Cohen-Macaulay morphisms decompose into clopens of pure relative dimension

Lemma 29.29.4. Let $f : X \to S$ be a morphism of schemes. Assume that

1. $f$ is flat,

2. $f$ is locally of finite presentation, and

3. for all $s \in S$ the fibre $X_ s$ is Cohen-Macaulay (Properties, Definition 28.8.1)

Then there exist open and closed subschemes $X_ d \subset X$ such that $X = \coprod _{d \geq 0} X_ d$ and $f|_{X_ d} : X_ d \to S$ has relative dimension $d$.

Proof. This is immediate from Algebra, Lemma 10.130.9. $\square$

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