# Questions tagged [witt-vectors]

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33
questions

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### Cartier duality and Frobenius on Witt vector schemes

Suppose for simplicity we are working over $\mathbb{F}_p$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $Spec(\mathbb{F}_p)$. Let $\mathbb{W}_p(-)$ denote ...

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424 views

### Witt vectors addition confusing

I raise this confusing because I try to understand the witt vectors for characteristic not equal to p.
Let us assume p=2. The Witt Polynomials is explicitly given by
$$
S_0=X_0+Y_0
$$
$$
S_1=X_1+...

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230 views

### Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?

Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...

**7**

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233 views

### Witt vectors and flat liftings of (non)perfect fields

My motivating question is as follows: Why does Theorem 2.1. in Deligne-Illusie's classical work on the Hodge degeneration (EUDML link), i.e. the decomposition theorem, does $k$ need to be a perfect ...

**14**

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**1**answer

797 views

### How to visualize a Witt vector?

As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.

**6**

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378 views

### Group schemes over ring of Witt vectors and their representing algebras

Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring ...

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194 views

### Specific unit in ring of Witt vectors

Let $\mathcal{O}$ be the ring of integers in a $p$-adic local field, totally ramified over $\mathbb{Q}_p$. We fix a uniformizer $\pi$ and form the ring of relative Witt vectors $W_{\mathcal{O}}(\...

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395 views

### $p$-adic completeness of the ring of Witt vectors

Let $R$ be a ring that is $p$-adically complete for a prime $p$ and let $W(R)$ denote the ring of $p$-typical Witt vectors. Is it true that $W(R)$ is $p$-adically complete? (A ring $A$ is $p$-adically ...

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### Witt-vector vectors

I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...

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172 views

### Universal Witt vectors in full complete closed p-adic space omega?

Is there a p-adic mathematical structure that incorporates the advantages of both universal Witt vectors (not p-typical-limited; implementing Frobenius and Verschiebung operations) and permitting ...

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121 views

### How is the p-adic norm calculated when using universal witt vectors?

How is the p-adic norm calculated when using UNIVERSAL WITT VECTORS?
Is the p-adic norm calculated in the familiar way, in the sense that we look to the last digit to the right, and the prime number ...

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506 views

### 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.
$\mathbb{Z}...

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62 views

### 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 ...

**4**

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510 views

### 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})) = \mathbb{Z}_{p}[x]/(x^{n})...

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177 views

### 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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418 views

### 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 $\sum a_i p^i$...

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288 views

### 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 ...

**12**

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**1**answer

663 views

### 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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**1**answer

471 views

### 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 ...

**7**

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**1**answer

561 views

### 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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283 views

### 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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416 views

### 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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661 views

### 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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**1**answer

353 views

### 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 $A\...

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629 views

### 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 $p_1,p_2,\dots$...

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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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352 views

### 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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**4**answers

4k views