Definition 42.7.6. Let $(S, \delta )$ as in Situation 42.7.1. For any scheme $X$ locally of finite type over $S$ and any irreducible closed subset $Z \subset X$ we define

where $\xi \in Z$ is the generic point of $Z$. We will call this the *$\delta $-dimension of $Z$*. If $Z$ is a closed subscheme of $X$, then we define $\dim _\delta (Z)$ as the supremum of the $\delta $-dimensions of its irreducible components.

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