Let $A$ be any unital associative, possibly non-commutative ring and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be the abelian group constructed by Hesselholt. In arXiv:1708.04065 it was proved that if $ p=2$ and $A$ is non commutative unital torsion free ring then there is no surjective continuous group homomorphism from $W(A) \to HH_0(E(A)): = E(A)/\overline{[E(A),E(A)]}$ which commutes with the Verschiebung operator and the Teichmüller map. In this paper we generalise this result to all primes $p$ and simplify the arguments used for $p=2$. We also prove that if $A$ a is non-commutative unital ring then there is no continuous map of sets $HH_0(E(A)) \to W(A)$ which commutes with the ghost maps.
We study Veronese and Segre morphisms between non-commutative projective spaces. We compute finite reduced Gröbner bases for their kernels, and we compare them with their analogues in the commutative case.
The purpose of this this paper is to generalize the functors arising from the theory of Witt vectors duto to Cartier. Given a polynomial $g(q)\in \mathbb Z[q]$, we construct a functor ${\overline {W}}^{g(q)}$ from the category of $\mathbb Z[q]$-algebras to that of commutative rings. When $q$ is specialized into an integer $m$, it produces a functor from the category of commutative rings with unity to that of commutative rings. In a similar way, we also construct several functors related to ${\overline { W}}^{g(q)}$. Functorial and structural properties such as induction, restriction, classific…
Let $A$ be any associative ring , possibly non-commutative, and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be that constructed by Hesselholt. The goal of this paper is to answer the following question by Hesselholt: Is $HH_0(E(A)) $ isomorphic to $W(A)$? We show that in the case $p=2$, there is no such isomorphism possible if one insists it to be compatible with the Verscheibung operator and the Teichmüller map.
The main theorem (Theorem 4.1) of this paper claims that any ring morphism from an Azumaya algebra of constant rank over a commutative ring to another one of the same constant rank and over a reduced commutative ring induces a ring morphism between the centers of these algebras. The second main theorem (Theorem 5.3) implies (through its Corollary 5.4) that a ring morphism between two Azumaya algebras of the same constant rank is an isomorphism if and only if it induces an isomorphism between their centers. As preliminary to these theorems we also prove a theorem (Theorem 2.8) describing the Br…
In this paper, we construct a $q$-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where $q$ ranges over the set of integers. When $q=1$, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. {70} (1988), 87-132). To achieve our goal we first show that there exists a $q$-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case $q=1$, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their…
For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that W(R) := W(R;R) is Morita invariant in R. For an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element $χ_f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The ass…
In this paper a further study is made of $H$-signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms $H$ may be replaced by a single form and that the $H$-signature is invariant under Morita equivalence of algebras with involution. The "prime ideals" of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz-Leicht. It follows that $H$-signatures canonically correspond to morphisms into the integers.
For $a,b\in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skewsymmetric) non-inner biderivations. As an applications, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.
To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism taking its values in Q[X], we deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber's formula. We also give non-commutative versions of these results: polynomials…
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $ϕ_σ : RG\to RG$ denote the involution defined by $ϕ_σ (\sum r_{g}g) = \sum r_{g} σ(g) g^{-1}$, where $σ:G\to \{\pm 1\}$ is a group homomorphism (called an orientation morphism). An element $x$ in $RG$ is said to be antisymmetric if $ϕ_σ (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute.
The main purpose of this paper is to study non-commutative ternary Nambu-Poisson algebras and their Hom-type version. We provide construction results dealing with tensor product and direct sums of two (non-commutative) ternary (Hom-)Nambu-Poisson algebras. Moreover, we explore twisting principle of (non-commutative) ternary Nambu-Poisson algebras along with an algebra morphism that lead to construct (non-commutative) ternary Hom-Nambu-Poisson algebras. Furthermore, we provide examples and a 3-dimensional classification of non-commutative ternary Nambu-Poisson algebras.
In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are inde…
This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebras to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show th…
Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G\_\infty$-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any $C\_\infty$-morphism $ϕ$ ({\rm i.e.} morphism of commutative, associative algebra up to homotopy) between $\g\_1$ and $\g\_2$, there exists a $G\_\infty$-morphism $Φ$ between $\g\_1$ and $\g\_2$ that restricts to $ϕ$. We also show that any $L\_\infty$-morphism ({\rm i.e.} morphism of Lie algebra up to homotopy), in particular the…
Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$.
