Lemma 42.42.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Write $\delta = \delta _{X/S}$ as in Section 42.7. The following are equivalent

1. There exists a decomposition $X = \coprod _{n \in \mathbf{Z}} X_ n$ into open and closed subschemes such that $\delta (\xi ) = n$ whenever $\xi \in X_ n$ is a generic point of an irreducible component of $X_ n$.

2. For all $x \in X$ there exists an open neighbourhood $U \subset X$ of $x$ and an integer $n$ such that $\delta (\xi ) = n$ whenever $\xi \in U$ is a generic point of an irreducible component of $U$.

3. For all $x \in X$ there exists an integer $n_ x$ such that $\delta (\xi ) = n_ x$ for any generic point $\xi$ of an irreducible component of $X$ containing $x$.

The conditions are satisfied if $X$ is either normal or Cohen-Macaulay1.

Proof. It is clear that (1) $\Rightarrow$ (2) $\Rightarrow$ (3). Conversely, if (3) holds, then we set $X_ n = \{ x \in X \mid n_ x = n\}$ and we get a decomposition as in (1). Namely, $X_ n$ is open because given $x$ the union of the irreducible components of $X$ passing through $x$ minus the union of the irreducible components of $X$ not passing through $x$ is an open neighbourhood of $x$. If $X$ is normal, then $X$ is a disjoint union of integral schemes (Properties, Lemma 28.7.7) and hence the properties hold. If $X$ is Cohen-Macaulay, then $\delta ' : X \to \mathbf{Z}$, $x \mapsto -\dim (\mathcal{O}_{X, x})$ is a dimension function on $X$ (see Example 42.7.4). Since $\delta - \delta '$ is locally constant (Topology, Lemma 5.20.3) and since $\delta '(\xi ) = 0$ for every generic point $\xi$ of $X$ we see that (2) holds. $\square$

[1] In fact, it suffices if $X$ is $(S_2)$. Compare with Local Cohomology, Lemma 51.3.2.

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