Christopher Deninger, Young-Tak Oh
We determine the universal deformation over reduced base rings of the Witt ring scheme enhanced by a Frobenius lift and Verschiebung. It agrees with a q-deformation earlier introduced by the second author, for which we also give a simpler description. In the appendix we discuss a Witt vector theory for ind-rings which may be of independent interest.
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A closer look at Witt rings for forms of higher degree
该研究重新探讨Harrison和Pareigis提出的d次非退化形式Witt环的定义与特例，完成H-形式分类，定义了d次对角形式Witt环及两种新的d次非退化形式Witt环。
This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability. We call such partial orders presentable. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of quadratically presentable fields, that is fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. As an application, we show that Witt rings of …
$q$-deformation of Witt-Burnside rings
本文构造了交换环上投射有限群的Witt-Burnside环的一个$q$ -变形，其中$q$的值域为整数集.当$q=1$时，它与A.衣服和C。Siebeneicher（Adv. Math. {70}（1988），87-132）.为了实现我们的目标，我们首先证明了存在一个$q$ -变形的项链环上的一个投射有限群的交换环。在经典的情况下，即，表壳$q=1$、q变形的Witt-Burnside戒指和项链戒指总是配备有感应和限制。我们还研究了它们的性质。作为一个副产品，我们证明了一个猜想，由于Lenart（J代数。199（1998），703-732）。最后，在$G$是交换投射有限群的情况下，我们将$\mathbb W_G^q$分类为严格自然同构.
Normal Quaternionic Matrices and Finitely Generated Witt Rings
本文通过四元数矩阵方法验证了抽象威特环的初等类型猜想，利用2- torsion部分验证了阿罗森-福斯特主定理，并通过计算找到n至7的矩阵结构。
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 simplif…
We prove that every isometry of between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the broader class of semilocal rings, but with certain restrictions on the base modules, which cannot be removed in general. Our result implies that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. This improves a cancellation result of Quebbemann, Scharlau and Schulte, which applies to quadratic spaces over hermitian categories. Comb…
Some Galois extensions of quadratic extensions associated with Witt rings
确定了一个Galois域扩张$E/F$，其伽罗瓦群是指数至多为8的亲2-群，幂零类至多为4，使得它包含了关于$F$的所有二次扩张的Witt环的基本信息.
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…
Mixed Witt rings of algebras with involution
本文通过将双线性形式与厄米特形式结合，为第一类带对合代数定义了混合Witt环，并推导了其基于对合迹形式的乘积公式与互反公式等基本性质。
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 $\mathbb{F}_2[x]/(x^n)$.
Poincare-Birkhoff-Witt Deformations of Calabi-Yau Algebras
最近，Bocklandt证明了Van den Bergh在其分次版本中提出的一个猜想，指出一个3维Calabi-Yau的分次代数（有关系）是由齐次势W定义的。本文证明了，如果我们给W加上任意一个较小次的势，我们得到A的Poincare-Birkhoff-Witt形变。这种PBW变形是卡-丘变形，并且在A的所有PBW变形中具有特征。提出了各种例子。
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
Structure theorems for AP rings
在“新证明结构定理的维特环”，刘易斯表明如何标准环理论的结果可以推导出一个快速和基本的方式从事实上，维特环的一个领域是完整的，并从具体性质的显式零化多项式，他提供。我们将在本文中表明，同样的结构结果适用于更大类的交换环，而不仅仅是Witt环。我们将为这些环构造零化多项式。
On the presentation of the Grothendieck-Witt group of symmetric bilinear forms over local rings
本文证明了剩余域不为F2的交换局部环上内积空间的链引理，并借此确立了该环上对称双线性型Grothendieck-Witt群的通常展示成立。
In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.
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.
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…
An Artin-Schreier-type theory for signatures of hermitian forms over involutions
本文研究了第一类对合代数上埃尔米特型的特征理论，证明了Witt环素谱与序空间的相互决定关系可通过混合Witt环推广至该情形。
Global Deformations of the Witt Algebra of Krichever-Novikov Type
通过考虑由尺规作图法给出的Witt（和Virasoro）代数的非平凡全局变形，证明了尽管它们具有无穷小的形式刚性，但它们不是全局刚性的.这表明需要明确指出所考虑的变形类型。出现的家庭被构造为家庭的代数的Krichever-Novikov型。它们以一种自然的方式出现在二维共形域论量子化的整体算符方法中。此外，给出了Witt代数的无穷小刚性和形式刚性的一个证明。
A natural eñe product construction of the Big Witt ring
通过多项式零点作用定义的eñe积，给出大Witt环的一个直接、自包含且自然的构造，避免了经典构造中突兀的公式。
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 note, we introduce a notion of multi-parameter formal deformations of ternary hom-Nambu-Lie algebras. Within this framework, we construct formal deformations of the three-dimensional Jacobian determinant and of the cross-product in four-dimensional Euclidean space. We also conclude that the previously defined ternary q-Virasoro-Witt algebra is a formal deformation of the ternary Virasoro-Witt algebra.
We classify all invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring) with values in $A$, that it to say functions $I^n(K)\rightarrow A(K)$ compatible with field extensions, in the cases where $A(K)=W(K)$ is the Witt ring and $A(K)=H^*(K,μ_2)$ is mod 2 Galois cohomology. This is done in terms of some invariants $f_n^d$ that behave like divided powers with respect to sums of Pfister forms, and we show that any invariant of $I^n$ can be written uniquely as a (possibly infinite) combination of those $f_n^d$. This in particular allows to lift operations defined on mod …
A universal group-theoretic characterisation of $p$-typical Witt vectors
本文通过群论方法，证明p-典型威利特向量群的性质本质特征，适用于非交换环的推广。
Deformations of Lie Algebras using $σ$-derivations
本文基于$σ$ -导子给出了Witt代数和Virasoro代数的变形.我们表明，$σ$ -扭曲Jacobi型恒等式持有这样的变形的发电机。对于这种构造所模拟的李代数的$σ$ -扭曲推广，我们发展了一个中心扩张理论。我们表明，我们的方法可以用来构建新的变形的李代数和他们的中心扩展，其中特别是自然包括$q$ -变形的Witt和Virasoro代数相关的$q$ -差分算子，也提供相应的q-变形的雅可比恒等式。
We introduce an $A_\infty$-algebra structure on the Hochschild cohomology of the endomorphism bimodule of a finite-dimensional representation of an associative algebra. We prove that this structure determines a presentation for non-commutative deformations of the representation. From this, we deduce presentations of universal deformation rings of Galois representations. In turn, we apply these presentations in order to deduce universal deformation rings of Galois pseudorepresentations, supplying a a tangent and obstruction theory for pseudorepresentations. This generalizes the broadly used tan…
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