Timeline for Motivation for Henselian rings in algebraic geometry
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 21, 2023 at 13:10 | vote | accept | user267839 | ||
| Jun 18, 2021 at 23:01 | comment | added | LSpice | @Rachmaninow98, since you said that this answer was exactly what you were looking for, it would be a good idea to accept it. | |
| Mar 31, 2021 at 6:15 | comment | added | ali | of course that is true. | |
| Mar 30, 2021 at 23:02 | comment | added | user267839 | alright now, thank you again. little nitpick, when you consider the thickering as above I think you mean the map in other direction than that one you wrote, ie tho canonical projection $\mathbb{Z}_p/P^n \to F_p$ not $F_p\to \mathbb{Z}_p/P^n$? | |
| Mar 30, 2021 at 22:56 | history | bounty ended | user267839 | ||
| Mar 30, 2021 at 6:09 | comment | added | ali | no there is no relation other than that often Witt vector is complete for p-adic topology. completeness is the important thing. the relation between $F(\bar{A})$ and $f(A/m)$ is true for all ring not only henselian ring I think a good start to read about deformation theory is part 3 of the book FGA explained, there is also the stack project and illusie thesis which are more technical. | |
| Mar 29, 2021 at 20:09 | comment | added | user267839 | and finally: could you recommend a book/script on defo theory where this relation of $F(\overline{A}) $ and $F(A/m)$ for Henselian $A$ is discussed in detail? | |
| Mar 29, 2021 at 20:05 | comment | added | user267839 | so in general there is no direct connection between formal smoothness and the concept of Witt vectors? | |
| Mar 29, 2021 at 20:03 | history | edited | ali | CC BY-SA 4.0 |
added 8 characters in body
|
| Mar 29, 2021 at 20:01 | comment | added | ali | for your second question I Just mean all the power series that satisfy a polynomial equation over $\mathbb{C}$ which is as I said the Henselisation of $(C[X],X)$ | |
| Mar 29, 2021 at 19:59 | comment | added | ali | @Rachmaninow98 about your first question no I really meant $Z_p/p^n$ I just use witt vector as the force of habit. if you have a complete local ring $A$ with residue field $k$ then of course you should use $A/m^n$ not $W_n(k)$. | |
| Mar 29, 2021 at 19:58 | comment | added | user267839 | Secondly: what do you mean by $\mathbb C[[x]]^{alg}$? I know this notation only for fields. | |
| Mar 29, 2021 at 19:58 | comment | added | user267839 | Thank you for your answer. This one is exactly was looking for! Two questions on your notations: In your first part you wrote $ F_p \to W_n(F_p)$ where as I know $W_n$ means the Witt ring wrt $F_p$, an in this case that's $\mathbb{Z}_p$. Was the choice of Witt-notation intended to indicate in which direction the notion of formal smoothness generalizes the usual Henselian property in your example with curve $C= V(f)$ to $A \to W_n(A)$ where $A$ is is an arbitrary ring? | |
| Mar 29, 2021 at 10:20 | history | answered | ali | CC BY-SA 4.0 |