Local cohomology I

Fix a noetherian local ring ${(A, \mathfrak{m})}$ .

Let ${C \in D(A)}$ (for ${D(A)}$ the derived category of ${A}$ , or preferably its higher-categorical analog). Let us define the local cohomology functor

$\displaystyle \Gamma_{\mathfrak{m}}: D(A) \rightarrow D(A), \quad \Gamma_{\mathfrak{m}}(C) = \varinjlim \mathbf{Hom}({A}/\mathfrak{m}^i, C).$

We can think of this at a number of levels: for instance, it is the (derived) functor of the ordinary functor on ${A}$ -modules which sends an ${A}$ -module ${M}$ to its submodule

$\displaystyle \Gamma_{\mathfrak{m}}(M) = \varinjlim\hom(A/\mathfrak{m}^i, M)$

of ${\mathfrak{m}}$ -power torsion elements. From this point of view, we can think of the cohomology groups

$\displaystyle H^i_{\mathfrak{m}}(M) \stackrel{\mathrm{def}}{=} H^i (\Gamma_{\mathfrak{m}}(M))$

as defining “cohomology with supports” for the pair ${(\mathrm{Spec} A, \mathrm{Spec} A \setminus \left\{\mathfrak{m}\right\})}$ with coefficients in the sheaf ${M}$ . 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 ${\Gamma_{\mathfrak{m}}}$ in the regular case, in terms of dualizing objects. So, let’s suppose ${A}$ is regular local on, of dimension ${d = \dim A}$ . In this case, each ${A/\mathfrak{m}^i}$ lives in the smaller perfect derived category ${\mathrm{D}_{\mathrm{perf}}(A)}$ , and we will use the duality in that category.

Namely, recall that we have a functor ${D: \mathrm{D}_{\mathrm{perf}}(A) \rightarrow \mathrm{D}_{\mathrm{perf}}(A)^{op} }$ given by ${\mathbf{Hom}(\cdot, A)}$ , which induces a duality on the perfect derived category of ${A}$ , as we saw yesterday.

Let ${K = \varinjlim DA/\mathfrak{m}^i}$ . We saw in the previous post that ${K}$ is cohomologically concentrated in the degree ${d}$ , and it is a shift of the module ${Q = \varinjlim \mathrm{Ext}^d(A/\mathfrak{m}^i, A)}$ : we saw that ${Q}$ was the injective envelope of ${k}$ . The next result will reduce the computation of ${\Gamma_{\mathfrak{m}}}$ to an ${\mathrm{Ext}}$ computation.

Theorem 5 (Local duality) If ${C \in \mathrm{D}_{\mathrm{perf}}(A)}$ and ${A}$ is regular, then we have a canonical isomorphism in ${D(A)}$ ,

$\displaystyle \Gamma_{\mathfrak{m}}(C) \simeq \mathbf{Hom}( D C, K).$

Proof: In fact, we can do this using purely formal manipulations. We have:

where at the last stage we used the fact that ${DC}$ is a compact object. $\Box$

In particular, since ${Q}$ is an injective module, we can now calculate the cohomology groups ,

$\displaystyle H^i( \Gamma_{\mathfrak{m}}(C)) = H^i \mathbf{Hom}( D C, K) = \hom_A( H^{d-i}(DC), Q).$

In the previous post, we saw that the functor ${\hom(\cdot, Q)}$ defined a duality on the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$ -modules. In general, the functor ${\hom(\cdot, Q)}$ can be defined for any ${A}$ -module, though it is no longer a duality. We will denote it by ${^{\vee}}$ ; 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 ${A}$ -module ${M}$ over the regular local ring ${A}$ , there is a canonical isomorphism for all ${i}$ ,

$\displaystyle H^i_{\mathfrak{m}} (M) = \mathrm{Ext}^{d-i}(M, A)^{\vee}.$

2. Definitions of local cohomology

This is a powerful result. Although we have not said much about the functors ${H^i_{\mathfrak{m}}}$ , they have a number of interpretations:

