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Odd H-depth and H-separable extensions

Lars Kadison (2012)

Open Mathematics

Let C n(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B....

On ℤ/2ℤ-extensions of pointed fusion categories

Leonid Vainerman, Jean-Michel Vallin (2012)

Banach Center Publications

We give a classification of ℤ/2ℤ-graded fusion categories whose 0-component is a pointed fusion category. A number of concrete examples are considered.

On a generalization of W*-modules

David P. Blecher, Jon E. Kraus (2010)

Banach Center Publications

a recent paper of the first author and Kashyap, a new class of Banach modules over dual operator algebras is introduced. These generalize the W*-modules (that is, Hilbert C*-modules over a von Neumann algebra which satisfy an analogue of the Riesz representation theorem for Hilbert spaces), which in turn generalize Hilbert spaces. In the present paper, we describe these modules, giving some motivation, and we prove several new results about them.

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Andrzej Skowroński (2003)

Open Mathematics

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote ℒ A to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by ℛ A the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with ℒ A ∪ ℛ A co-finite in ind A, quasi-tilted algebras and...

On Auslander–Reiten components for quasitilted algebras

Flávio Coelho, Andrzej Skowroński (1996)

Fundamenta Mathematicae

An artin algebra A over a commutative artin ring R is called quasitilted if gl.dim A ≤ 2 and for each indecomposable finitely generated A-module M we have pd M ≤ 1 or id M ≤ 1. In [11] several characterizations of quasitilted algebras were proven. We investigate the structure and homological properties of connected components in the Auslander-Reiten quiver Γ A of a quasitilted algebra A.

On corings and comodules

Hans-Eberhard Porst (2006)

Archivum Mathematicum

It is shown that the categories of R -coalgebras for a commutative unital ring R and the category of A -corings for some R -algebra A as well as their respective categories of comodules are locally presentable.

On generalized q.f.d. modules

Mohammad Saleh, S. K. Jain, Sergio R. López-Permouth (2005)

Archivum Mathematicum

A right R -module M is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module M is g.q.f.d. iff every direct sum of M -singular M -injective modules in σ [ M ] is weakly injective iff every direct sum of M -singular weakly tight is weakly tight iff...

On hereditary artinian rings and the pure semisimplicity conjecture: rigid tilting modules and a weak conjecture

José L. García (2014)

Colloquium Mathematicae

A weak form of the pure semisimplicity conjecture is introduced and characterized through properties of matrices over division rings. The step from this weak conjecture to the full pure semisimplicity conjecture would be covered by proving that there do not exist counterexamples to the conjecture in a particular class of rings, which is also studied.

On hereditary rings and the pure semisimplicity conjecture II: Sporadic potential counterexamples

José L. García (2015)

Colloquium Mathematicae

It was shown in [Colloq. Math. 135 (2014), 227-262] that the pure semisimplicity conjecture (briefly, pssC) can be split into two parts: first, a weak pssC that can be seen as a purely linear algebra condition, related to an embedding of division rings and properties of matrices over those rings; the second part is the assertion that the class of left pure semisimple sporadic rings (ibid.) is empty. In the present article, we characterize the class of left pure semisimple sporadic rings having finitely...

On large selforthogonal modules

Gabriella D'Este (2006)

Commentationes Mathematicae Universitatis Carolinae

We construct non faithful direct summands of tilting (resp. cotilting) modules large enough to inherit a functorial tilting (resp. cotilting) behaviour.

On Matlis dualizing modules.

Enochs, Edgar E., López-Ramos, J.A., Torrecillas, B. (2002)

International Journal of Mathematics and Mathematical Sciences

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