# Questions tagged [witt-vectors]

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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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### 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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### 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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### 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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### 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 ...