The Relationship Between Cohomological Dimension and Support

Question

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) \stackrel{\text{def}}{=} \sup(\{ i \in \mathbb{N} \mid {H_{I}^{i}}(M) \neq 0 \}). $$ From the fact that $ \operatorname{cd}(I,M) \leq \operatorname{cd}(I,R) $, one guesses that there is a relationship between cohomological dimension and the support of modules such as: $$ \operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff \operatorname{cd}(I,M) \leq \operatorname{cd}(I,N). $$

Can anyone prove this relationship or give a counterexample, please? You can add any assumption that helps, such as ‘being local’.

Answer 1

The closest result that I can find is the following:

Theorem: Let $ R $ be a commutative Noetherian unital ring and $ I $ an ideal of $ R $. If $ M $ and $ N $ are two finitely generated $ R $-modules such that $ \operatorname{supp}(M) \subseteq \operatorname{supp}(N) $, then $ \operatorname{cd}(I,M) \leq \operatorname{cd}(I,N) $.

The reference is K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological Dimension of Certain Algebraic Varieties, Proc. Amer. Math. Soc. 130 (2002), 3537-3544.

Answer 2

This is false; for instance, you can always sum $M$ with any injective $R$-module to potentially increase the support without changing the cohomological dimension. In fact, at least when $R$ is noetherian, there always exists an injective $R$-module $J$ with support all of $\text{Spec }R$, so you can even make the support of $M\oplus J$ everything.

In fact, this argument shows that your property, namely that for any two $R$-modules $N$ and $M$, $\text{supp } M \subseteq \text{supp } N$ if and only if $cd(I,M) \leq cd(I, N)$, is equivalent to $cd(I,M)=0$ for all $M$. Indeed, just take $N$ to be an injective $R$-module with full support, and you get $cd(I,M) \leq 0$. So, your question is really: when is $cd(I,M) = 0$ for all $M$? This is true precisely when the complement of $V(I)$ in $\text{Spec }R$ is affine.