# Article

Full entry | PDF   (0.1 MB)
Keywords:
associated graded algebra; \$QF\$ algebra; \$QF\$-3 algebra; upper Loewy series; lower Loewy series
Summary:
We prove that an associated graded algebra \$R_G\$ of a finite dimensional algebra \$R\$ is \$QF\$ (= selfinjective) if and only if \$R\$ is \$QF\$ and Loewy coincident. Here \$R\$ is said to be Loewy coincident if, for every primitive idempotent \$e\$, the upper Loewy series and the lower Loewy series of \$Re\$ and \$eR\$ coincide. \endgraf \$QF\$-3 algebras are an important generalization of \$QF\$ algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra \$R\$, the associated graded algebra \$R_G\$ is \$QF\$-3 if and only if \$R\$ is \$QF\$-3.
References:
[1] Auslander, M.: Representation dimension of Artin algebras. Queen Mary College Lecture Notes (1971). Zbl 0331.16026
[2] Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A. No. 150 (1958), 1-60. MR 0096700 | Zbl 0080.25702
[3] Nakayama, T.: On Frobeniusean algebras. II, Ann. Math. 42 (1941), 1-21. DOI 10.2307/1968984 | MR 0004237 | Zbl 0026.05801
[4] Tachikawa, H.: Quasi-Frobenius rings and generalizations. LNM 351 (1973). Zbl 0271.16004
[5] Tachikawa, H.: QF rings and QF associated graded rings. Proc. 38th Symposium on Ring Theory and Representation Theory, Japan 45-51.\hfil http://fuji.cec.yamanash.ac.jp/ring/oldmeeting/2005/reprint2005/abst-3-2.pdf MR 2264126
[6] Thrall, R. M.: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64 (1948), 173-183. MR 0026048 | Zbl 0041.01001

Partner of