Timeline for A question regarding base change of a smooth algebra via completion
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2021 at 20:33 | comment | added | Neil Epstein | @Anonymous Perfect. Thank you! Post it as an answer (along with your demonstration of non-flatness) and I'll accept it. | |
| Mar 1, 2021 at 20:32 | comment | added | Anonymous | With the current definition of "smoothness" (which I agree should be renamed regularity), the answer is easily no: take $S = \widehat{R}$, so $S=T=\widehat{R}$, but the map $\widehat{R} \otimes_R \widehat{R} \to \widehat{R}$ is usually not flat. For instance, if $R$ is the local ring of $\mathbf{A}^1$ at $0$, then this map is not flat. (For instance, one can see this using cotangent complexes, though it's probably more elementary than that.) | |
| Mar 1, 2021 at 20:08 | comment | added | Piotr Achinger | Often such a map is called a "regular morphism". | |
| Mar 1, 2021 at 19:52 | history | edited | Neil Epstein | CC BY-SA 4.0 |
I explained what I meant by the ambiguous term "smooth".
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| Mar 1, 2021 at 19:51 | comment | added | Neil Epstein | @JasonStarr I see. I guess this was a language problem. By "smooth", I meant "flat, with geometrically regular fibers". I am not assuming finite type. I will edit my question accordingly. | |
| Mar 1, 2021 at 19:26 | comment | added | Jason Starr | "How do you know that $\widehat{R}\otimes_R S$ is Noetherian?" If $S$ is smooth over $R$, then $\widehat{R}\otimes_R S$ is smooth over the Noetherian ring $\widehat{R}$. In particular, it is finitely presented over $\widehat{R}$. Thus it is Noetherian by the Hilbert Basis Theorem. | |
| Mar 1, 2021 at 19:17 | comment | added | Neil Epstein | @PiotrAchinger How do you know that $\hat{R} \otimes_R S$ is Noetherian? | |
| Mar 1, 2021 at 18:04 | comment | added | Piotr Achinger | Note that $T$ concides with the $m$-adic completion of the Noetherian $\widehat{R}\otimes_R S$, so the answer is positive. | |
| Mar 1, 2021 at 16:33 | history | asked | Neil Epstein | CC BY-SA 4.0 |