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
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.
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.
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
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