Section 19.3 Properties and Vanishing Theorems
Let’s collect some basic properties of local cohomology. We note that many (but not all) of the following facts can be proven quickly from more than one characterization of local cohomology.
Let \(R\) be a noetherian ring, \(I\) an ideal, and \(M\) an \(R\)-module. Let \(\psi: R \rightarrow S\) be a ring homomorphism, and \(N\) an \(S\)-module.
- \(\mathrm{H}_{I}^{i}(M)\) is \(I\)-torsion for every \(i \geq 0\text{.}\)
- If \(\sqrt{J}=\sqrt{I}\text{,}\) then \(\mathrm{H}_{I}^{i}(M)=\mathrm{H}_{J}^{i}(M)\) for every \(i \geq 0\text{.}\)
- (Invariance of base) \(\mathrm{H}_{I}^{i}\left({ }_{\psi} N\right)\) has a natural \(S\)-module structure, and \(\mathrm{H}_{I}^{i}\left({ }_{\psi} N\right) \cong \mathrm{H}_{\psi(I) S}^{i}(N)\) as \(S\)-modules. We often write \(\mathrm{H}_{I}^{i}(N)=\mathrm{H}_{I S}^{i}(N)\) to abbreviate this when the map is implicit.
- For any arbitrary direct limit system, \(\mathrm{H}_{I}^{i}\left(\lim _{\longrightarrow} M_{\lambda}\right) \cong \underline{\lim } \mathrm{H}_{I}^{i}\left(M_{\lambda}\right)\text{.}\)
- (Flat base change) If \(\psi\) is flat, then \(S \otimes_{R} \mathrm{H}_{I}^{i}(M) \cong \mathrm{H}_{I}^{i}\left(S \otimes_{R} M\right) \cong \mathrm{H}_{I S}^{i}\left(S \otimes_{R} M\right)\text{.}\)
- If \(\mathfrak{p} \in \operatorname{Spec}(R)\text{,}\) then \(\mathrm{H}_{I}^{i}(M)_{\mathfrak{p}} \cong \mathrm{H}_{I_{\mathfrak{p}}}^{i}\left(M_{\mathfrak{p}}\right)\text{.}\)
Proof. 1. Any submodule or quotient module of an \(I\)-torsion module is \(I\)-torsion. Thus, the cohomology of a complex of \(I\)-torsion modules, e.g., \(\Gamma_{I}\left(E^{\bullet}\right)\) for some injective resolution \(M \rightarrow E^{\bullet}\text{,}\) is also \(I\)-torsion.
- The functors \(\Gamma_{I}\) and \(\Gamma_{J}\) are the same, so they yield the same thing when applied to an injective resolution.
- For any \(x \in S\text{,}\) the \(S\)-linear map \(N \stackrel{\cdot x}{\longrightarrow} N\) is the same as the \(R\)-linear map \({ }_{\psi} N \stackrel{\cdot x}{\longrightarrow} \psi N\text{;}\) this gives a \(\operatorname{map} \mathrm{H}_{I}^{i}\left({ }_{\psi} N\right) \stackrel{\cdot x}{\longrightarrow} \mathrm{H}_{I}^{i}\left({ }_{\psi} N\right)\) by functoriality; it follows from functoriality that the map \(\cdot x y\) agrees with the composition of maps \(\cdot x\) and \(\cdot y\text{,}\) so this is an \(S\)-module action.
The noted isomorphism follows from the Čech cohomology isomorphism \(\check{H}^{i}\left(\underline{f} ;{ }_{\psi} N\right) \cong \check{H}^{i}(\underline{\psi(f)} ; N)\text{.}\)
- This again follows from the Čech cohomology statement.
- \(\left.\check{C} \bullet(\underline{f} ; M) \otimes_{R} S \cong \check{C} \bullet \underline{f \otimes 1} ; M \otimes_{R} S\right)\text{.}\) By flatness of \(S\text{,}\) cohomology commutes with taking the tensor product.
