Timeline for Are etale morphisms "strongly formally etale"?
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| Apr 17, 2020 at 4:35 | comment | added | Pavel Čoupek | I think that is OK because $Z \rightarrow W$ is bijective. Given $w \in W$, let $z$ be the preimage of $w$ in $Z$, and denote by $y, x$ their images in $Y, X,$ resp. (in particular, $y \mapsto x$). Then replacing "affine open" by "affine neighbourhood of [the respective fixed point]" in the above, one produces $W''$ containing $w$. Of course, you might not cover $Y$ this way, but you cover at least an open subscheme containing the full (set-theoretic) image of $Z \rightarrow Y$, which agrees with the image of any possible lift $W \rightarrow Y$ (again due to $Z \rightarrow W$ bijective). | |
| Apr 17, 2020 at 4:11 | comment | added | R. van Dobben de Bruyn | @PavelČoupek: the thing that's unclear to me is how is lifting of this local version going to imply the global version. The hardest part is probably showing that $W$ can be covered by such $W''$. But you might be right that this just works; I just didn't see an obvious argument. | |
| Apr 17, 2020 at 3:15 | comment | added | Pavel Čoupek | I thought that after choosing affine $X'$ in $X$, you just choose some affine $Y'$ of $Y$ in the preimage of $X'$. Then $Y' \rightarrow X'$ should be still w. étale since w. étale is local on the source and target. After that, some yoga on the LHS: take similarly $W'$ affine in $W$ in preimage of $X'$, then intersect the preimages of $Y', W'$ in $Z$ and choose affine open $Z'$ inside this intersection. Image of $Z'$ in $W$ is open in $W'$, so after possibly takings even smaller affine $W''$ inside, taking preimage $Z''$ of $W''$; then $Z'' \rightarrow W''$ is univ. homeo. Is this incorrect? | |
| Apr 17, 2020 at 2:54 | comment | added | R. van Dobben de Bruyn | @PavelČoupek even in the étale case I'm not sure how to do this. When you choose an affine in $X$, you want its preimage in $Y$ to still be affine, or at least to contain an affine containing the image of $Z$. I see no a priori reason why this should be true. (Like in my answer, you may assume $W = X$ if you want, but I don't see how that helps.) | |
| Apr 17, 2020 at 2:50 | comment | added | Pavel Čoupek | Is it necessary for the weakly étale morphisms to be affine? It seems to me that when a weakly étale morphism $Y\rightarrow X$ gets tested against a universal homeomorphism $Z \rightarrow W$, by shrinking neighborhoods around chosen points repeatedly, one can make all the schemes involved affine (while maintaining that the restriction $Z'\rightarrow W'$ is a universal homeomorphism, using that univ. homeo's are affine). After obtaining the lifts in the affine situation, uniqueness of the lifts should allow them to glue to a global one, no? | |
| Apr 16, 2020 at 23:44 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |