of is one less. We have an exact sequence
When , induction shows that the group to the far right vanishes, which implies that multiplication by is surjective on . Nakayama’s lemma shows that the group vanishes.
There is an objection here: if , there does not necessarily exist regular on , as the depth of is not necessarily its dimension. We can reduce to this case, however, by using the fact that has a finite filtration whose subquotients are of the form , where is an associated prime of (so that has dimension ). It suffices to prove the vanishing for modules of the form , but these admit regular elements.
Anyway, the point of the above discussion was to show that the three different (and equivalent) definitions can be used, in different ways, to prove results we want. For the rest of Grothendieck’s theorem, certain definitions will be better suited than others.