- Special case of flat base change.
We will find that we have the best understanding of local cohomology modules \(\mathrm{H}_{I}^{i}(M)\) in two situations: when \(I\) is a maximal ideal, or when \(M=R\) is a regular ring. Apropos the first of these settings:
Proposition 3.11. Let \(R\) be a noetherian ring, \(I\) an ideal, and \(M\) an \(R\)-module. Let \(\mathfrak{m}\) be a maximal ideal of \(R\text{.}\)
- If \(M\) is finitely generated, then \(\mathrm{H}_{\mathfrak{m}}^{i}(M)\) is artinian for all \(i\text{.}\)
- \(\mathrm{H}_{\mathfrak{m}}^{i}(M) \cong \mathrm{H}_{\mathfrak{m}}^{i}(M)_{\mathfrak{m}} \cong \mathrm{H}_{\mathfrak{m} R_{\mathfrak{m}}}^{i}\left(M_{\mathfrak{m}}\right)\text{.}\)
- If \((R, \mathfrak{m})\) is local, then \(\mathrm{H}_{\mathfrak{m}}^{i}(M) \cong \mathrm{H}_{\mathfrak{m} \widehat{R}}^{i}\left(\widehat{R} \otimes_{R} M\right)\text{.}\) In particular, if \(M\) is finitely generated, then \(\mathrm{H}_{\mathfrak{m}}^{i}(M) \cong \mathrm{H}_{\mathfrak{m} \widehat{R}}^{i}(\widehat{M})\text{.}\)
- If \((R, \mathfrak{m}) \rightarrow(S, \mathfrak{n})\) is a local homomorphism, and \(S\) is module-finite over the image of \(R\text{,}\) then \(\mathrm{H}_{\mathfrak{m}}^{i}(S)=\mathrm{H}_{\mathfrak{n}}^{i}(S)\text{.}\)
Proof. 1. By Problem #6 of HW #1, if \(M \rightarrow E^{\bullet}\) is an injective resolution,
\begin{equation*}
\Gamma_{\mathfrak{m}}\left(E^{\bullet}\right)=0 \rightarrow E_{R}(R / \mathfrak{m})^{\mu(0, \mathfrak{m})} \rightarrow E_{R}(R / \mathfrak{m})^{\mu(1, \mathfrak{m})} \rightarrow \cdots
\end{equation*}
The Bass number \(\mu(i, \mathfrak{m})(M)\) is finite, so the complex above is a complex of artinian modules. Thus, \(H_{\mathfrak{m}}^{i}(M)\) is a quotient of a submodule of a finite direct sum of copies of \(E_{R}(R / \mathfrak{m})\text{,}\) hence, artinian.
- If \(M\) is \(\mathfrak{m}\)-torsion, then any element outside of \(\mathfrak{m}\) acts on \(M\) invertibly (check!). Then the second is a special case of 6 above.
