scholarly journals On lifting stable diagrams in Frobenius categories

2007 ◽  
Vol 9 (1) ◽  
pp. 163-183
Author(s):  
Matthias Künzer
Keyword(s):  
Frobenius Categories

scholarly journals Frobenius categories

2006 ◽  
Vol 303 (1) ◽  
pp. 309-357 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Frobenius Categories

scholarly journals Frobenius categories, Gorenstein algebras and rational surface singularities

2014 ◽  
Vol 151 (3) ◽  
pp. 502-534 ◽  
Author(s):  
Martin Kalck ◽  
Osamu Iyama ◽  
Michael Wemyss ◽  
Dong Yang
Keyword(s):  
Sufficient Conditions ◽  
Rational Surface ◽  
Gorenstein Ring ◽  
Projective Modules ◽  
Rational Singularity ◽  
Gorenstein Projective Modules ◽  
Surface Singularities ◽  
Frobenius Category ◽  
Rational Surface Singularity ◽  
Frobenius Categories

AbstractWe give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

On relative counterpart of Auslander’s conditions

2021 ◽  
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
J. R. García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
New Approach ◽  
Gorenstein Rings ◽  
Semidualizing Module ◽  
Gorenstein Projective Modules ◽  
Frobenius Category ◽  
Relative Notion ◽  
Frobenius Categories

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

ABSTRACT ALGEBRAIC INTEGRALS AND FROBENIUS CATEGORIES

2013 ◽  
Vol 24 (10) ◽  
pp. 1350081 ◽  
Author(s):  
MIODRAG CRISTIAN IOVANOV
Keyword(s):  
Invariant Measure ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Existence And Uniqueness ◽  
Compact Groups ◽  
Finite Dimensional ◽  
Frobenius Categories ◽  
Finite Dimensional Algebras

We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim ( Hom C(C, M)) = dim (M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

scholarly journals Ordinary Grothendieck Groups of a Frobenius P-Category

2011 ◽  
Vol 18 (01) ◽  
pp. 1-76 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Characteristic Zero ◽  
Inverse Limit ◽  
Grothendieck Group ◽  
Characteristic P ◽  
Direct Sum ◽  
Grothendieck Groups ◽  
Frobenius Categories

In [7] we have introduced the Frobenius categories [Formula: see text] over a finite p-group P, and we have associated to [Formula: see text] — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from [Formula: see text], determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from [Formula: see text], determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in [Formula: see text] of a suitable set of representatives of the [Formula: see text]-classes of elements of P.

scholarly journals A note on model structures on arbitrary Frobenius categories

2017 ◽  
Vol 67 (2) ◽  
pp. 329-337
Author(s):  
Zhi-Wei Li
Keyword(s):  
Frobenius Categories ◽  
Model Structures

Derived equivalences from cohomological approximations and mutations of Φ-Yoneda algebras

2013 ◽  
Vol 143 (3) ◽  
pp. 589-629 ◽  
Author(s):  
Wei Hu ◽  
Steffen Koenig ◽  
Changchang Xi
Keyword(s):  
Triangulated Categories ◽  
Endomorphism Rings ◽  
Infinite Dimensional ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
New Construction ◽  
Yoneda Algebras ◽  
Frobenius Categories ◽  
Finite Dimensional Algebras

A new construction of derived equivalences is given, which relates different endomorphism rings and, more generally, cohomological endomorphism rings, including higher extensions, of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable to a wide variety of situations, covering finite-dimensional algebras as well as certain infinite-dimensional algebras, Frobenius categories and n-Calabi–Yau categories.

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