Timeline for Künneth formula for $\pi_1$-proper morphisms
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2022 at 18:23 | comment | added | Peter Scholze | Sure! I claim that if $R$ is a normal integral domain with fraction field $K$, then finite etale $X\times \mathrm{Spec}(R)$-schemes embed fully faithfully into finite etale $X\times \mathrm{Spec}(K)$-schemes. (In particular, connectedness is preserved.) For the claim, you can use h-descent in $X$ to reduce to the case that $X$ is normal and affine, in which case the inverse functor is given by taking the normalization in the finite etale scheme over the generic fibre. | |
| Oct 17, 2022 at 15:22 | comment | added | Benedikt | Thanks a lot for your answer! May I ask for some clarification on "comparing with the generic fibre"? While taking the generic fiber certainly helps showing faithfulness and the fraction field being algebraically closed helps with essential surjectivity as described above, does it also help with fullness? Is it possible that after the reductions you describe, connectedness is preserved under taking the generic fiber, similar to the case of a discrete valuation ring, see e.g. Tag 055J? | |
| Oct 11, 2022 at 17:56 | comment | added | Peter Scholze | Alternatively: Reduce first to the case that the strictly henselian ring is a domain (say, by noetherian approximation to have only finitely many generic points, and then induction on the number of components). Then pass up finite covers to reduce to the case that the fraction field is algebraically closed. Then compare to the generic fibre. | |
| Oct 11, 2022 at 17:49 | comment | added | Peter Scholze | Ah, this seems to be another situation where analytic adic spaces are, a priori, easier than schemes, but it can be made to work for schemes as well. The trick is to use v-descent to reduce to the case where the strictly henselian ring is in fact a valuation ring with algebraically closed fraction field. In that case, I think one can argue by comparing to the generic fibre. | |
| Oct 10, 2022 at 17:10 | history | edited | Benedikt | CC BY-SA 4.0 |
deleted 1 character in body
|
| Oct 10, 2022 at 16:57 | history | edited | Benedikt | CC BY-SA 4.0 |
added 5 characters in body
|
| S Oct 10, 2022 at 16:53 | review | First questions | |||
| Oct 10, 2022 at 17:32 | |||||
| S Oct 10, 2022 at 16:53 | history | asked | Benedikt | CC BY-SA 4.0 |