Higher Auslander-Reiten sequences via morphisms determined by objects

Let (C ,E, s) be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we define two additive subcategories Cr and Cl of C in terms of the representable functors from the stable category of C to the category of finitely generated k-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten n-exangles via determined morphisms. These results unify and extend their works by Jiao–Le for exact categories, Zhao–Tan– Huang for extriangulated categories, Xie–Liu–Yang for n-abelian categories.

[1]  H. Nakaoka,et al.  n-exangulated categories (I): Definitions and fundamental properties , 2020 .

[2]  C. Ringel.  The Auslander bijections: how morphisms are determined by modules , 2013, 1301.1251.

[3]  M. Auslander,et al.  Almost split sequences in subcategories , 1981 .

[4]  Tiwei Zhao,et al.  Almost split triangles and morphisms determined by objects in extriangulated categories , 2020, 2005.06690.

[5]  I. Reiten,et al.  Representation theory of artin algebras IV1)Invariants given by almost split sequences , 1977 .

[6]  H. Krause.  Morphisms Determined by Objects in Triangulated Categories , 2011, 1110.5625.

[7]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[8]  Yann Palu,et al.  Extriangulated categories, Hovey twin cotorsion pairs and model structures , 2019 .

[9]  Idun Reiten,et al.  Noetherian hereditary abelian categories satisfying Serre duality , 2002 .

[10]  C. Ringel.  Morphisms determined by objects: The case of modules over artin algebras , 2011, 1110.6734.

[11]  Dieter Happel,et al.  Triangulated categories in the representation theory of finite dimensional algebras , 1988 .

[12]  Panyue Zhou,et al.  $n$-exact categories arising from $n$-exangulated categories , 2021, 2109.12954.

[13]  Sondre Kvamme,et al.  An introduction to higher Auslander–Reiten theory , 2016, Bulletin of the London Mathematical Society.

[14]  Amit Shah.  Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories , 2018, Journal of Pure and Applied Algebra.

[15]  Shiping Liu.  Auslander-Reiten theory in a Krull-Schmidt category , 2010 .

[16]  H. Lenzing,et al.  Auslander-Reiten duality for abelian categories , 2004 .

[17]  H. Krause.  Morphisms determined by objects and flat covers , 2014, 1403.6039.

[18]  G. Jasso,et al.  n-Abelian and n-exact categories , 2014, 1405.7805.

[19]  Jiaqun Wei,et al.  (n + 2)-Angulated Quotient Categories , 2019 .

[20]  Pengjie Jiao,et al.  The Auslander–Reiten duality via morphisms determined by objects , 2016, 1612.02591.

[21]  Steffen Oppermann,et al.  n-angulated categories , 2010, 1006.4592.

[22]  Panyue Zhou,et al.  Two new classes of n-exangulated categories , 2021, Journal of Algebra.

[23]  Xiao-Wu Chen,et al.  A note on morphisms determined by objects , 2013, 1311.1854.

[24]  Ning Bian.  Right Minimal Morphisms in Arbitrary Krull-Schmidt Categories , 2009 .

[25]  이화영.  X , 1960, Chinese Plants Names Index 2000-2009.

[26]  Pengjie Jiao.  The generalized Auslander–Reiten duality on an exact category , 2016, Journal of Algebra and Its Applications.

[27]  I. Reiten,et al.  Representation theory of artin algebras iii almost split sequences , 1975 .

[28]  Auslander Maurice,et al.  Representation Theory of Artin Algebras I , 1974 .

[29]  Dongdong Zhang,et al.  On the existence of Auslander-Reiten $n$-exangles in $n$-exangulated categories , 2021 .

[30]  Yann Palu,et al.  Auslander–Reiten theory in extriangulated categories , 2018, Transactions of the American Mathematical Society, Series B.