# Tagged Questions

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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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### Is there a universal property for Witt vectors?

Do the Witt vectors satisfy a universal property?
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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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### 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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### 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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### 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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### $p$-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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### 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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### 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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### 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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### 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 ...