- By flat base change, \(\widehat{R} \otimes_{R} \mathrm{H}_{\mathfrak{m}}^{i}(M) \cong \mathrm{H}_{\mathfrak{m} \widehat{R}}^{i}\left(\widehat{R} \otimes_{R} M\right)\text{.}\) Since \(\mathrm{H}_{\mathfrak{m}}^{i}(M)\) is \(\mathfrak{m}\)-torsion, it is already the same as \(\widehat{R} \otimes_{R} \mathrm{H}_{\mathfrak{m}}^{i}(M)\text{.}\)
- \(\mathrm{H}_{\mathfrak{m}}^{i}(S)=\mathrm{H}_{\mathfrak{m} S}^{i}(S)=\mathrm{H}_{\mathfrak{n}}^{i}(S)\text{,}\) since \(\sqrt{\mathfrak{m} S}=\mathfrak{n}\text{.}\)
Remark 3.12. Both \(\# 1\) and \(\# 2\) of the previous proposition fail for the module \(\mathrm{H}_{(x, y)}^{2}\left(\frac{K[x, y, u, v]}{(x u-y v)}\right)\) from worksheet \(\# 3\text{,}\) with \(\mathfrak{m}\) replaced by \(\mathfrak{p}=(x, y)\text{.}\)
We want to note now that if \(R\) is a graded ring, \(M\) a graded module, and \(I\) a homogeneous ideal (i.e., an ideal generated by homogeneous elements \(f_{1}, \ldots, f_{t}\) ) then the local cohomology modules \(\mathrm{H}_{I}^{i}(M)\) are graded as well. This can be seen from the Čech complex:
\begin{equation*}
0 \rightarrow M \rightarrow \bigoplus_{1 \leq i \leq t} M_{f_{i}} \rightarrow \bigoplus_{1 \leq i<j \leq t} M_{f_{i} f_{j}} \rightarrow \cdots \rightarrow \bigoplus_{1 \leq i \leq t} M_{f_{1} \cdots \widehat{f}_{i} \cdots f_{t}} \rightarrow M_{f_{1} \cdots f_{t}} \rightarrow 0
\end{equation*}
which is evidently a complex of graded modules in this case.
We could also see the grading from the other descriptions of local cohomology. This clear for the Koszul description, and is easily seen to give the same grading. There is also a classification of which graded modules are injective as objects in the category of graded modules. This endows the other two descriptions of local cohomology with a grading, and the Fundamental Theorem is compatible with this extra structure. We won’t develop this approach (unless I change my mind!), but note that it is carried out throughly in Bruns & Herzog as well as Brodmann & Sharp.
Our goal now is to understand, for an ideal \(I\) in a noetherian \(\operatorname{ring} R\text{,}\) and an \(R\)-module \(M\text{,}\) what are the least and greatest \(i\text{,}\) respectively, for which \(\mathrm{H}_{I}^{i}(M) \neq 0\text{.}\)
There is an obvious guess (and obvious lower bound) for the least nonvanishing index: the depth of \(I\) on \(M\text{.}\) This guess turns out to be correct.
Theorem 3.13. Let \(R\) be a noetherian ring, \(I\) an ideal, and \(M\) a finitely generated \(R\)-module. Then,
\begin{equation*}
\min \left\{i \mid \mathrm{H}_{I}^{i}(M) \neq 0\right\}=\operatorname{depth}_{I}(M) .
\end{equation*}
Our convention is that \(\min (\varnothing)=\operatorname{depth}_{I}(M)=\infty\) if \(I M=M\text{.}\) In particular, if \(\mathfrak{p}\) is prime and \(R\) is Cohen-Macaulay, then
\begin{equation*}
\min \left\{i \mid \mathrm{H}_{\mathfrak{p}}^{i}(R) \neq 0\right\}=\operatorname{ht}(\mathfrak{p}) .
\end{equation*}
Proof. If \(I M=M\text{,}\) then there is some \(f \in I\) that acts as the identity on \(M\text{,}\) so \(I^{n} \mathrm{H}_{I}^{i}(M)=\) \(f^{n} \mathrm{H}_{I}^{i}(M)=\mathrm{H}_{I}^{i}(M)\) for all \(n\text{.}\) On the other hand, \(\mathrm{H}_{I}^{i}(M)\) is \(I\)-torsion, so it must be the zero module.
We now assume that \(I M \neq M\) and argue by induction on the depth.
