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We extend categorical Morita equivalence to finite tensor categories graded by a finite group G. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided G-crossed tensor categories. [ABSTRACT FROM AUTHOR]

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra R c o A ̲ with a certain subalgebra of the smash product R # Â. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context. [ABSTRACT FROM AUTHOR]

Abstract: Let be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools—a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting—to study semiorthogonal decompositions of the bounded derived category . Together with results in the theory of quadratic forms, we apply these tools in the case where and has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X. [Copyright &y& Elsevier]

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5], and their categories of ϵ-hermitian forms. For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum. [ABSTRACT FROM PUBLISHER]

We characterize those semigroups with local units which are Morita equivalent to semigroups with local units having commuting idempotents. A key element in this characterization is a description of those inverse categories which are equivalent to Cauchy completions of inverse semigroups. [ABSTRACT FROM PUBLISHER]

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories. Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider the particular case in which those dg categories are just ordinary algebras. [ABSTRACT FROM AUTHOR]

We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes. [ABSTRACT FROM AUTHOR]

Abstract: A Morita context is constructed for any comodule of a coring and, more generally, for an bicomodule Σ for a coring extension of . It is related to a 2-object subcategory of the category of k-linear functors . Strictness of the Morita context is shown to imply the Galois property of Σ as a -comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an bicomodule Σ—implying strictness of the associated Morita context—is introduced. It is shown to be equivalent to being a Galois -comodule and isomorphic to , in the category of left modules for the ring and right comodules for the coring , i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. [Copyright &y& Elsevier]

Abstract: A recollement of triangulated categories describes one such category as being “glued together” from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a differential graded algebra in terms of two other such categories. [Copyright &y& Elsevier]

Abstract: We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon. [Copyright &y& Elsevier]

For any left R-module P with endomorphism ring S, the adjoint pair of functors P⊗S− and HomR(P,−) induce an equivalence between the categories of P-static R-modules and P-adstatic S-modules. In particular, this setting subsumes the Morita theory of equivalences between module categories and the theory of tilting modules. In this paper we consider, more generally, any adjoint pair of covariant functors between complete and cocomplete Abelian categories and describe equivalences induced by them. Our results subsume the situations mentioned above but also equivalences between categories of comodules. [Copyright &y& Elsevier]

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard''s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg–Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. [Copyright &y& Elsevier]

AbstractWe present two typical cases among five Japanese students who have complained of hearing music. They were found among 638 student case reports during 4 years at the mental health service center of a preparatory school. We diagnosed the complaint of hearing music as a symptom of obsession. Morita therapy was an effective treatment for this symptom. We discuss the characteristics of the symptom as an obsession and the therapy in the light of Morita theory. [ABSTRACT FROM AUTHOR]

In this paper we consider ultrametric Banach modules over commutative ultrametric Banach algebras with unit. We study the descent problem along a morphism f: R → S of such algebras and show that descent morphisms coincide with weak retracts. We give further conditions for having an effective descent morphism or for having a Morita equivalence between the corresponding categories of ultrametric Banach modules. [ABSTRACT FROM AUTHOR]

Given a partial action α = (A g , α g) g ∈ G of a connected groupoid G on a ring A and an object x of G , the isotropy group G (x) acts partially on the ideal A x of A by the restriction of α. In this paper we investigate the following reverse question: under which conditions a partial group action of G (x) on an ideal of A can be extended to a partial groupoid action of G on A ? The globalization problem and some applications to the Morita and Galois theories are also considered. [ABSTRACT FROM AUTHOR]