Talk:Witt vector

Witt polynomials?

What is the precise relation between Witt vectors and Witt polynomials? Thanks. Jakob.scholbach 20:27, 3 April 2007 (UTC)

Finite dimensional?

Isn't there some finite-dimensional algebra of Witt n-vectors, of which this is presumably the limit as n goes to infinity? That's the impression I got from looking at Serre's article "Sur la topologie des varietes algebriques en caracteristique p."

What's confusing is that the allegedly different Teichmüller representatives, if defined as the solutions of x^p − x = 0 in F_p, are just F_p. And if we take {0, 1, 2, ..., p − 1} as the representatives of F_p, then nothing has changed! What's really going on is that we want the solutions of x^p − x = 0 in Z_p, and each such solution mod p gives a distinct element in F_p. We can then pick our usual representatives for F_p to represent the Teichmüller representatives, but the addition and multiplication arithmetic would be different, as the rest of the article describes. 128.105.2.11 (talk) 21:10, 9 June 2011 (UTC)

A concrete example would really help here. What are the Teichmüller representatives for F₅, for example? The 5-1'th roots of unity are 1, -1, i and -i, but how does i represent an element of F₅? There appears to be no natural way to determine whether the class containing 2 should be represented by i or -i. –Henning Makholm (talk) 16:34, 8 September 2011 (UTC)

The Teichmüller rep.s of Z_5 are the 4th roots of unity in Z_5, plus zero. I wouldn't call one of them "i" since i is usually meant to be a 4th root of unity in C, and Z_p is not (naturally) a subring of C - maybe call it "i_p". i_p can be approximated by Hensel's lemma, as a zero of X^4-1 (discarding the "trivial solutions 1 and -1, and choosing one of the remaining two). This gives a representation i_p=a_0+a_1p+a_2p^2+..., with a_i between 0 and p-1. The term a_0 will be either 2 or 3, and this gives you the correspondence to F_5. --Roentgenium111 (talk) 14:40, 18 November 2011 (UTC)

Re-write

I just re-wrote the entire section called "details", hopefully making it much more clear as to what is going on. I did not change the flow or development of that section, but sharpened the focus, tightened up the language, removed much of the ambiguity. I had found it painfully impossible to understand as originally written; I hope its OK now. Well, the last half of the argument could use some more cleanup, but I'm exhausted for now. 67.198.37.16 (talk) 19:49, 5 July 2016 (UTC)

Ghost Components - misleading?

The sentence "The ghost components can be thought of as an alternative coordinate system ..." strikes me as deeply misleading, since when the ring you start with has characteristic p as it usually will, the map to ghost components is definitely not injective.

Universal property?

Let R be perfect of characteristic p. Since $\mathbb {F} _{p^{n}}\to \mathbb {F} _{p}$ has nilpotent kernel, deformation theory tells us that the category of formally étale algebras over $\mathbb {F} _{p^{n}}$ and $\mathbb {F} _{p}$ are equivalent, and the functor is given by nothing but $W_{n}(R)$. Furthermore, the Witt vector $W(R):=\lim _{\longleftarrow }W_{n}(R)$ is the unique p-adically complete p-torsionfree $\mathbb {Z} _{p}$-algebra lifting R. --Fourier-Deligne Transgirl (talk) 08:49, 17 November 2022 (UTC)