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 xp &minus; x = 0 in Fp, are just Fp. And if we take {0, 1, 2, ..., p &minus; 1} as the representatives of Fp, then nothing has changed! What's really going on is that we want the solutions of xp &minus; x = 0 in Zp, and each such solution mod p gives a distinct element in Fp. We can then pick our usual representatives for Fp 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)