Timeline for Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jan 5 at 16:04 | comment | added | R. van Dobben de Bruyn | The Isom scheme exists when $X$ and $Y$ are flat and projective over $S$ (see e.g. FGA Explained, Thm. 5.23), in particular in the finite étale case. For finite étale morphisms, it can also be constructed by descent: a morphism $X\to S$ is finite étale if and only if there is an étale cover $T\to S$ such that $X_T\cong\coprod_{i\in I}T$ for a finite set $I$. If this holds for $X$ and $Y$, choose a cover $T\to S$ trivialising both, so $\mathbf{Isom}_T(X_T,Y_T)$ is also a disjoint union of copies of $T$. Galois descent then constructs $\mathbf{Isom}_S(X,Y)$ as a finite étale scheme over $S$. | |
| Jan 5 at 15:48 | comment | added | LSpice | When you refer generally to the comments, do you mean specifically this one? | |
| Jan 5 at 15:48 | history | edited | LSpice | CC BY-SA 4.0 |
Tidying
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| Jan 5 at 14:48 | history | edited | user267839 | CC BY-SA 4.0 |
added 40 characters in body
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| Jan 5 at 14:38 | history | asked | user267839 | CC BY-SA 4.0 |