Language：English   Publisher：Transactions of the American Mathematical Society Series B  

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated cate-gories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of suf-ficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre du-ality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category C of an extriangu-lated category C is a τ-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if C has enough projectives, almost split extensions and source morphisms. This gives various consequences on C, including Igusa– Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of C via the complete mesh category of the Auslander–Reiten species of C. Finally we prove that any locally finite symmetrizable τ-quiver (=val-ued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.

DOI： 10.1090/btran/159

Scopus

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Journal of Algebra  

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories by means of Hovey twin cotorsion pairs, and the localization of exact categories by two-sided admissibly percolating subcategories.

DOI： 10.1016/j.jalgebra.2022.08.008

Scopus

Language：English   Publisher：Journal of Algebra  

In n-Exangulated Categories (I), we introduced the notion of n-exangulated categories for each positive integer n. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a common generalization of n-exact categories in the sense of Jasso and (n+2)-angulated categories in the sense of Geiss-Keller-Oppermann. In this second article we introduce the notion of n-cluster tilting subcategories for extriangulated categories with enough projectives and injectives and show that under certain conditions such n-cluster tilting subcategories are n-exangulated.

DOI： 10.1016/j.jalgebra.2021.11.042

Scopus

Language：English   Publisher：Communications in Algebra  

Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras (Formula presented.) indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of (Formula presented.) On the way we will see that the Avella-Alaminos–Geiss invariants of (Formula presented.) are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.

DOI： 10.1080/00927872.2021.2008412

Web of Science

Scopus

Language：English   Publisher：Journal of Algebra  

DOI： 10.1016/j.jalgebra.2020.11.017

Web of Science

Scopus