The Stacks project

Lemma 51.16.2. Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset A$ be an ideal. If $H^ d_{V(I)}(M) = 0$ for some finite $A$-module whose support contains all the irreducible components of dimension $d$, then $\text{cd}(A, I) < d$.

Proof. By Lemma 51.4.7 we know $\text{cd}(A, I) \leq d$. Thus for any finite $A$-module $N$ we have $H^ i_{V(I)}(N) = 0$ for $i > d$. Let us say property $\mathcal{P}$ holds for the finite $A$-module $N$ if $H^ d_{V(I)}(N) = 0$. One of our assumptions is that $\mathcal{P}(M)$ holds. Observe that $\mathcal{P}(N_1 \oplus N_2) \Leftrightarrow (\mathcal{P}(N_1) \wedge \mathcal{P}(N_2))$. Observe that if $N \to N'$ is surjective, then $\mathcal{P}(N) \Rightarrow \mathcal{P}(N')$ as we have the vanishing of $H^{d + 1}_{V(I)}$ (see above). Let $\mathfrak p_1, \ldots , \mathfrak p_ n$ be the minimal primes of $A$ with $\dim (A/\mathfrak p_ i) = d$. Observe that $\mathcal{P}(N)$ holds if the support of $N$ is disjoint from $\{ \mathfrak p_1, \ldots , \mathfrak p_ n\} $ for dimension reasons, see Lemma 51.4.7. For each $i$ set $M_ i = M/\mathfrak p_ i M$. This is a finite $A$-module annihilated by $\mathfrak p_ i$ whose support is equal to $V(\mathfrak p_ i)$ (here we use the assumption on the support of $M$). Finally, if $J \subset A$ is an ideal, then we have $\mathcal{P}(JM_ i)$ as $JM_ i$ is a quotient of a direct sum of copies of $M$. Thus it follows from Cohomology of Schemes, Lemma 30.12.8 that $\mathcal{P}$ holds for every finite $A$-module. $\square$

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