Regular sequences in Cohen-Macaulay local rings are characterized by cutting out something of the correct dimension.

Lemma 10.104.2. Let $R$ be a Noetherian local Cohen-Macaulay ring with maximal ideal $\mathfrak m$. Let $x_1, \ldots , x_ c \in \mathfrak m$ be elements. Then

$x_1, \ldots , x_ c \text{ is a regular sequence } \Leftrightarrow \dim (R/(x_1, \ldots , x_ c)) = \dim (R) - c$

If so $x_1, \ldots , x_ c$ can be extended to a regular sequence of length $\dim (R)$ and each quotient $R/(x_1, \ldots , x_ i)$ is a Cohen-Macaulay ring of dimension $\dim (R) - i$.

Proof. Special case of Proposition 10.103.4. $\square$

Comment #916 by Matthieu Romagny on

Suggested slogan: Regular sequences of Cohen-Macaulay local rings

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