Lemma 77.3.3. In Situation 77.2.1. Let $(Y', y') \to (Y, y)$ be a morphism of pointed algebraic spaces. If $Y' \to Y$ is quasi-finite at $y'$ and $\mathcal{F}$ is pure above $y$, then $\mathcal{F}_{Y'}$ is pure above $y'$.
Proof. It $(T \to Y', t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}_{Y'}$ above $y'$ with $T \to Y'$ quasi-finite at $t$, then $(T \to Y, t' \to t, \xi )$ is an impurity of $\mathcal{F}$ above $y$ with $T \to Y$ quasi-finite at $t$, see Morphisms of Spaces, Lemma 67.27.3. Hence the lemma follows immediately from the definition of purity. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)