The Stacks project

Lemma 10.130.8. Let $R$ be a ring. Let $R \to S$ be a ring map which is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write $S = S_0 \times \ldots \times S_ n$ as a product of $R$-algebras $S_ d$ such that each $S_ d$ satisfies (a), (b), (c) and has all fibres equidimensional of dimension $d$.

Proof. For each integer $d$ denote $W_ d \subset \mathop{\mathrm{Spec}}(S)$ the set defined in Lemma 10.130.4. Clearly we have $\mathop{\mathrm{Spec}}(S) = \coprod W_ d$, and each $W_ d$ is open by the lemma we just quoted. Hence the result follows from Lemma 10.24.3. $\square$

Comments (2)

Comment #1562 by kollar on

I find the formulation confusing. I suggest:

Let R be a ring. Let R→Sd be ring maps. Assume that R→S0×…×Sn is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then each R→Sd satisfies (a), (b), (c) and has all fibres equidimensional of dimension d.

Comment #1580 by on

OK, I tried to clear up the statement. Thanks! The change is here.

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