The Stacks project

Proposition 10.103.4. Let $R$ be a Noetherian local ring, with maximal ideal $\mathfrak m$. Let $M$ be a Cohen-Macaulay module over $R$ whose support has dimension $d$. Suppose that $g_1, \ldots , g_ c$ are elements of $\mathfrak m$ such that $\dim (\text{Supp}(M/(g_1, \ldots , g_ c)M)) = d - c$. Then $g_1, \ldots , g_ c$ is an $M$-regular sequence, and can be extended to a maximal $M$-regular sequence.

Proof. Let $Z = \text{Supp}(M) \subset \mathop{\mathrm{Spec}}(R)$. By Lemma 10.60.13 in the chain $Z \supset Z \cap V(g_1) \supset \ldots \supset Z \cap V(g_1, \ldots , g_ c)$ each step decreases the dimension at most by $1$. Hence by assumption each step decreases the dimension by exactly $1$ each time. Thus we may successively apply Lemma 10.103.3 to the modules $M/(g_1, \ldots , g_ i)$ and the element $g_{i + 1}$.

To extend $g_1, \ldots , g_ c$ by one element if $c < d$ we simply choose an element $g_{c + 1} \in \mathfrak m$ which is not in any of the finitely many minimal primes of $Z \cap V(g_1, \ldots , g_ c)$, using Lemma 10.15.2. $\square$

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