Morita Theorems Research Articles

Representations of some associative pseudoalgebras

Abstract In this paper, we generalize Schur–Weyl duality and Morita Theorem on associative algebras to those on associative H-pseudoalgebras. Meanwhile, we get a plenty of associative H-pseudoalgebras over a cocommutative Hopf algebra H

Morita equivalence for graded rings

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.

Morita theorem for hereditary Calabi-Yau categories

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic d-Calabi-Yau triangulated category with a d-cluster tilting object T such that its shifted sum T⊕⋯⊕T[−(d−2)] has hereditary endomorphism algebra H is triangle equivalent to the orbit category Db(modH)/τ−1/(d−1)[1] of the derived category of H for a naturally defined (d−1)-st root τ1/(d−1) of the AR translation, provided H is of non-Dynkin type. We also show that heredity of H follows from that of T when d=3, that of T⊕T[−1] when d=4, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of T. Our result therefore generalizes the established theorems by Keller–Reiten and Keller–Murfet–Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.

A hermitian analogue of the Morita theorems

ABSTRACTLet F1 be an equivalence of type I between two categories of modules with hermitian forms. In other words, suppose F is an equivalence between the two underlying categories of modules, and F1 is an equivalence, given by and F1(f) = F(f), such that the non-singularity of a free hyperbolic module of hyperbolic rank 1 is preserved by F1. Then every equivalence generated by a set of hermitian Morita equivalence data is of type I and every equivalence of type I arises from a set of hermitian Morita equivalence data.

A Morita theorem for modular finite W-algebras

We consider the Lie algebra $$\mathfrak {g}$$ of a simple, simply connected algebraic group over a field of large positive characteristic. For each nilpotent orbit $$\mathcal {O}\subseteq \mathfrak {g}$$ we choose a representative $$e\in \mathcal {O}$$ and attach a certain filtered, associative algebra $$\widehat{U}(\mathfrak {g},e)$$ known as a finite W-algebra, defined to be the opposite endomorphism ring of the generalised Gelfand–Graev module associated to $$(\mathfrak {g}, e)$$ . This is shown to be Morita equivalent to a certain central reduction of the enveloping algebra of $$U(\mathfrak {g})$$ . The result may be seen as a modular version of Skryabin’s equivalence.

The Weak b-principle: Mumford Conjecture

AbstractIn this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b–principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen–Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b–principle, namely, a version of the b–principle for oriented colored broken submersions together with the Harer stability theorem and Miller–Morita theorem implies the Madsen–Weiss theorem.

Morita theorems for partially ordered monoids

Two partially ordered monoids S and T are called Morita equivalent if the categories of right S-posets and right T -posets are Pos-equivalent as categories enriched over the category Pos of posets. We give a description of Pos-prodense biposets and prove Morita theorems I, II, and III for partially ordered monoids.

A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak ∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401–2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak ∗ -continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak ∗ Morita equivalence bimodule. We also develop the theory of the W ∗ -dilation, which connects the non-selfadjoint dual operator algebra with the W ∗ -algebraic framework. In the case of weak ∗ Morita equivalence, this W ∗ -dilation is a W ∗ -module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W ∗ -dilation is a key part of the proof of our main theorem.

Global fixed points for centralizers and Morita’s Theorem

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3.

Comments on a paper “A Hermitian Morita theorem for algebras with anti-structure”

In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], End P A should be replaced by ( End P A ) o . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa.

Morita Context and Morita-Like Equivalence for the xst-Rings

The notion of the xst-rings was introduced by Garcia and Marin [5] in 1999. In this paper, we consider Morita context, Morita-like equivalence and the exchange property for the xst-rings. The results of the first Morita theorem are generalized to the xst-rings. So we obtain an important Morita-like equivalence of the xst-rings, from which, as an immediate consequence, we deduce the main result of Xu-Shum-Turner [4] and the standard Morita equivalence, A ∼ Mn(A), for a unital ring A. Moreover, we describe the properties of those well-known intermediate matrix rings, and show that the exchange property of a unital ring A coincides with the one for any Mn(A) as well as any intermediate matrix ring sitting between FMΓ(A) and FCΓ(A), which is an extension of a well-known result obtained by Nicholson [7].

Stable model categories are categories of modules

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.

Modules and Morita theorem for operads

Associative rings A, B are called Morita equivalent when the categories of left modules over them are equivalent. We call two classical linear operads P, Q Morita equivalent if the categories of algebras over them are equivalent. We transport a part of Morita theory to the operadic context by studying modules over operads. As an application of this philosophy, we consider an operadic version of the sheaf of linear differential operators on a (super)manifold M and give a comparison theorem between algebras over this sheaf on M and M red .

A Morita theorem for algebras of operators on Hilbert space

We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule.

