Applied Categorical Structures – Official Blog

by Asensio, M. J.; Năstăsescu, C.; Torrecillas, B.

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕
_σ∈G

R

_σ
with finite support if R

_e
has Gabriel dimension then $R\hbox{-}{\rm gr}$ has Gabriel dimension. Moreover, adding some lattice results, we prove that if $R\hbox{-}{\rm mod}$ has Gabriel dimension then $R\hbox{-}{\rm gr}$ also has Gabriel dimension.

DOI: 10.1007/s10485-006-9042-7
Online Date: 11/18/2006
Print publication date: 12/1/2006
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by Caenepeel, Stefaan

DOI: 10.1007/s10485-006-9055-2
Online Date: 11/18/2006
Print publication date: 12/1/2006
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by Böhm, Gabriella; Brzeziński, Tomasz

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid ℋ = (ℋ

_L
, ℋ

_R
)) is cleft if and only if it is ℋ

_R
-Galois and has a normal basis property relative to the base ring L of ℋ

_L
. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined.

DOI: 10.1007/s10485-006-9043-6
Online Date: 11/18/2006
Print publication date: 12/1/2006
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by Iovanov, Miodrag-Cristian

In this note we give a new and elementary proof of a result of Năstăsescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra $C^*$ is a direct summand in M (the splitting problem for coalgebras).

DOI: 10.1007/s10485-006-9050-7
Online Date: 11/18/2006
Print publication date: 12/1/2006
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by Kadison, Lars

A depth two extension A | B is shown to be weak depth two over its double centralizer V

_A
(V

_A
(B)) if this is separable over B. We consider various examples and non-examples of depth one and two properties. Depth two and its relationship to direct and tensor product of algebras as well as cup product of relative Hochschild cochains is examined. Section 6 introduces a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End
^D

C

^D
forms a right bialgebroid over the centralizer subalgebra g
^* : D
^* → C
^* of the dual algebra C
^*.

DOI: 10.1007/s10485-006-9051-6
Online Date: 11/17/2006
Print publication date: 12/1/2006
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by Bulacu, Daniel; Torrecillas, Blas

For a quasi-Hopf algebra H, an H-bicomodule algebra $\mathbb{A}$ and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules ${}^C_H{\cal M}_{\mathbb{A}}^H$ is equivalent to the category of right–left generalized Yetter–Drinfeld modules ${}^C{\cal YD}(H)_{\mathbb{A}}$. Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.

DOI: 10.1007/s10485-006-9045-4
Online Date: 11/17/2006
Print publication date: 12/1/2006
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by Gómez-Torrecillas, J.

We investigate which aspects of recent developments on Galois corings and comodules admit a formulation in terms of comonads. The general theory is applied to the study of Galois comodules over corings over firm rings.

DOI: 10.1007/s10485-006-9049-0
Online Date: 11/17/2006
Print publication date: 12/1/2006
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by Knox, Michelle L.; McGovern, Warren Wm.

In the article (Martinez and Zenk, Algebra Universalis, 50, 231–257, 2003.), the authors studied several conditions on an algebraic frame L. In particular, four properties called Reg(1), Reg(2), Reg(3), and Reg(4) were considered. There it was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between Reg(3) and Reg(4). In the main section of the article we apply our notions to the frame of multiplicative filters of ideals in a commutative ring with unit and give characterizations of several well-known classes of commutative rings.

DOI: 10.1007/s10485-006-9038-3
Online Date: 11/17/2006
Print publication date: 4/1/2007
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by Brzeziński, Tomasz; Turner, Ryan B.

A theory of monoids in the category of bicomodules of a coalgebra C or C-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix
C
-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider’s Theorem II [Schneider, Isr. J. Math., :167–195, 1990], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.

DOI: 10.1007/s10485-006-9044-5
Online Date: 11/16/2006
Print publication date: 12/1/2006
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by Ferreira, Maria João; Picado, Jorge

In pointfree topology, the point-finite covers introduced by Dowker and Strauss do not behave similarly to their classical counterparts with respect to tran- sitive quasi-uniformities, contrarily to what happens with other familiar types of interior-preserving covers. The purpose of this paper is to remedy this by modifying the definition of Dowker and Strauss. We present arguments to justify that this modification turns out to be the right pointfree definition of point-finiteness. Along the way we place point-finite covers among the classes of interior-preserving and closure-preserving families of covers that are relevant for the theory of (transitive) quasi-uniformities, completing the study initiated with Ferreira and Picado, Kyungpook Math. J.,
415–442, 2004.

DOI: 10.1007/s10485-006-9039-2
Online Date: 11/16/2006
Print publication date: 4/1/2007
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