Andrés Cañas, Víctor Marín, Héctor Pinedo
In this article we construct the inverse semigroup of equivalence classes of partial Galois abelian extensions of a commutative ring R with same group G, called the Harrison partial inverse semigroup.
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Nilpotent and abelian Hopf-Galois structures on field extensions
设$L/K$是具有群$Γ$的域的有限Galois扩张.当$Γ$是幂零的时，我们证明了在$L/K$上计数所有幂零Hopf-Galois结构的问题可以归结为相应的$Γ$的Sylow子群的问题.我们用它来枚举所有的幂零（分别）。阿贝尔）Hopf-Galois结构。当$Γ$是交换的时，给出了$L/K$上的交换Hopf-Galois结构具有$Γ$型的条件.给出了$n$的一个判别准则，使得$n$的循环扩张上的\emph{every} Hopf-Galois结构具有循环型.
We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group G', a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G, +) and associative, commutative nilpotent algebra structures A on (G, +) of Caranti, et. al., we show that if A gives rise to a H-Hopf Galois structure on L/K, then the K-subHopf algebras of H correspond to the ideals of A. Among …
On the Dec group of finite abelian Galois extensions over global fields
若K/F是其伽罗瓦群指数为t的整体域的有限交换伽罗瓦扩张，则存在一个短正合序列，其结果是：若t无平方，则Dec（K/F）= Br_{t}（K/F），并以此证明具有整体剩余域的Hensel值域上的素指数除代数同构于循环代数的张量积.最后，我们构造了一个反例来证明高指数除代数的结果。
Indecomposable p-algebras and Galois subfields in generic abelian crossed products
设F是一个Hensel值域，char（F）= p，D是一个半分歧的，“非强退化”的p-代数.我们证明了D的所有伽罗瓦子域都是惯性的。以此为工具，我们研究了一般交换交叉积p-代数，证明了由非退化矩阵定义的非循环一般交换交叉积p-代数是不可分解的p-代数.为了构造这些指数为p且指数为大的不可分解p-代数的例子，我们研究了定义交换交叉积的矩阵的退化性与Severi-Brauer簇CH ^2中的挠之间的关系。
Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${\rm GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${\rm GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${\rm GR}(p^r,s)[G]$. A general f…
Galois corings applied to partial Galois theory
部分伽罗瓦扩展最近介绍了Dokuchaev，费雷罗和帕克。利用Galois核理论，我们引入了非交换环的部分Galois扩张。我们将Morita上下文与环上的部分作用相关联。
Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …
For a partial Galois extension of commutative rings we give a seven terms sequence, which is an analogue of the Chase-Harrison-Rosenberg sequence.
Galois Correspondence for Partial Groupoid Actions
本文证明了群胚在交换环上正交部分作用的伽罗瓦对应定理，并将结果推广到非正交情形，分别建立了强伽罗瓦部分群胚作用与全局群胚作用的两种对应关系。
Partial groupoid actions: globalization, Morita theory and Galois theory
本文引入了环上广群的部分作用的概念，给出了它的整体化存在的一个判据，构造了与可整体化的部分广群作用相联系的Morita上下文，并引入了部分Galois扩张的概念，它与这种上下文的严格性有关.
On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions
本文证明了可分阿贝尔扭群可实现为拟局部域的Brauer群。它描述了主拟局部域的Brauer群的同构类，并解决了关于两种基本类型的Henselian值绝对稳定域的Brauer群的约化分支的类似问题。对于拟局部域E和有限可分扩张R/E，我们得到了范数群等式$N(R/E) = N(R_{0}/E)$成立的两个充分条件，其中R_{0}是E在R中的极大交换扩张.这是用于推导信息产生的具体关系之间的伽罗瓦群和有限伽罗瓦扩展E的范群。
Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension
本研究确定了可解扩张上作为Hopf-Galois结构类型的有限非阿贝尔单群，同时给出了乘法群可解的斜brace中作为加法群的这类单群的完整列表。
Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras. On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field F is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of F and the isomorphism class of a G-Galois extension of F. This connection is used to classify the simple G-Galois extensions of F in terms of …
Let $\operatorname{G}$ be a finite groupoid and $α=(S_g,α_g)_{g\in \operatorname{G}}$ a unital partial action of group-type of $\operatorname{G}$ on a commutative ring $S=\oplus_{y\in\operatorname{G}_0}S_y$. We shall prove a Galois correspondence between a class of wide subgroupoids of $\operatorname{G}$ and a class of subrings of $S$. We recover known results for global groupoid actions and we give several examples to illustrate the correspondence.
This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$ on the subalgebra of invariants $S^{α_H}$ of $S$ such that $S^{α_H}$ is an $α_{G/H}$-partial Galois extension of $S^{α_G}$. Second, assuming that $G$ is abelian, we construct a commutative inverse semigroup $T_{par}(G,R)$, whose elements are equivalence classes of $α_G$-part…
We are considering iterative derivations on the function field L of abelian schemes in positive characteristic p>0, and give conditions when the torsion group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are the ID-Galois groups of L over certain ID-subfields. For an explicit example, we even give a construction of (a family of) such iterative derivations.
Partial Hopf-Galois theory
本文发展了部分H-模代数的部分Hopf-Galois理论，并恢复了Hopf代数经典结果的类似物。
Partial (co)actions of Hopf algebras and partial Hopf-Galois theory
引入了霍普夫代数$H$在代数上的部分（余）作用。为此，我们首先介绍的概念，宽松的核心，推广Wisbauer的概念弱核心。我们还有松懈环的双重概念。给出了部分$H$ -余代数的几个对偶结果，发展了部分$H$ -余代数的伽罗瓦论.
Galois descent of splendid Rickard equivalences for blocks of $p$-nilpotent groups
本文通过证明链复形的一般Galois下降准则，加强了p-幂零群块等价及辉煌Rickard等价的Galois下降结果，从而验证了Broué阿贝尔亏群猜想的一个精细化版本。
Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case
本文完成了当伽罗瓦群 Sylow $p$-子群为初等阿贝尔群时，$p^{2}q$ 次扩张的 Hopf-Galois 结构及相应斜括号的分类，通过伽马函数与全形正则子群的一一对应发展了更概念化的枚举方法。
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