Timeline for Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2021 at 18:47 | comment | added | Peter Scholze | @Z.M Which solidification to take doesn't actually matter: It's enough to take the one for $\mathbb Z$, the result will already be solid over $A$. It's part of the magic :-). And at least in my thesis, I found it psychologically quite useful to avoid $A_{\mathrm{inf}}$; I think it's no coincidence that the tilting equivalence was first proved without the intermediary of $A_{\mathrm{inf}}$, even if with it it's quite simple. These big rings are just a bit confusing... | |
| Nov 4, 2021 at 18:44 | comment | added | Peter Scholze | @pupshaw There's no natural map, in fact the isomorphism is not canonical. (A "Breuil-Kisin twist" appears.) But part 3) works nonetheless: Up to (non-unique) isomorphism, there's a unique $p$-adically complete deformation. | |
| Nov 4, 2021 at 18:33 | comment | added | Z. M | Another question: I am not sure about the meaning of the solidifcation of $L_{A/\mathbb Z}$ when $A$ is a $p$-complete ring. For example, when $A$ is $p$-completion of a discrete ring, seemingly the "natural" one to consider is the $(A,A)_\blacksquare$-solidification rather than $(A,\mathbb Z)_\blacksquare$. | |
| Nov 4, 2021 at 18:28 | comment | added | Dustin Clausen | @Z.M Well, I guess I just meant it's nicer from the perspective of making this discussion independent of the usual definition of B_dR. I'm a an of A_inf too. | |
| Nov 4, 2021 at 17:47 | comment | added | Z. M | @pupshaw Another way to rephrase this map is that this is essentially the first Breuil–Kisin twist. See the first line on page 9 of arxiv.org/abs/2111.01830 | |
| Nov 4, 2021 at 17:46 | comment | added | Z. M | @DustinClausen I am not sure how it is nicer to avoid $A_{\operatorname{inf}}$. I mean, Frobenius on this prism looks like a rotation, but I have no intuition on $\mathcal O_F$ and $\mathcal O_F^\flat$ without that picture. | |
| Nov 4, 2021 at 16:54 | comment | added | pupshaw | Marvellous to watch this playing out in real time, I'm so glad I asked. If you'll pardon the very basic question, what is the natural map $L_{R/\mathbb{Z}} \to R[1]$? | |
| Nov 4, 2021 at 10:47 | comment | added | Dustin Clausen | Thanks, Peter! I didn't know one could avoid A_inf. This is much nicer than what I was thinking of. | |
| Nov 4, 2021 at 8:06 | history | answered | Peter Scholze | CC BY-SA 4.0 |