Timeline for answer to Reduction step to $k=\bar{k}$ in the proof of rigidity lemma by R. van Dobben de Bruyn
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| Mar 10, 2023 at 1:17 | vote | accept | user267839 | ||
| Feb 15, 2023 at 0:38 | comment | added | R. van Dobben de Bruyn | There is probably also a commutative algebra argument, but your suggestion of a diagrammatic proof led me to the one presented above. (The local computation is that checking whether $C \to A \otimes B$ lands in $k \otimes B$ can be done over $\bar k$, which should be clear by considering the map of $k$-vector spaces $C \to (A/k) \otimes B$. But reducing to the affine case is a bit annoying, because the inverse image of an affine open in $Z$ need not be of the form $U \times V$ for $U \subseteq X$ and $V \subseteq Y$ open.) | |
| Feb 14, 2023 at 23:55 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |