The Stacks project

Lemma 66.33.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian algebraic spaces over $S$ which is flat, locally of finite type and of relative dimension $d$. For every point $x$ in $|X|$ with image $y$ in $|Y|$ we have $\dim _ x(X) = \dim _ y(Y) + d$.

Proof. By definition of the dimension of an algebraic space at a point (Properties of Spaces, Definition 65.9.1) and by definition of having relative dimension $d$, this reduces to the corresponding statement for schemes (Morphisms, Lemma 29.29.6). $\square$


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