Definition 66.33.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Let $d, r \in \{ 0, 1, 2, \ldots , \infty \}$.

1. We say the dimension of the local ring of the fibre of $f$ at $x$ is $d$ if the equivalent conditions of Lemma 66.22.5 hold for the property $\mathcal{P}_ d$ described in Descent, Lemma 35.33.6.

2. We say the transcendence degree of $x/f(x)$ is $r$ if the equivalent conditions of Lemma 66.22.5 hold for the property $\mathcal{P}_ r$ described in Descent, Lemma 35.33.7.

3. We say $f$ has relative dimension $d$ at $x$ if the equivalent conditions of Lemma 66.22.5 hold for the property $\mathcal{P}_ d$ described in Descent, Lemma 35.33.8.

Comment #2098 by Chiara Damiolini on

I think there's a typo in (3). Probably it should be "We say that f has relative dimension [...]" or "We say the relative dimension of f at x is d if [...]".

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