Questions tagged [witt-vectors]
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41 questions
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Are Teichmuller elements needed for universal Witt vectors?
Are Teichmuller elements used with or needed for universal Witt vectors?
If so, what prime number p is used for determining the Teichmuller elements?
And, if so, why are they needed? (It seems that ...
2
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144
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Lifting Galois groups to Witt vectors
Let $L/K$ be an arbitrary finite Galois extension of fields. This induces an injection of Cohen-Witt rings $W_C(K) \to W_C(L)$. Cohen-Witt rings are a generalization of Witt rings - in general they ...
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Explicit construction for Cohen’s $p$-ring with imperfect residual field
Apologize if this is a below-research-level question. Asked in stack exchange but no response yet.
Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of ...
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Higher Artin-Schreier homomorphism?
For the additive group over a characteristic $p$ field one has a short exact sequence of abelian algebraic groups
$$\{1\} \to {\mathbb Z}/p \to {\mathbb G}_a \to {\mathbb G}_a \to \{1\},$$
where the ...
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1
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Splitting the Witt vectors of $\overline{\mathbb{F}_p}$
Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\...
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Image of the ghost map of $p$-typical Witt vectors and $A$-ring structure of $W(A)$
For all ring with unit element $A$ let $W(A)$ be the ring of $p$-typical Witt vectors. Denote by $$\phi\;:\;W(A)\to A^{\mathbb{N}}$$
the ghost map, which is given by
$$\phi(a_0,a_1,a_2,\ldots)\;=\;(\...
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How to prove the map of rings $\mathcal{R} \to \mathcal{R'}$ is flat?
We fix a finite extension $K$ of $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $\kappa$. Consider the ring of witt vectors $W(\kappa)$ over the residue field $\...
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Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?
Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f_*\mathcal{O}_X$ and $R^2f_*\mathcal{O}_X$ are both ...
user158636
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Cartier duality and Frobenius on Witt vector schemes
Suppose for simplicity we are working over $\mathbb{F}_p$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $Spec(\mathbb{F}_p)$. Let $\mathbb{W}_p(-)$ denote ...
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Witt vectors addition confusing
I raise this confusing because I try to understand the witt vectors for characteristic not equal to p.
Let us assume p=2. The Witt Polynomials is explicitly given by
$$
S_0=X_0+Y_0
$$
$$
S_1=X_1+...
- 397
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265
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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 \...
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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 ...
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1
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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.
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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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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 $W_{\mathcal{O}}(\...
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$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 ...
- 2,663
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2
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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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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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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 ...
3
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1
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698
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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.
$\mathbb{Z}...
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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})) = \mathbb{Z}_{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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605
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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 $\sum a_i p^i$...
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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 ...
- 2,022
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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 ...
- 4,482
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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 rings 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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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 ...
- 1,098
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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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Is there a universal property for Witt vectors?
Do the Witt vectors satisfy a universal property?
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