Duality Theorems for Crossed Products over Rings

On the regularity of crossed products

We study some generalizations of the notion of regular crossed products K*G. For the case when K is an algebraically closed field, we give necessary and sufficient conditions for the twisted group ring K*G to be an n-weakly regular ring, a $ξ^* N$-ring or a ring without nilpotent elements.

Epsilon-strongly graded rings, separability and semisimplicity

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…

Weak crossed-product orders over valuation rings

Let $F$ be a field, let $V$ be a valuation ring of $F$ of arbitrary Krull dimension (rank), let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Let $f:G\times G\mapsto K\setminus \{0\}$ be a normalized two-cocycle such that $f(G\times G)\subseteq S\setminus \{0\}$, but we do not require that $f$ should take values in the group of multiplicative units of $S$. One can construct a crossed-product $V$-order $A_f=\sum_{σ\in G}Sx_σ$ with multiplication given by $x_σsx_τ=σ(s)f(σ,τ)x_{στ}$ for $s\in S$, $σ,τ\in G$. We characterize semiheredita…

Maximal Crossed Product Orders over Discrete Valuation Rings

The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abel…

素数群格环及其在部分交叉积和莱维特路径代数中的应用

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。

Free Division Rings of Fractions of Crossed Products of Groups With Conradian Left-Orders

Let $D$ be a division ring of fractions of a crossed product $F[G,η,α]$ where $F$ is a skew field and $G$ is a group with Conradian left-order $\leq$. For $D$ we introduce the notion of freeness with respect to $\leq$ and show that $D$ is free in this sense if and only if $D$ can canonically be embedded into the endomorphism ring of the right $F$-vector space $F((G))$ of all formal power series in $G$ over $F$ with respect to $\leq$. From this we obtain that all division rings of fractions of $F[G,η,α]$ which are free with respect to at least one Conradian left-order of $G$ are isomorphic and …

Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

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…

Crossed product Leavitt path algebras

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb Z$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded…

Epsilon-strongly groupoid graded rings, the Picard inverse category and cohomology

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…

Normality of algebras over commutative rings, crossed pairs, and the Teichmueller class

Let S be a commutative ring, Q a group that acts on S, and let R be the subring of S fixed under Q. A Q-normal S-algebra consists of a central S-algebra A and a homomorphism s from Q to the group Out(A) of outer automorphisms of A that lifts the Q-action on S. We associate to a Q-normal S-algebra (A,s) a crossed 2-fold extension which, in turn, represents a class, the Teichmueller class of (A,s), in the third cohomology group of Q with coefficients in the group U(S) of units of S, endowed with the obvious Q-module structure. Suitable equivalence classes of Q-normal Azumaya S-algebras constitut…

Partial crossed products as equivalence relation algebras

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.

Partial generalized crossed products and a seven-term exact sequence

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

On a class of hereditary crossed-product orders

In this brief note, we revisit a class of crossed-product orders over discrete valuation rings introduced by D. E. Haile. We give simple but useful criteria, which involve only the two-cocycle associated with a given crossed-product order, for determining whether such an order is a hereditary order or a maximal order.

Twists, crossed products and inverse semigroup cohomology

Twisted étale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists…

Simplicity of Partial Crossed Products

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…

矩阵环上的非常好分次是 epsilon-强

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。

Partial generalized crossed products and a seven term exact sequence (expanded version)

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…

Weak crossed product orders arising from partitions of a finite group

Let R be a complete discrete valuation ring with quotient field K, L a finite Galois extension of K with Galois group G and S the integral closure of R in L. In this article, using elements of the monoid Sl(G), the set of semilinear maps of G on the set of natural numbers introduced in [8], we construct certain crossed product orders in the case that the extension S/R is unramified. We associate an element of Sl(G) to every crossed product order and give a criterion for inflating idempotent 2-cocycles.

非阿贝尔群oid的扩展及其群oid环

Non-Abelian extensions of groupoids and their groupoid rings

本文提出了一种群oid扩展的分类理论，并展示了群oid交叉积的构造及其分类意义。

Artinian 环的余倾斜对偶

Cotilting duality for Artinian rings

本文推广 Morita-Azumaya 经典对偶定理至余倾斜双模，通过有界导出范畴分析其表示的对偶。

结构矩阵环上的优良分次

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。

Noncommutative Pierce Duality between Steinberg Rings and Ample Ringoid Bundles

Classic work of Pierce and Dauns-Hofmann shows that biregular rings are dual to simple ring bundles over Stone spaces. We extend this duality to Steinberg rings, a purely algebraic generalisation of Steinberg algebras, and ringoid bundles over ample groupoids. We base this largely on an even more general extension of Lawson's noncommutative Stone duality, specifically between Steinberg semigroups, a generalisation of Boolean inverse semigroups, and category bundles over ample groupoids.

Crossed Products and Coding Theory

Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main feature of ambient code spaces which is used to evaluate the efficiency of their various codes. This note aims at classifying the ambient spaces of crossed products up to Hamming-isometry. We establish a criterion for two crossed products of a group G over a base ring R to be isometric in terms of a certain G-automorphism action on the second cohomology of G w…

Duality and Rational Modules in Hopf Algebras over Commutative Rings

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$.

Maximal Graded Orders over Crystalline Graded Rings

Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two constructions are given for producing maximal graded orders. On the way, a new concept appears, so-called, spectrally twisted group. Some general properties of it are studied. At the end of the paper several examples are considered.

如何在Morita环上构造相对于完全对偶对的Gorenstein投射模

How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

该研究针对满足特定张量零化条件的Morita环，利用两侧环的（完全）对偶对构造环上模的（完全）对偶对，推广了三角矩阵环上的相关结果，并构造了对应完全对偶对的Gorenstein投射模，给出了在Ding投射模中的应用。

完备离散赋值环上对称代数的Tate对偶与转移

Tate Duality and Transfer for Symmetric Algebras Over Complete Discrete Valuation Rings

本文证明了完备离散赋值环上对称代数的Hochschild上同调中对偶转移映射与Tate对偶可交换，并在Calabi-Yau三角范畴的更广泛背景下解释了该结果。

具有有限 Gorenstein 弱整体维数的环上的同伦等价与 Grothendieck 对偶

Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

本文证明具有有限 Gorenstein 弱整体维数的环满足同伦范畴的三角等价，并将其应用于 Ding-Chen 环和 Gorenstein n-相干环的 Grothendieck 对偶。

Prime Modules and Associated Primes of Induced Modules over Rings Graded by Unique Product Monoids

We study prime ideals, prime modules, and associated primes of graded modules over rings $S$ graded by a unique product monoid. We consider two situations in detail: (a) the case where $S$ is strongly group-graded and (b) the case where $S$ is a crossed product and the ideal or module is induced from the identity component $R$ of $S$. We give explicit conditions for ideals and modules of $R$ to induce prime ideals of or prime modules over $S$ in these two cases. We then describe the set of associated prime ideals of an arbitrary induced module. One of our main interests is to give necessary an…

Pseudo-Sylvester domains and skew Laurent polynomials over firs

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with free-by-{infinite cyclic} and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof o…

Crossed products by twisted partial actions and graded algebras

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 …

Tannaka-Krein duality for Hopf algebroids

We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.

Duality Preserving Gray Maps for Codes over Rings

Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.

诺特世界中的特征倾斜模与Ringel对偶

Characteristic tilting modules and Ringel duality in the Noetherian world

本文研究交换诺特环上分裂拟遗传代数的Ringel对偶，刻画了包含所有标准模的最小子解消子范畴，并证明其间的正合等价可提升为保持拟遗传结构的Morita等价。

On a class of semihereditary crossed-product orders

Let $F$ be a field, let $V$ be a valuation ring of $F$ of arbitrary Krull dimension (rank), let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Let $f:G\times G\mapsto K\setminus \{0\}$ be a normalized two-cocycle such that $f(G\times G)\subseteq S\setminus \{0\}$, but we do not require that $f$ should take values in the group of multiplicative units of $S$. One can construct a crossed-product $V$-algebra $A_f=\sum_{σ\in G}Sx_σ$ in a natural way, which is a $V$-order in the crossed-product $F$-algebra $(K/F,G,f)$. If $V$ is unramified …

Duality for partial group actions

Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e.G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*G with the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions in the sense of R.Exel. In particular we show that the smash product (A*_αG)#k[G]^* of the partial skew group…

Relative nonhomogeneous Koszul duality

This book contains a detailed exposition of the nonhomogeneous Koszul duality theory in the relative situation over a noncentral, noncommutative, nonsemisimple base ring, as announced in Section 0.4 of arXiv:0708.3398. We prove the Poincare-Birkhoff-Witt theorem in this context and construct the triangulated equivalences of derived Koszul duality. The duality between the ring of differential operators and the de Rham DG-algebra, with the ring of functions as the base ring, is the thematic example. The moderate generality level makes the exposition in this book more accessible than the very hea…