If \(\operatorname{depth}_{I}(M)=0\text{,}\) then every element of \(I\) is a zerodivisor on \(M\text{,}\) so \(I\) is contained in the union of the associated primes of \(M\text{,}\) and hence in some associated prime \(\mathfrak{p}\) of \(M\) by prime avoidance. Then, the copy of \(R / \mathfrak{p}\) in \(M\) is killed by \(I\text{,}\) hence nonzero in \(\mathrm{H}_{I}^{0}(M)\text{.}\)
Now, if the depth is \(d>0\text{,}\) take a regular element \(x \in I\) on \(M\text{.}\) The SES
\begin{equation*}
0 \rightarrow M \stackrel{x}{\longrightarrow} M \rightarrow M / x M \rightarrow 0
\end{equation*}
yields the LES
\begin{equation*}
\cdots \rightarrow H_{I}^{i-1}(M) \rightarrow H_{I}^{i-1}(M / x M) \rightarrow H_{I}^{i}(M) \stackrel{x}{\longrightarrow} H_{I}^{i}(M) \rightarrow \cdots
\end{equation*}
For \(i<d\text{,}\) we obtain that \(x\) is a nzd on \(H_{I}^{i}(M)\text{,}\) but this module is \(I\)-torsion (hence \(x\)-torsion), so it must be zero. Then, the LES shows that \(0 \neq H_{I}^{d-1}(M / x M)\) injects into \(H_{I}^{i}(M)\text{,}\) so the latter is nonzero.
We turn our attention to the top nonvanishing local cohomology module. This will be of recurring interest, so we give it some nomenclature.
Definition 3.14. Let \(R\) be a ring, \(I\) an ideal, and \(M\) an \(R\)-module. The cohomological dimension of \(I\) on \(M\) is
\begin{equation*}
\operatorname{cd}(I, M):=\max \left\{i \mid \mathrm{H}_{I}^{i}(M) \neq 0\right\}
\end{equation*}
This is closely related to another invariant of independent interest.
Definition 3.15. Let \(R\) be a ring, \(I\) an ideal. The arithmetic rank of \(I\) is
\begin{equation*}
\operatorname{ara}(I):=\min \left\{t \mid \text { there exist } f_{1}, \ldots, f_{t} \text { such that } \sqrt{I}=\sqrt{\left(f_{1}, \ldots, f_{t}\right)}\right\} \text {. }
\end{equation*}
The following fact, now easy to prove, is one of the key points to many applications of local cohomology.
Theorem 3.16. Let \(R\) be a noetherian ring, \(I\) an ideal. Then \(\operatorname{ara}(I) \geq \operatorname{cd}(I, M)\) for all \(R\) modules \(M\text{.}\)
Proof. Let \(\sqrt{\left(f_{1}, \ldots, f_{t}\right)}=\sqrt{I}\text{.}\) Then \(\mathrm{H}_{I}^{i}(M)=\check{H}^{i}\left(f_{1}, \ldots, f_{t} ; M\right)\text{,}\) and the latter vanishes for \(i>t\text{,}\) since the Čech complex on \(t\) elements lives only in cohomological degrees less than or equal to \(t\text{.}\)
This Theorem is useful both for understanding rank and for understanding cohomological dimension. We will employ it for the latter purpose first.
Example 3.17. Let \(K\) be a field, and \(T=\frac{K[x, y, u, v]}{(x u-y v)}\text{.}\) Let \(I=(x, y)\text{.}\) We saw in worksheet #3 that \(\mathrm{H}_{I}^{2}(T) \neq 0\text{.}\) We claim that the support of \(\mathrm{H}_{I}^{2}(T)\) is \(\mathcal{V}((x, y, u, v))\text{,}\) so \(\operatorname{Ass}\left(\mathrm{H}_{I}^{2}(T)\right)=\{(x, y, u, v)\}\text{.}\) If \(\mathfrak{p}\) is a prime that does not contain \(x\) or \(y\text{,}\) then \(I_{\mathfrak{p}}\) is the unit ideal, so \(\mathrm{H}_{I}^{2}(T)_{\mathfrak{p}}=\mathrm{H}_{I_{\mathfrak{p}}}^{2}\left(T_{\mathfrak{p}}\right)=0\text{.}\) If \(\mathfrak{p}\) does not contain \(u\text{,}\) then \(x=\frac{y v}{u}\) in \(T_{\mathfrak{p}}\text{,}\) so \(I_{\mathfrak{p}}=(y)\text{.}\) Thus, \(\mathrm{H}_{I}^{2}(T)_{\mathfrak{p}}=\mathrm{H}_{I_{\mathfrak{p}}}^{2}\left(T_{\mathfrak{p}}\right)=0\text{.}\) Similarly, if \(\mathfrak{p}\) does not contain \(v\text{,}\) the same vanishing occurs.