${H^i_{\mathfrak{m}} }$ is a special case of a sheaf-theoretic construction of “cohomology with supports:” that is, given a space ${X}$ and a closed subspace ${Z}$ , one can consider the functor
$\displaystyle \Gamma_Z: \mathbf{Sh}(X) \rightarrow \mathbf{Ab}$

sending a sheaf ${\mathcal{F}}$ to its group of sections supported on ${Z}$ . The derived functors of ${\Gamma_Z}$ are the local cohomology functors, and a special case of them (for ${X = \mathrm{Spec}A, Z = \left\{\mathfrak{m}\right\}}$ ) is given by the ${H^i_{\mathfrak{m}}}$ above. From this point of view, the ${H^i_{\mathfrak{m}}}$ are analogous to relative cohomology groups.
${H^i_{\mathfrak{m}}}$ can be computed using a Koszul-type complex tensored with ${M}$ , and is related to questions of depth. Namely, choose generators ${x_1, \dots, x_n }$ of ${\mathfrak{m}}$ (or, more generally, of a power of ${\mathfrak{m}}$ ). For ${x \in A}$ , let ${K(x)}$ be the complex
$\displaystyle A \rightarrow A_x .$

Then one has an isomorphism

$\displaystyle H^i_{\mathfrak{m}}(M) \simeq H^i( M \otimes K(x_1) \otimes \dots \otimes K(x_n)).$

These complexes ${K(x_i)}$ are colimits of Koszul complexes, so the depth of ${M}$ can be used to show vanishing of the groups ${H^i_{\mathfrak{m}}(M)}$ . 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 ${\mathrm{Ext}}$ 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 ${\vee}$ . Nonetheless, in particular examples duality functors can often be written down explicitly. For instance, for $\mathbb{Z}_p$ , we can take $\mathbb{Z}[p^{-1}]/\mathbb{Z}$ 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 ${(R, \mathfrak{m})}$ be a local ring (not necessarily regular) of dimension ${d}$ . Let ${M}$ be any ${R}$ -module, non-zero. We have the following fundamental vanishing and non-vanishing theorem.

Theorem 7 (Grothendieck) Let ${M}$ be a finitely generated module over ${R}$ . The groups ${H^i_{\mathfrak{m}}( M)}$ are zero for ${i \notin [\mathrm{depth} M, \dim M]}$ , 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

$\displaystyle H^i_{\mathfrak{m}}(M) = 0 , \quad i > \dim M.$

The rest of the bounds will be sketched in later posts.

By replacing ${R}$ by ${R / \mathrm{ann} (M)}$ (which does not affect the local cohomology groups), we can assume ${M}$ has support on all of ${\mathrm{Spec} R}$ , and thus reduce to the vanishing assertion

$\displaystyle H^i_{\mathfrak{m}}(M) = 0, \quad i > \dim R. \ \ \ \ \ (1)$

There are several ways we could prove this:

Method 1

A classical vanishing theorem of Grothendieck asserts that on an ${n}$ -dimensional noetherian space ${X}$ , we have ${H^i(X, \mathcal{F}) = 0}$ for ${i > n}$ and any sheaf of abelian groups on ${X}$ . The local cohomology groups can be reconstructed from sheaf cohomology: that is, we can use the interpretation of “cohomology with supports.”

$\displaystyle 0 \rightarrow H^0_Z(\mathcal{F}) \rightarrow H^0(X, \mathcal{F}) \rightarrow H^0(X \setminus Z, \mathcal{F}) \rightarrow H^1_Z(\mathcal{F} ) \rightarrow \dots.$

Here ${H^0_Z}$ is the functor which assigns to a sheaf the group of sections supported on ${Z}$ . Using this exact sequence, we find that if ${X}$ has dimension ${n}$ , then ${H^i_Z \equiv 0}$ for ${i > n}$ . Since local cohomology can be described in this way (as the ${H^i_Z}$ for ${Z = \left\{\mathfrak{m}\right\}}$ ), the vanishing result (1) follows.

