Lemma 51.17.4 (0EBY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.17: Frobenius action
Lemma 51.17.4 (cite)

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See [Lech-inequalities] and [Lemma 3 page 300, MatCA].

Lemma 51.17.4. Let $(A, \mathfrak m)$ be a local ring. If $\mathfrak m = (x_1, \ldots , x_ r)$ and $x_1^{e_1}, \ldots , x_ r^{e_ r}$ are independent for some $e_ i > 0$ , then $\text{length}_ A(A/(x_1^{e_1}, \ldots , x_ r^{e_ r})) = e_1\ldots e_ r$ .

Proof. Use Lemmas 51.17.2 and 51.17.3 and induction. $\square$

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