Proposition 3.18. Let \(R\) be a noetherian ring, \(I\) an ideal. For any \(R\)-module, \(\operatorname{cd}(I, M) \leq \operatorname{cd}(I, R)\text{.}\)
Proof. Let \(c=\operatorname{cd}(I)\text{.}\) Since local cohomology commutes with direct sums, we have that \(\mathrm{H}_{I}^{i}(F)=0\) for any free module \(F\) and any \(i>c\text{.}\) Now, given an arbitrary module \(M\text{,}\) take a short exact sequence
\begin{equation*}
0 \rightarrow M^{\prime} \rightarrow F \rightarrow M \rightarrow 0
\end{equation*}
with \(F\) free. The LES of local cohomology gives isomorphisms \(\mathrm{H}_{I}^{i+1}\left(M^{\prime}\right) \cong \mathrm{H}_{I}^{i}(M)\) for all \(i>c\text{.}\) Repeating this, we find for all \(a>0\text{,}\) that there is some module \(L\) such that \(\mathrm{H}_{I}^{i}(M) \cong \mathrm{H}_{I}^{i+a}\left(M^{(a)}\right)\text{.}\) But, since \(I\) is finitely generated, \(\operatorname{ara}(I)\) is finite, and \(\operatorname{cd}\left(I,\left(M^{(a)}\right)\right.\) is less than this number for all modules. Hence, we must have that \(\mathrm{H}_{I}^{i}(M)=0\text{.}\)
Following the last proposition, we write \(\operatorname{cd}(I)\) for \(\operatorname{cd}(I, R)\text{.}\) When \(I=\mathfrak{m}\) is a maximal ideal, we can say a bit more about cohomological dimension. Recall that, by the dimension of a module, we mean simply the dimension of \(R / \operatorname{ann}(M)\text{.}\) We note that, if \(M\) is finitely generated, this agrees with the dimension of the support of \(M\) as a subset of \(\operatorname{Spec}(R)\text{,}\) but this is not true if \(M\) is arbitrary (e.g. take \(M=E_{R}(k)\) for \(R\) of positive dimension).
Proposition 3.19. Let \((R, \mathfrak{m})\) be local, and \(M\) an \(R\)-module. Then \(\operatorname{cd}(\mathfrak{m}, M) \leq \operatorname{dim}(M)\text{.}\)
Proof. By the invariance of base property, we may replace \(R\) by \(R / \operatorname{ann}(M)\text{,}\) and \(\mathfrak{m}\) by its image there, thus we can assume that \(\operatorname{dim}(M)=\operatorname{dim}(R)\text{.}\) The maximal ideal of \(R\) is generated by \(\operatorname{dim}(R)\) elements up to radical, so \(\operatorname{cd}(\mathfrak{m}, M) \leq \operatorname{dim}(R)\text{,}\) as required.
Corollary 3.20. Let \((R, \mathfrak{m})\) be local. The ring \(R\) is Cohen-Macaulay if and only if \(\mathrm{H}_{\mathfrak{m}}^{i}(R)=0\) for all \(i \neq \operatorname{dim}(R)\text{.}\)
We want to now extend our cohomological dimension bounds to all ideals.
