In this note we study Morita contexts and Galois extensions for corings. For a coring $\QTR{cal}{C}$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\QTR{cal}{M}^{\QTR{cal}{C}}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called \QTR{em}{cleft}$C$\QTR{em}{-Galois extensions} over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) $H$-Galois extensions.
El Kaoutit and Gómez Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings, as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the corin…
Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler, Doi, and Cohen, Fischman and Montgomery. We discuss when the contexts are strict. We apply our theory corings arising from entwining structures, and this leads us to the notion of cleft entwining structure.
We study co-Frobenius and more generally Quasi-co-Frobenius corings over arbitrary baserings and over PF baserings in particular. We generalize some results about (Quasi-) co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterisations for co-Frobenius and Quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properies of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterises (Quasi-) co-F…
By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $Σ$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $Σ$ is finitely generated and projective as an $A$-module. That is, we o…
A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $Σ$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors $\Mm^\Cc\to\Mm^\Dd$. Strictness of the Morita context is shown to imply the Galois property of $Σ$ as a $\cC$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an $L$-$\cC$ bicomodule $Σ$ -- implying strictness of the associated Morita context -- is introduced. It is sho…
A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of wh…
Partial Galois extensions were recently introduced by Dokuchaev, Ferrero and Paques. We introduce partial Galois extensions for noncommutative rings, using the theory of Galois corings. We associate a Morita context to a partial action on a ring.
We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the $G$-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra.
A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element. Biseparable corings are defined, and a conjecture relating them to Frobenius corings is proposed.
A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with surjective counits. Under this equivalence, comodules over a coring correspond to integrable connections or quasi-cohesive curved modules, while contramodules over a coring correspond to a specific class of curved modules introduced and termed Z-divergences in here.
We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.
We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras.
We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. Gómez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.
Given a ring $A$ and an $A$-coring $\cC$ we study when the forgetful functor from the category of right $\cC$-comodules to the category of right $A$-modules and its right adjoint $-\otimes_A\cC$ are separable. We then proceed to study when the induction functor $-\otimes_A\cC$ is also the left adjoint of the forgetful functor. This question is closely related to the problem when $A\to {}_A{\rm Hom}(\cC,A)$ is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of $A$ fixed under the coaction of $\cC$ is an equivalence. We also …
We extend Morita theory to abelian categories by using wide Morita contexts. Several equivalence results are given for wide Morita contexts between abelian categories, widely extending equivalence theorems for categories of modules and comodules due to Kato, Muller and Berbec. In the case of Grothendieck categories we derive equivalence results by using quotient categories. We apply the general equivalence results to rings with identity, rings with local units, graded rings, Doi-Hopf modules and coalgebras.
Properties of (most general) non-commutative torsors or A-B torsors are analysed. Starting with pre-torsors it is shown that they are equivalent to a certain class of Galois extensions of algebras by corings. It is shown that a class of faithfully flat pre-torsors induces equivalences between categories of comodules of associated corings. It is then proven that A-B torsors correspond to monoidal functors (and, under some additional conditions, equivalences) between categories of comodules of bialgebroids.
We develop a general framework for the study of strong Morita equivalence in which $C^*$-algebras and hermitian star products on Poisson manifolds are treated in equal footing. We compare strong and ring-theoretic Morita equivalences in terms of their Picard groupoids for a certain class of unital $^*$-algebras encompassing both examples. Within this class, we show that both notions of Morita equivalence induce the same equivalence relation but generally define different Picard groups. For star products, this difference is expressed geometrically in cohomological terms.
A.S. Dugas and R. Martínez-Villa proved in \cite[Corollary 5.1]{dm} that if there exists a stable equivalence of Morita type between the $k$-algebras $Λ$ and $Γ$, then it is possible to replace $Λ$ by a Morita equivalent $k$-algebra $Δ$ such that $Γ$ is a subring of $Δ$ and the induction and restriction functors induce inverse stable equivalences. In this note we give an affirmative answer to a question of Alex Dugas about the existence of a $Γ$-coring structure on $Δ$. We do this by showing that $Δ$ is a Frobenius extension of $Γ$.
We show that the category of corings over a fixed base ring with local units is equivalent to the category of comonads in (right) unital modules whose underlying functors preserve inductive limits. Changing base rings, we prove a bi-equivalence of bicategories. A base ring extension of corings by adjunctions is also introduced.
The monoidal version of classical Morita theory is a theory of bialgebroids. To make this explicit we construct a bicategory the objects of which are the bialgebroids and in which equivalence of objects means that the corresponding module categories are monoidally equivalent. The module categories of bialgebroids and of Frobenius Hopf algebroids are characterized by the existence of strong comonoid progenerators and strong Frobenius progenerators, respectively.
Let $R$ be a commutative ring. An Azumaya coring consists of a couple $(S,\Cc)$, with $S$ a faithfully flat commutative $R$-algebra, and an $S$-coring $\Cc$ satisfying certain properties. If $S$ is faithfully projective, then the dual of $\Cc$ is an Azumaya algebra. Equivalence classes of Azumaya corings form an abelian group, called the Brauer group of Azumaya corings. This group is canonically isomorphic to the second flat cohomology group. We also give algebraic interpretations of the second Amitsur cohomology group and the first Villamayor-Zelinsky cohomology group in terms of corings.
In this work we investigate partial actions of a Hopf algebra H on nonunital algebras and the associated partial smash products. We show that our partial actions correspond to nonunital algebras in the category of partial representations of H. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and w…
Let $H$ be a Hopf algebra, and $A,B$ be $H$-Galois extensions. We investigate the category ${}_A\mathcal{M}_B^H$ of relative Hopf bimodules, and the Morita equivalences between $A$ and $B$ induced by them.
We describe how to construct all inverse semigroups Morita equivalent to a given inverse semigroup. This is done by taking the maximum inverse images of the regular Rees matrix semigroups over the inverse semigroup where the sandwich matrix satisfies what we call the McAlister conditions.
We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister's theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.
Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.
We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.
Up to Morita equivalence, every quasi-hereditary algebra is the dual algebra of a directed bocs or coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way is given.
We generalize the notion of stable equivalence of Morita type and define what is called "singular equivalence of Morita type with level". Such an equivalence of induces an equivalence between singular categories. We will also prove that a derived equivalence of standard type induces a singular equivalence of Morita type with level.
Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita $^*$-equivalence of rings with involution, and strong Morita equivalence of $C^*$-algebras. Second, in the general setting of $^*$-algebras over $\ring C$, we define `closed' $^*$-ideals as the ones occuring as kernels of $^*$…
The primary purpose of this thesis is to show every ultragraph Leavitt path algebra is Morita equivalent, as a ring, to a graph Leavitt path algebra. Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde showed every ultragraph $C^{*}$-algebra is Morita equivalent, in the $C^{*}$-sense, to a graph $C^{*}$-algebra; our result is an algebraic analog of this fact. Further, we will use our result to give an alternate proof for established conditions which guarantee the simplicity of an ultragraph Leavitt path algebra over a field.
To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-m…
The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.
In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for $σ$-unital rings having a $σ$-unit consisting of von Neumann regular elements. The following aim is to show that a property is Morita invariant if it is invariant under taking corners and under taking matrices. The previous results are used to check the Morita invariance of certain ring properties (being locally left/right artinian/noetherian, being categorically left/right…
This paper is investigative work into the properties of a family of graded algebras recently defined by Varagnolo and Vasserot, which we call VV algebras. We compare categories of modules over KLR algebras with categories of modules over VV algebras, establishing various Morita equivalences. Using these Morita equivalences we are able to prove several properties of certain classes of VV algebras such as (graded) affine cellularity and affine quasi-heredity.
We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.
Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left $\mathbb{Z}_2$-graded modules over all real and complex Clifford superalgebras are described explicitly.
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
Brauer graph algebras are closed under stable equivalence of Morita type
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University（北京师范大学数学科学学院，数学与复杂系统重点实验室）；School of Mathematics, Liaoning Normal University（辽宁师范大学数学学院）
Motivated by understanding the Broué's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants …
We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalenc…
Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M} \to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\otimes_R - : {}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The l…
In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $Φ$ and any generator $X$ of the $A$-module category, the $Φ$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idemp…