Wide Morita contexts and equivalences of comodule categories

We introduce a purely categorical notion of Morita context between abelian categories, which extends the classical notion of Morita context for module categories as well as the Takeuchi's notion of Morita context for comodule categories (Takeuchi, 1977). We prove an extension both of Morita's theorem on categories of modules and of Takeuchi's theorem on comodule categories for our general notion of Morita context. As an application of our theory, we obtain an equivalence between certain comodule categories defined by any Takeuchi's Morita context similar to that obtained by Nicholson-Watters for module categories at (Nicholson and Watters, 1988).

Finite-to-one mappings and large transfinite dimension

Pol (1996) and Arenas (1996) independently introduced transfinite extensions of finite order of mappings by the use of the length of a partially ordered set and Borst's order, respectively. By use of the transfinite order of mappings, Arenas introduced a transfinite dimension O-dim based on the Morita's theorem and proved that every countable-dimensional compact metric space has O-dim. Then he asked whether the converse is true. In the present note, we shall show that both the transfinite extensions given by Pol and Arenas are the same if we ignore the values, and give an affirmative answer to Arenas' question as follows: a metrizable space X has the order dimension O-dim X if and only if X has large transfinite dimension Ind X. Furthermore, we shall prove that if a metrizable space X has the order dimension O-dim, then Ind X ⩽ O-dim X and O-dim S α = α for every ordinal number α < ω 1, where S ga is Smirnov's compactum.

Strong Morita Equivalence and a Generalisation of the Rees Theorem

In our earlier work we associated a natural category to a semigroup with local units and proved semigroup analogues of the celebrated Morita theorems for rings. In this article we use the notion of a Morita context to define an equivalence relation on a far wider class of semigroups. We give a semigroup analogue of Morita I and show that Morita equivalent semigroups can be constructed with ease. Our construction is based on the classical Rees theorem and a generalisation of this theorem by Hotzel. We then use properties of equivalent categories to deduce properties of this construction. Finally, we give a new generalisation of the Rees theorem and applications of this generalisation.

Deriving DG categories

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a triangulated analogue (4.3) of a of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a Morita theorem (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.

Bimodules, the Brauer group, Morita equivalence, and cohomolgy

The purpose of this note is to focus attention on an equivalence relation which we call ‘congruence of k-algebras’ and define in terms of the categories of bimodules. (Here k is a commutative unital ring.) Many familiar Morita invariant properties are trivially seen to be congruence invariant. Examples include the center, the lattice of two-sided ideals, and the Hochschild and cyclic cohomologies. In Section 3 we prove that Morita equivalence implies congruence, thereby obtaining new, uniform, and more efficient proofs of the Morita invariance of these properties. (Actually, the invariance of cohomology seems to have been previously noticed for only one type of Morita equivalence: that between an algebra A and the r X r matrix algebra hlr(A).) We also give new cochain maps inducing the cohomology isomorphism. In Section 2 we use congruence to answer Zelinsky’s concluding question in [ll] and to give an economical description of the Brauer group as consisting of the Morita equivalence classes of congruence-trivial algebras (under tensor product). This definition provides a categorical interpretation of the Brauer group and is available, without introducing separability, immediately after proving the Eilenberg-Watts theorem characterizing functors between module categories. (The Morita theorems, which are invoked, at least tacitly, in most treatments of the Brauer group, follow easily

Endomorphism rings and category equivalences

The use of category equivalences for the study of endomorphism rings stems from the Morita theorem. In a sense, this theorem can be viewed as stating that if P is a finitely generated projective generator of R-mod and S = End( RP), then properties of P correspond to properties of S through the equivalence between the categories R-mod and S-mod given by the functor Hom,(P, -). Generalizations of this theorem were given in [4, 51, In [S], P is only assumed to be finitely generated and projective, and Hom,(P, -) gives in this case an equivalence between S-mod and a quotient category of R-mod, while in [S] it is shown that if P is a finitely generated quasiprojective self-generator, then the equivalence induced by the same functor is now defined between the category a[P] of all the R-modules subgenerated by P and S-mod. Later on, other category equivalences were constructed, in an analogous way to those already mentioned, by replacing S-mod by a certain quotient category of S-mod. Thus, in [14] Morita contexts are used to obtain a category equivalence between quotient categories of both R-mod and S-mod for an arbitrary R-module M. On the other hand, if M is a C-quasiprojective module, then it is shown in [8 3 that the functor Hom,(M, -) induces an equivalence between quotient categories of o[M] and S-mod, and the latter quotient category coincides with S-mod when M is finitely generated. All the above constructions can be considered as particular cases of the following: if V is a locally finitely generated Grothendieck category and M is an object of V with S = End,(M), then the class of the M-distinguished objects of g (in the sense of [lo]) is the torsionfree class of a torsion