On the comparison of two constructions of Witt vectors of non-commutative rings

Witt vector rings and quotients of monoid algebras

In a previous paper Cuntz and Deninger introduced the ring $C(R)$ for a perfect $\mathbb{F}_p$-algebra $R$. The ring $C(R)$ is canonically isomorphic to the $p$-typical Witt ring $W(R)$. In fact there exist canonical isomorphisms $α_n \colon \mathbb{Z}R/I^n \xrightarrow{\sim} W_n(R)$. In this paper we give explicit descriptions of the isomorphisms $α_n$ for $n\geq 2$ if $p\geq n$.

Grothendieck–Witt 环上的幂结构与动机 Euler 特征

Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic

本文在域 $k$ 的 Grothendieck–Witt 环上构造了与簇的对称幂和动机 Euler 特征相容的幂结构，并证明在维数 $0$ 的簇上该结构与已知幂结构相容。

On the 2-typical de Rham-Witt complex

In this paper we introduce the 2-typical de Rham-Witt complex for arbitrary commutative, unital rings and log-rings. We describe this complex for the rings \Z and \Z_{(2)}, for the log-ring (\Z_{(2)},M) with the canonical log-structure, and we describe its behaviour under polynomial extensions. In an appendix we also describe the $p$-typical de Rham-Witt complex of (\Z_{(p)},M) for p odd.

Formalizing the Ring of Witt Vectors

The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring of characteristic $0$ out of a ring of characteristic $p > 1$, and it can be used more specifically to construct from a finite field containing $\mathbb{Z}/p\mathbb{Z}$ the corresponding unramified field extension of the $p$-adic numbers $\mathbb{Q}_p$ (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, …

On the additive and multiplicative structures of the exceptional units in finite commutative rings

Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then as an application we find a necessary and sufficient condition under which $R$ is generated by its exceptional units.

A Witt-Burnside ring attached to a pro-dihedral group

The ring of classic Witt vectors is a fundamental object in mixed characteristic commutative algebra which has many applications in number theory. There is a significant generalization due to Dress and Siebeneicher which for any profinite group G produces a ring valued functor W_G, where the classic Witt vectors are recovered as the example G = Z_p. This article explores the structure of the image of this functor where G is the pro-2 group formed by taking the inverse limit of 2-power dihedral groups, and the image of W_G is taken on a field of characteristic 2.

An alternative to Witt vectors

The ring of Witt vectors associated to a ring R is a classical tool in algebra. We introduce a ring C(R) which is more easily constructed and which is isomorphic to the ring of Witt vectors W(R) for a perfect F_p-algebra R. It is obtained as the completion of the monoid ring ZR, for the multiplicative monoid R, with respect to the powers of the kernel of the natural map from ZR to R.

Zeta functions, Grothendieck groups, and the Witt ring

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products of a variety.

An embedding problem of Noetherian rings into the Witt vectors

The aim of this article is to prove some results on the existence of an integral extension domain of a complete local Noetherian domain in mixed characteristic $p>0$ having certain distinguished properties with respect to the Frobenius map. We prove the main results by constructing required extension domains via Witt vectors and the method of maximal étale extensions. It is worth remarking that the resulting algebras have deep connections with the homological conjectures and the rings in $p$-adic Hodge theory.

Witt vectors, semirings, and total positivity

We extend the big and $p$-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the $p$-typical case, it uses positivity with respect to an apparently new basis of the $p$-typical symmetric functions. We also give explicit descriptions of the big Witt vectors of the natural numbers and of the nonnegative reals, the second of which is a restatement of Edrei's theorem on totally positive power series. Finally…

On the structure of Witt-Burnside rings attached to pro-p groups

The classical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor W_G attached to any profinite group G. The classical Witt vectors are those attached to the p-adic integers. Here we examine the ring structure of W_G(k) for several examples of pro-p groups G and fields k of characte…

可换环中的筛法定理的类比研究

An analogue of Rogers' theorem on sieving in commutative rings

本文研究了在可换环中筛法定理的类比条件，证明了当且仅当环是Dedekind环时成立，并探讨了有限可换环中该定理成立的条件。

On the de Rham-Witt complex over perfectoid rings

Fix an odd prime $p$. The results in this paper are modeled after work of Hesselholt and Hesselholt-Madsen on the $p$-typical absolute de Rham-Witt complex in mixed characteristic. We have two primary results. The first is an exact sequence which describes the kernel of the restriction map on the de Rham-Witt complex over $A$, where $A$ is the ring of integers in an algebraic extension of $\mathbb{Q}_p$, or where $A$ is a $p$-torsion-free perfectoid ring. The second result is a description of the $p$-power torsion (and related objects) in the de Rham-Witt complex over $A$, where $A$ is a $p$-t…