10 Citations
Exact structures and degeneration of Hall algebras
- MathematicsAdvances in Mathematics
- 2022
Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
- Mathematics
- 2019
We show that the mesh mutations are the minimal relations among the $\mathbf{g}$-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to…
Relations for Grothendieck groups of n-cluster tilting subcategories
- MathematicsJournal of Algebra
- 2021
On the lattice of weakly exact structures
- Mathematics
- 2020
The study of exact structures on an additive category $\mathcal{A}$ is closely related to the study of closed additive sub-bifunctors of the maximal extension bifunctor $\mbox{Ext}^1$ on…
Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories
- MathematicsAlgebras and Representation Theory
- 2021
We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only…
Exact subcategories, subfunctors of $\operatorname{Ext}$, and some applications
- Mathematics
- 2022
. Let ( A , E ) be an exact category. We establish basic results that allow one to identify sub(bi)functors of Ext E ( − , − ) using additivity of numerical functions and restriction to…
Auslander's defects over extriangulated categories: An application for the general heart construction
- MathematicsJournal of the Mathematical Society of Japan
- 2021
The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension…
References
SHOWING 1-10 OF 28 REFERENCES
Relations for Grothendieck groups of Artin algebras
- Mathematics
- 1984
M. C. R. Butler has shown that if A is an artin algebra of finite representation type, then the almost split sequences generate the relations for the Grothendieck group of A. This paper is primarily…
Classifications of exact structures and Cohen–Macaulay-finite algebras
- MathematicsAdvances in Mathematics
- 2018
The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras
- Mathematics
- 2000
In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have…
Relations for Grothendieck groups of Gorenstein rings
- Mathematics
- 2016
We consider the converse of the Butler, Auslander-Reiten's Theorem which is on the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the…
The Homological Theory of Maximal Cohen-Macaulay Approximations
- Mathematics
- 1989
Soclete Mathematlque de FranceMemoire n° 38, 1989, p.5-37.THE HOMOLOGICAL THEORYOFMAXIMAL COHEN-MACAULAY APPROXIMATIONSbyMaurice Auslander (Brandeis) and Ragnar-Olaf Buchweitz (Toronto)Summary. Let R…
MODULES OF FINITE PROJECTIVE DIMENSION FOR STANDARDLY STRATIFIED ALGEBRAS
- Mathematics
- 2001
This paper deals with standardly stratified algebras, a generalization of quasihereditary algebras. As for quasihereditary algebras we show that there is a tilting module naturally associated with…
$n$-quasi-abelian categories vs $n$-tilting torsion pairs
- MathematicsDocumenta Mathematica
- 2021
Rump and successively Bondal and Van den Bergh provide an equivalence between the notion of quasi-abelian category studied by Schneiders and that of tilting torsion pair on an abelian category. Any…