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.

  • $\begingroup$ also asked here $\endgroup$ – user 1 Mar 14 '16 at 18:29
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    $\begingroup$ I'm not really sure what you are looking for, are you only looking at things relative to maps. I could give you a long list of properties of rings commonly measured by local cohomology (like depth and dimension...) On the other hand, the complex $R \Gamma_m(R)$ is Matlis dual to the dualizing complex, so you can measure any property you could measure from the cohomology of the dualizing complex, which a huge number of properties. $\endgroup$ – Karl Schwede Mar 23 '16 at 22:08
  • $\begingroup$ @Karl Schwede Can you please give list of properties of rings commonly measured by local cohomology (like depth and dimension...) ? I only know dim, depth and filter-depth. things relative to maps also welcome. $\endgroup$ – user 1 Apr 7 '16 at 12:09
  • $\begingroup$ @user1 It might not be so convenient to build a full list, as new perspectives keep the list growing. For example, in the graded setting, local cohomology helps to calculate regularity and the a-invariant. Also, considering the section ring of a line bundle, local cohomology is essentially the same thing as sheaf cohomology -- and there are tons of invariants there. Maybe you could sharpen your question to narrow down the potential list or suggest the types of problems your interested in and better advice can be given. $\endgroup$ – lemiller Apr 25 '16 at 3:50
  • $\begingroup$ I appreciate all answers, but actually I dont want answers about sheaf. $\endgroup$ – user 1 Apr 25 '16 at 6:59

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)

  • $\begingroup$ Pham Hung Quy thanks. can you tell more about "The Faltings annihilator theorem give us the answer in general case"? $\endgroup$ – user 1 Apr 26 '16 at 14:08
  • $\begingroup$ I mean the Faltings annihilator implies Theorem 1. By Faltings annihilator we have $\mathfrak{p} \in Var(\mathfrak{a})$ if and only if $$\mathrm{depth} R_{\mathfrak{p}} + \dim R/\mathfrak{p} < d.$$ The last one is equivalent to $\mathfrak{p} \in nCM(R)$. $\endgroup$ – Pham Hung Quy Apr 27 '16 at 15:30

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.


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