Theorem 3.21. If \(I\) is a proper ideal in a local ring \((R, \mathfrak{m}, k)\) of dimension \(d\text{,}\) then \(\operatorname{ara}(I) \leq d\text{.}\)
Proof. If \(I=\mathfrak{m}\text{,}\) this is standard, so we assume that the height of \(I\) is less than \(d\text{.}\)
Let \(\mathcal{P}_{n}=\{\mathfrak{p} \in \operatorname{Spec}(R) \mid \operatorname{height}(\mathfrak{p})=n\) and \(\mathfrak{p} \nsupseteq I\}\text{.}\) We will inductively find
\begin{equation*}
r_{0}, \ldots, r_{n} \in I \text { such that }\left(r_{0}, \ldots, r_{n}\right) \nsubseteq \mathfrak{p} \text { for any } \mathfrak{p} \in \mathcal{P}_{n}
\end{equation*}
Once we have done this for \(n=d-1\text{,}\) we have \(d\) elements such that \(\mathcal{V}\left(\left(r_{0}, \ldots, r_{d-1}\right)\right)=\mathcal{V}(I)\text{,}\) and we are done.
To choose \(r_{0}\text{,}\) we only need to avoid a subset of the minimal primes of \(R\text{,}\) which is finite, so we can do this by prime avoidance.
Suppose that we have elements satisfying the specified condition for all \(i<n\text{.}\) The set of minimal primes of \(\left(r_{0}, \ldots, r_{n}\right)\) of height \(n+1\) not containing \(I, \operatorname{Min}\left(\left(r_{0}, \ldots, r_{n}\right)\right) \cap \mathcal{P}_{n+1}\text{,}\) is finite (and \(I\) is clearly not any such \(I\) ), so we can choose an element \(r_{n+1}\) in \(I\) not in any of these.
Now, suppose that \(\left(r_{0}, \ldots, r_{n}, r_{n+1}\right)\) is contained in some \(\mathfrak{p}\) in \(\mathcal{P}_{n+1}\text{.}\) Then, \(\left(r_{0}, \ldots, r_{n}\right) \subseteq \mathfrak{p}\) as well. If \(\mathfrak{p}\) is not minimal over \(\left(r_{0}, \ldots, r_{n}\right)\text{,}\) then take some \(\mathfrak{q}\) in between. The height of \(\mathfrak{q}\) is less than \(n+1\text{,}\) so \(\mathfrak{q}\) contradicts the induction hypothesis. If \(\mathfrak{p}\) is minimal over \(\left(r_{0}, \ldots, r_{n}\right)\text{,}\) then \(\mathfrak{p} \in \operatorname{Min}\left(\left(r_{0}, \ldots, r_{n}\right)\right) \cap \mathcal{P}_{n+1}\text{,}\) and \(r_{n+1} \in \mathfrak{p}\) contradicts the choice of \(r_{n+1}\text{.}\)
Corollary 3.22. If \(I\) is an ideal in a noetherian ring, and \(M\) an \(R\)-module, then \(\operatorname{cd}(I, M) \leq\) \(\operatorname{dim}(M)\text{.}\)
Proof. Given a counterexample, we can localize at an associated prime of \(\mathrm{H}_{I}^{i}(M)\text{,}\) and since \(\mathrm{H}_{I}^{i}(M)_{\mathfrak{p}}=\mathrm{H}_{I_{\mathfrak{p}}}^{i}\left(M_{\mathfrak{p}}\right)\) for all \(\mathfrak{p}\text{,}\) and \(\operatorname{dim}\left(M_{\mathfrak{p}}\right) \leq \operatorname{dim}(M)\text{,}\) it suffices to assume that \(R\) is local. Moreover, by invariance of base, we can replace \(R\) by \(R / \operatorname{ann}(M)\text{,}\) and assume that \(\operatorname{dim}(R)=\operatorname{dim}(M)\text{.}\) Then, applying the previous theorem, \(\operatorname{cd}(I, M) \leq \operatorname{ara}(I) \leq \operatorname{dim}(R)=\operatorname{dim}(M)\text{.}\)
