invariants that can be measured by Local Cohomology

Question

What invariants can be measured by Local Cohomology (and what application it has)?

As an example of what I mean:

Local Cohomology can measure invariants like depth and dim. So in some cases Local Cohomology help us detect Cohen-Macaulay-ness: Let $(R,m)$ and $(S,n)$ be local rings and $S$ is an $R$-Algebra via homomorphism $f:R\to S.$ Assume that for all $i$ we have following isomorphism $H^i_n(S)\cong H^i_m(R)$ of $R$-modules. Then knowing Cohen-Macaulay-ness of $R$ we know if $S$ is Cohen-Macaulay.

Thank you.

Answer 1

There are many invariants can be measured by local cohomology. Here are a few examples.

I assume $(R, \mathfrak{m})$ is a complete local ring of dimension $d$. The non Cohen-Macaulay locus of $R$ is defined as follows

$nCM(R) = \{\mathfrak{p} \in \mathrm{Spec}(R) \ | \ R_{\mathfrak{p}} \text{ is not Cohen-Macaulay}\}.$

We can ask some questions about the non Cohen-Macaulay locus as: Is $nCM(R)$ a closed subset of $\mathrm{Spec}(R)$? What is the dimension of $nCM(R)$? Local cohomology give us the answers of these questions.

Indeed, let $\mathfrak{a}_i = \mathrm{Ann}(H^i_{\mathfrak{m}}(R))$, $i \ge 0$, and $\mathfrak{a} = \mathfrak{a}_0 \ldots \mathfrak{a}_{d-1}$. It is easy to see that if $R$ is Cohen-Macaulay i.e. $nCM(R) = \emptyset$, then $\mathfrak{a} = R$ since $H^i_{\mathfrak{m}}(R) = 0$ for all $i = 0, \ldots, d-1$. The Faltings annihilator theorem give us the answer in general case (see Brodmann-Sharp: local cohomology)

Theorem 1. Suppose $R$ is equidimensional. Then we have

(i) $nCM(R) = Var(\mathfrak{a})$ is a closed subset of $\mathrm{Spec}(R)$.

(ii) The dimension of $nCM(R)$ is $\dim R/\mathfrak{a}$.

We can define a generalization of the class of Cohen-Macaulay ring as follows.

Definition 2. The ring $R$ is called generalized Cohen-Macaulay if $\dim R/\mathfrak{a} \le 0$, that is $H^i_{\mathfrak{m}}(R)$ has finite length for all $i = 0, \ldots, d-1$.

It should be noted that the affine cone of any nonsingular projective variaty is generalized Cohen-Macaulay. Working with generalized Cohen-Macaulay rings are much more difficult than Cohen-Macaulay rings but we have some chances to do so with $\ell(H^i_{\mathfrak{m}}(R))$. For example, let $\mathfrak{q}$ be a parameter ideal of $R$. Then we always have $\ell(R/\mathfrak{q}) \ge e(\mathfrak{q})$ the multiplicity of $q$. Moreover $R$ is Cohen-Macaulay if and only if $\ell(R/\mathfrak{q}) = e(\mathfrak{q})$ for some (and any) $\mathfrak{q}$. When $R$ is generalized Cohen-Macaulay, the different $\ell(R/\mathfrak{q}) - e(\mathfrak{q})$ are bounded above by an invariant in terms of local cohomology. More precisely we have

$$\ell(R/\mathfrak{q}) - e(\mathfrak{q}) \le \binom{d-1}{i} \ell(H^i_{\mathfrak{m}}(R))$$

(see Trung's paper: towards a theory of generalized Cohen-Macaulay modules, Nagoya, 1986)

Answer 2

Just a couple of examples, if you are looking for applications in algebraic geometry (definitely not an exhaustive list).

If $(X, \mathcal{O}_X(1))$ is a (say) smooth projective variety, and $A$ its affine cone, then the vanishings of the local cohomology groups $H^i_{\mathfrak{m}}(A, \mathcal{O}_A)$ are related to the important concept of Arithmetically Cohen-Macaulay-ness for $X$ (see here, and one has as well a generalizations on ACM bundles). In general the more vanishing we have on the local cohomology, the easier is to describe the (generalized) deformation theory of the affine cone $A$, especially in terms of $X$ and its Hodge Theory.

If $X$ is a singular projective hypersurface with isolated singularities, the local cohomology of the Jacobian ring encodes several informations on $X$ itself. See for example this paper by E.Sernesi.