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 \} $.

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.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.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.

## Comments (2)

Comment #2098 by Chiara Damiolini on

Comment #2126 by Johan on