Rejection lemma and almost split sequences

  • Yu. A. Drozd Inst. Math. Acad. Sci. Ukraine, Kiev
Keywords: lattices, orders, bijective lattices, almost split sequences, Auslander-Reiten quivers, stable categories

Abstract

UDC 512.55

We study the behavior of almost split sequences and Auslander – Reiten quivers of an order under rejection of bijective modules as defined in [Ю. А. Дрозд, В. В. Кириченко, О квазибассовых порядках, Изв. АН СССР. Сер. мат., 36, 328 – 370 (1972)]. In particular, we establish relations of stable categories and almost split sequences for an order $A$ and the order $A\prime$ obtained from $A$ by such a rejection. These results are refined for the Gorenstein and Frobenius cases.

References

M. Auslander, I. Reiten, Almost split sequences for Cohen – Macaulay modules, Math. Ann., 277, № 2б 345 – 349 (1987), https://doi.org/10.1007/BF01457367 DOI: https://doi.org/10.1007/BF01457367

M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge Univ. Press (1997).

W. Bruns, J. Herzog, Cohen – Macaulay rings, Cambridge Univ. Press (1993).

Ch. W. Curtis, I. Reiner, Methods of representation theory, Vol. 1, John Wiley & Sons (1981).

V. Dlab, C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc., 173 (1976), https://doi.org/10.1090/memo/0173 DOI: https://doi.org/10.1090/memo/0173

Yu. A. Drozd, V. V. Kirichenko, O kvazibassovy`kh poryadkakh, Izv. AN SSSR. Ser. mat., 36, 328 – 370 (1972).

Yu. A. Drozd, V. V. Kirichenko, Primarny`e poryadki s konechny`m chislom nerazlozhimy`kh predstavlenij, Izv. AN SSSR. Ser. mat., 37, 715 – 736 (1973).

Yu. A. Drozd, V. V. Kirichenko, Konechnomerny`e algebry`, Vishha shk., Kiev (1994).

Yu. A. Drozd, V. V. Kirichenko, A. V. Rojter, O nasledstvenny`kh i bassovy`kh poryadkakh, Izv. AN SSSR. Ser. mat., 31, 1415 – 1436 (1967).

Yu. Drozd, A. Plakosh, Cohomologies of the Kleinian 4-group, Arch. Math., 115, no. 2, 139 – 145 (2020), https://doi.org/10.1007/s00013-020-01451-6 DOI: https://doi.org/10.1007/s00013-020-01451-6

H. Hijikata, K. Nishida, Bass orders in non semisimple algebras, J. Math. Kyoto Univ., 34, no. 4, 797 – 837 (1994), https://doi.org/10.1215/kjm/1250518887 DOI: https://doi.org/10.1215/kjm/1250518887

H. Hijikata, K. Nishida, Primary orders of finite representation type, J. Algebra, 192, no. 2, 592 – 640 (1997), https://doi.org/10.1006/jabr.1996.6946 DOI: https://doi.org/10.1006/jabr.1996.6946

B. Keller, Derived categories and their use, Handbook Algebra, vol. 1, 671 – 701 (1996)б https://doi.org/10.1016/S1570-7954(96)80023-4 DOI: https://doi.org/10.1016/S1570-7954(96)80023-4

T. Y. Lam, A first Course in noncommutative rings, Springer (1991)б https://doi.org/10.1007/978-1-4684-0406-7 DOI: https://doi.org/10.1007/978-1-4684-0406-7

E. Matlis, Injective modules over Noetherian rings, Pacif. J. Math., 8, 511 – 528 (1958). DOI: https://doi.org/10.2140/pjm.1958.8.511

H. Matsumura, Commutative algebra, The Benjamin/Cummings Publ. Com. (1980).

К. W. Roggenkamp, Gorenstein orders of finite representation type and bijective lattices, Lect. Notes Math., 1178, 243 – 270 (1986), https://doi.org/10.1007/BFb0075298 DOI: https://doi.org/10.1007/BFb0075298

A. V. Roiter, Analog odnoj teoremy` Bassa dlya modulej predstavlenij nekommutativny`kh poryadkov, Dokl. AN SSSR, 168, 1261 – 1264 (1966).

Published
18.06.2021
How to Cite
DrozdY. A. “Rejection Lemma and Almost Split Sequences”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 6, June 2021, pp. 780 -98, doi:10.37863/umzh.v73i6.6580.
Section
Research articles