Method 2

The ring ${R}$ has dimension ${d}$ , so there exist ${x_1, \dots, x_d \in \mathfrak{m}}$ which generate an ideal whose radical is ${\mathfrak{m}}$ : that is, they form a system of parameters for ${R}$ . Then, the Koszul-type complex described earlier can be used to compute ${H^i_{\mathfrak{m}}(M)}$ ; since this complex has length ${d}$ , we get the result.

Method 3

Finally, we can also prove vanishing using local duality. First, we can replace ${R}$ with its completion. This will change nothing, since ${\hat{R}}$ is flat over ${R}$ , and the local cohomology modules are all ${\mathfrak{m}}$ -power torsion anyway (so tensoring up to ${\hat{R}}$ has no effect). In particular, $\displaystyle H^i_{\mathfrak{m}}(M) = H^i_{\hat{\mathfrak{m}}}(\hat{M}).$

Thus, we can assume ${R}$ complete at the outset. It follows that ${R}$ is a quotient of a complete regular local ring ${S \twoheadrightarrow R}$ (by Cohen’s structure theorem). We may as well compute the ${H^i}$ over ${S}$ instead of ${R}$ . In fact, we can just throw out the old ${R}$ and assume that it is a complete regular local ring to begin with, though we then lose the hypothesis ${\dim M = \dim R}$ and have to prove the stronger statement. In other words, we have to show that if ${R}$ is regular local (even complete) and ${M}$ a finitely generated ${R}$ -module, then

$\displaystyle \mathrm{Ext}^{d-i}(M, R) = 0, \quad i > \dim M. \ \ \ \ \ (2)$

This will then imply the vanishing result we want, as the dual of ${\mathrm{Ext}^{d-i}(M, R)}$ is ${H^i_{\mathfrak{m}}(M)}$ .

We can prove (2) by induction on ${\dim M}$ . When ${\dim M = 0}$ , it suffices to show that

$\displaystyle \mathrm{Ext}^j(M, R) = 0 , \quad j < d$

for ${M}$ of finite length over ${R}$ . We can reduce by exact sequences to the case where ${M = k}$ , in which case we can use the Koszul resolution of ${k}$ by choosing a regular system of parameters for ${R}$ . Then self-duality of the Koszul complex, together with the fact that ${R}$ is regular, gives the result.

Now, suppose ${\dim M > 0}$ . Suppose ${x \in \mathfrak{m}}$ is regular on ${M}$ (see below); then we have a short exact sequence

$\displaystyle 0 \rightarrow M \stackrel{x}{\rightarrow} M \rightarrow M/xM \rightarrow 0,$

and the dimension of ${M/xM}$ is one less. We have an exact sequence

$\displaystyle \mathrm{Ext}^j(M/xM, R) \rightarrow \mathrm{Ext}^j(M, R) \stackrel{x}{\rightarrow} \mathrm{Ext}^j(M, R) \rightarrow \mathrm{Ext}^{j+1}(M/xM, R).$

When ${j < d - \dim M}$ , induction shows that the group to the far right vanishes, which implies that multiplication by ${x}$ is surjective on ${\mathrm{Ext}^j(M, R)}$ . Nakayama’s lemma shows that the group vanishes.

There is an objection here: if ${\dim M > 0}$ , there does not necessarily exist ${x \in \mathfrak{m}}$ regular on ${M}$ , as the depth of ${M}$ is not necessarily its dimension. We can reduce to this case, however, by using the fact that ${M}$ has a finite filtration whose subquotients are of the form ${R/\mathfrak{p}}$ , where ${\mathfrak{p}}$ is an associated prime of ${M}$ (so that ${R/\mathfrak{p}}$ has dimension ${\leq \dim M}$ ). It suffices to prove the vanishing for modules of the form ${R/\mathfrak{p}}$ , 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.

Climbing Mount Bourbaki