M. Dokuchaev, A. Paques, H. Pinedo, I. Rocha
For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.
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Given a partial action αof a group G on an associative algebra A we consider the crossed product A x_αG. Using the algebras of multipliers of ideals of A we prove that A x_αG is associative, provided that all ideals of A are idempotent. This generalizes a previous result on the associativity of A x_αG in the context of C*-algebras. We also give a criteria for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group al…
For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial C*-crossed products as groupoid C*-algebras. For completeness we include a new proof of the C* result for free partial actions.
In this article, we consider a twisted partial action $α$ of a group $G$ on a ring $R$ and it is associated partial crossed product $R*_α^wG$. We study necessary and sufficient conditions for the commutativity and simplicity of $R*_α^wG$. Let $R=C(X)$ the algebra of continuous functions of a topological space $X$ on the complex numbers and $C(X)*_αG$ the partial skew group ring, where $α$ is a partial action of a topological group $G$ on $C(X)$. We study some topological properties to obtain results on the algebra $C(X)$. Also, we study the simplicity of $C(X)*_αG$ using topological properties…
Given a non-necessarily commutative unital ring $R$ and a unital partial representation $Θ$ of a group $G$ into the Picard semigroup $\mathbf{PicS} (R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(Θ/R) $ formed by the isomorphism classes of partial generalized crossed products related to $Θ$ and identify an appropriate second partial cohomology group of $G$ with a naturally defined subgroup $\mathcal{C}_0(Θ/R) $ of $\mathcal{C}(Θ/R).$ Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact seque…
For a twisted partial action Θof a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_ΘG is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_ΘG for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is …
Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant, compactly supported functions on $X$, taking values in a given field $K$. We then study the ideal structure of the algebraic partial crossed product $L_c(X)\rtimes G$. After developping a theory of induced ideals, we show that every ideal in $L_c(X)\rtimes G$ may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic ver…
In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.
Prime group graded rings with applications to partial crossed products and Leavitt path algebras
本文研究了素数群格环的性质，并将其应用于部分交叉积和莱维特路径代数中，扩展了相关经典结果。
Partial generalized crossed products, Brauer groups and a comparison of seven-term exact sequences
该研究针对群在交换环上的部分作用，刻画了部分广义交叉积同构类群，证明部分Galois扩张下含R为极大交换子代数的Azumaya代数的结构，建立相对Brauer群与该类群的商关系，并沟通了两类七项正合列。
Let A#_{α, ω}H be a partial crossed product. In this paper, we first generalize the theorem about the existence of an enveloping action to twisted partial actions. Second, we construct a Morita context between the partial crossed product and the crossed product related to the enveloping action. Furthermore, we discuss equivalences of partial crossed products Finally, we investigate when A\subset A#_{α, ω}H becomes a separable extension.
The twisted partial group algebra and (co)homology of partial crossed products
本文引入扭曲部分群代数来刻画群的部分射影表示，并构造收敛到部分交叉积 Hochschild 同调与上同调的 Grothendieck 谱序列。
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case tha…
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded…
Given an inverse semigroup $S$ endowed with a partial action on a topological space $X$, we construct a groupoid of germs $S\ltimes X$ in a manner similar to Exel's groupoid of germs, and similarly a partial action of $S$ on an algebra $A$ induces a crossed product $A\rtimes S$. We then prove, in the setting of partial actions, that if $X$ is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs $S\ltimes X$ is isomorphic to the crossed product $A_R(X)\rtimes S$, where $A_R(X)$ is the Steinberg algebra of $X$. We also prove that the converse holds,…
Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group graded situation to the groupoid graded case, of the bijection from a certain second cohomology group, defined by the grading and the fun…
For a partial Galois extension of commutative rings we give a seven terms sequence, which is an analogue of the Chase-Harrison-Rosenberg sequence.
Very good gradings on matrix rings are epsilon-strong
本文研究矩阵环上的群分次，引入非常好分次并证明其必为 epsilon-强，同时给出 epsilon-交叉积的充分条件及 IBN 情形下的刻画。
Very good gradings on structural matrix rings
研究结构矩阵环Mₙ(ρ,R)上的优良G分次，证明其与预序ρ上的分次在平凡、对称等性质上等价，给出epsilon-强性判据，还建立了其与G自由偏作用的一一对应关系及分类结果。
We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity o…
The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz conformal subalgebra. For this purpose, an unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when $R$ is a free $\mathbb{C}[…
Algebraic constructions for left-symmetric conformal algebras
本文引入左对称共形代数的统一积概念，构造对象 $\mathcal{H}^2_R(Q,R)$ 以分类直和上的代数结构，并详细探讨了秩为1及特殊积的情形。
Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underlineΘ : \G \to \Picar{R}$ to the Picard group of $R$. We study under which conditions $\underlineΘ$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \subseteq S$ with the same set of local units and assuming that $\underlineΘ$ is induced by a homomorphism of groups $\G \to \Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exac…
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