Timeline for Flatness of tensor products of analytic germs
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2023 at 15:32 | comment | added | Chris | You're right, I took for granted that LHS was again a Noetherian local ring. I can't see at the moment why it's Noetherian | |
| Jan 1, 2023 at 12:43 | comment | added | Pelle Steffens | I'm aware that flatness of maps of Noetherian local rings is detected by passing to completions; however I'm quite sure the algebra $\mathcal{O}(\mathbb{C}^n)\otimes_{\mathbb{C}} \mathcal{O}(\mathbb{C}^m)$ is only local for $m$ or $n$ equal 0. Of course, since localization is flat and preserves Noetherianness, this wouldn't be an issue if we knew the tensor product was Noetherian, but I don't see this. Is it just Weierstrass preparation? | |
| Jan 1, 2023 at 12:00 | comment | added | Chris | By Tag 00HJ in Stacks Project, and the fact that for any Noetherian local ring the map $R\to\hat{R}$ is faithully flat, it suffices to check your statement on completions. The map induced on completions is an iso, hence flat. | |
| Dec 31, 2022 at 19:38 | history | asked | Pelle Steffens | CC BY-SA 4.0 |