Related papers: An alternative to Witt vectors

The cone of a classical group $G$ is an affine $G\times G$-variety. The aim of this note is to initiate its combinatorial study in the cases when $G$ is the complex orthogonal or symplectic group. The coordinate ring of the cone of $G$ is a…

We recall some basic computations in the Milnor-Witt K-theory of a field, following Morel. We then focus on the Witt K-theory of a field of characteristic two and give an elementary proof of the fact that it is isomorphic as a graded ring…

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…

Let $C$ denote a closed convex cone $C$ in $\mathbb{R}^d$ with apex at 0. We denote by $\mathcal{E}'(C)$ the set of distributions having compact support which is contained in $C$. Then $\mathcal{E}'(C)$ is a ring with the usual addition and…

Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras…

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for…

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in…

Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f :…

Let $\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\mathbb{F}$, let $W_k(\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\mathbb{F}$. We show that the group…

In this note we recall some recent progress in understanding the representation theory of *-algebras over rings C = R(i) where R is ordered and i^2 = -1. The representation spaces are modules over auxiliary *-algebras with inner products…

We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable…

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of…

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a $\mathbb{Z}$-algebra which is a finitely generated $\mathbb{Z}$-module, we assume that $R$ is explicitly given by a system of $\mathbb{Z}$-module generators $G$, its relation module ${\rm…

We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by min{v(x),v(y)}=1. It has a natural structure of an R-tree, induced by the order relation v is less than v' iff v(f) is less than v'(f) for all f. It…

Let $\mathscr{C}$ be a small category. For every commutative ring $R$ with unity, we associate an $R\mathrm{-linear}$ abelian category with the universal homotopy category of $\mathscr{C}$, where we can do the corresponding homological…

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…

We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary…

Let K be a field of any characteristic and let R be an algebra generated by two elements satisfying quadratic equations. Then R is a homomorphic image of F=K<x,y | x^2+ax+b=0,y^2+cy+d=0> for suitable a,b,c,d in K. We establish that F can be…

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the…

This paper aims at the following results: \begin{enumerate} \item The class of all $*$-regular rings forms a variety. \item A subdirectly irreducible $*$-regular ring $R$ is faithfully representable (i.e. isomorphic to a subring of an…