The Stacks project

Situation 81.2.1. Here $S$ is a scheme and $B$ is an algebraic space over $S$. We assume $B$ is quasi-separated, locally Noetherian, and universally catenary (Decent Spaces, Definition 67.25.4). Moreover, we assume given a dimension function $\delta : |B| \longrightarrow \mathbf{Z}$. We say $X/B$ is good if $X$ is an algebraic space over $B$ whose structure morphism $f : X \to B$ is quasi-separated and locally of finite type. In this case we define

\[ \delta = \delta _{X/B} : |X| \longrightarrow \mathbf{Z} \]

as the map sending $x$ to $\delta (f(x))$ plus the transcendence degree of $x/f(x)$ (Morphisms of Spaces, Definition 66.33.1). This is a dimension function by More on Morphisms of Spaces, Lemma 75.32.2.


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