Lemma 76.26.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is locally of finite presentation and Cohen-Macaulay. Then there exist open and closed subschemes $X_ d \subset X$ such that $X = \coprod _{d \geq 0} X_ d$ and $f|_{X_ d} : X_ d \to Y$ has relative dimension $d$.
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
with étale vertical arrows and $U$ and $V$ schemes. Then $U \to V$ is locally of finite presentation and Cohen-Macaulay (immediate from our definitions). Thus we have a decomposition $U = \coprod _{d \geq 0} U_ d$ into open and closed subschemes with $f|_{U_ d} : U_ d \to V$ of relative dimension $d$, see Morphisms, Lemma 29.29.4. Let $u \in U$ with image $x \in |X|$. Then $f$ has relative dimension $d$ at $x$ if and only if $U \to V$ has relative dimension $d$ at $u$ (this follows from our definitions). In this way we see that $U_ d$ is the inverse image of a subset $X_ d \subset |X|$ which is necessarily open and closed. Denoting $X_ d$ the corresponding open and closed algebraic subspace of $X$ we see that the lemma is true. $\square$
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