Strongly Quasi-Hereditary Algebras and Rejective Subcategories

Abstract

Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra A is strongly quasi-hereditary if and only if A is a Nakayama algebra.2010 Mathematics Subject Classification. 16G10, 18A40.

Characterizations of right rejective chains

“…Finally, we show the following proposition. This proposition is proven in [15, Theorem 3.6] and [22, Theorem 4.1]. In this paper, we give a proof using Theorem 3.9.…”

“…Hence, Theorem 3.12 is a refinement of their results. Moreover, Theorem 3.12(3) is proven in [22, Theorem 4.1].…”

“…It is known that special right rejective chains called total right rejective chains, are characterized by right-strongly quasi-hereditary algebras. Proposition 1.1 ([22,Theorem 3.22]) Let A be an artin algebra and (e 1 , e 2 , . .…”

Gabriel-Roiter measure, representation dimension and rejective chains

“…right) strongly quasihereditary ring. It turns out that the notion of complete total rejective chain is, in a specific sense, equivalent to that of a strongly quasihereditary ring, as observed in [33,Theorem 3.22]. In particular, the quasihereditary algebra defined in Section 2 is associated to a rejective chain (this will be clarified in Section 4).…”

“…Here LL(X) denotes the Loewy length of a module X in mod Λ. The ADR algebra must arise from a rejective chain, as it is a strongly quasihereditary algebra ( [9,33]). We clarify the latter assertion.…”