The witt-vectors tag has no usage guidance.

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### Witt vectors and maps of $\lambda$-rings

Consider the ring $W(\mathbb{F}_p)$ of big Witt vectors of $\mathbb{F}_p$. This has a natural structure of a $\lambda$-ring (in the strong sense) since rings of big Witt vectors always do.
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### Do any specializations of variables give valid equalities of series and products involving Witt symmetric functions?

Formally, Witt symmetric functions $w_n(x_1,x_2,...)$ ($n\geqslant1$) can be defined by
$$
\prod_n(1-w_nt^n)=1+\sum_k(-1)^ke_kt^k=\prod_j(1-x_jt),
$$
where $e_k(x_1,x_2,...)$ are the elementary ...

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### Explicit description of rings of Witt vectors

I have some basic questions on the rings of Witt vectors. The first example one looks at is $W(\mathbb{F}_{p})= \mathbb{Z}_{p}$. Is it known if $W(\mathbb{F}_{p}[x]/(x^{n})) = ...

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### Efficiently computing (plethysm-like?)substitutions of symmetric functions

This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...

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### Power series defined by Witt vectors / Teichmüller representatives of p-adics

Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i ...

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### Ring of Witt Vectors and Tensor product of Fields

Let $p > 2$ be a prime, and let $\textbf{F}_{p} =
\textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over
$\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic
$p$. Then we have ...

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### Vanishing theorems in positive characteristic

In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...

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### Vanishing cohomology of de-Rham Witt complex

Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand,
there is a surjective morphism from
$\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which ...

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### Criteria for ghost-Witt vectors: looking for history and references

I am looking for references (both of the readable and of the historical kind!) for the following result (which I formulate in one of its least general forms, so as not to complicate the discussion). I ...

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### Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...

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### Has anyone used this theorem of P. Cartier?

In "Groupes Algebriques et Groupes Formels", Conf. au coll. sur la theorie des groupes algebriques, Bruxelles 1962, P. Cartier proves the following in Section 9, Theoreme 1:
(What follows is my ...

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### Polynomials for addition in the Witt vectors

The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the ...

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### Deformation space of non-ordinary abelian varieties

It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...

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### Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius

This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ ...

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### About Frobenius of Witt vectors

Let $k$ be a characteristic $p$ alg. closed field, Let $W(k)$ be the Witt vectors， Let $\sigma$ be the Frobenius, then we also have $\sigma: W(k)^{\times} \to W(k)^{\times}$, where $W(k)^{\times}$ are ...

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### Tensor Product of Witt Vectors

Let $A$, $B$, and $C$ be commutative rings such that $A\otimes_C B$ makes sense. If $W_n(A\otimes_C B), W_n(A), W_n(C),$ and $W_n(B)$ are the length $n$ Witt vectors of the rings $A,B,C,$ and ...

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### Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties?

Let $\Lambda$ denote the ring of symmetric functions in variables $x_1,x_2,\dots$ and with coefficients in $\mathbf{Q}$. Then $\Lambda$ is freely generated as an $\mathbf{Q}$-algebra by ...

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### To what extent does Spec R determine Spec of the Witt vector ring over R?

Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from ...

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### Ghost-Witt sequences vs. ghost-Polya-Burnside sequences?

If you're in a hurry scroll down until the questions:
First the known part:
A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Witt sequence if there exists a sequence ...

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### Ghost components of a Witt vector - Motivation

I'd like to know if anyone has a good explanation for where the ghost components that are used to define Witt vectors come from. A lot of sources I've read take the ghost components for their ...

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### Do n-th Witt polynomials generate {P | P' is divisible by n} ?

EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting.
Let $P\in\mathbb{Z}\left[\Xi\right]$ be a polynomial (where $\Xi$ is a family of symbols that we use ...

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