Quasi-hereditary algebras.

*(English)*Zbl 0666.16014This article is the first of a series on quasi-hereditary algebras. This notion was introduced by E. Cline, B. Parshall and L. Scott [see J. Algebra 117, 504-521 (1988; Zbl 0659.18011)] and B. Parshall and L. Scott [see “Derived categories, quasi-hereditary algebras and algebraic groups”, Proc. Ottawa-Moosonee Workshop Algebra, Carleton Univ. Notes, No.3 (1988)]. Let A be a semiprimary ring, that is, A is associative, with 1, with nilpotent Jacobson radical N such that A/N is semisimple artinian. An ideal J of A is a heredity ideal if \(J^ 2=J\), \(JNJ=0\) and \(J_ A\) is a projective module. The ring A is called quasi-hereditary if it contains a heredity chain, that is, a chain \(0=J_ 0\subset J_ 1\subset...\subset J_ m=A\) of ideals of A such that, for all i, \(J_ i/J_{i-1}\) is a heredity ideal of \(A/J_{i-1}\). Typical examples are the finite dimensional hereditary algebras. In this article, the authors show that a semiprimary ring is hereditary if and only if any chain of idempotent ideals of A can be refined to a heredity chain.

It was proved by Parshall and Scott that every quasi-hereditary algebra has finite global dimension. Here the authors show that every finite dimensional algebra of global dimension 2 is quasi-hereditary and give an example of a (Nakayama) algebra of global dimension 4 which is not quasi- hereditary (since then, examples of algebras of global dimension 3 which are not quasi-hereditary were also found). The same example shows that the class of quasi-hereditary algebras is not closed under tilting. Finally, in a long appendix, the authors prove various statements on heredity ideals, giving examples to illustrate the necessity of some of the assumptions made. They also give (optimal) bounds for the global dimension and the Loewy length of a semiprimary quasi-hereditary ring.

It was proved by Parshall and Scott that every quasi-hereditary algebra has finite global dimension. Here the authors show that every finite dimensional algebra of global dimension 2 is quasi-hereditary and give an example of a (Nakayama) algebra of global dimension 4 which is not quasi- hereditary (since then, examples of algebras of global dimension 3 which are not quasi-hereditary were also found). The same example shows that the class of quasi-hereditary algebras is not closed under tilting. Finally, in a long appendix, the authors prove various statements on heredity ideals, giving examples to illustrate the necessity of some of the assumptions made. They also give (optimal) bounds for the global dimension and the Loewy length of a semiprimary quasi-hereditary ring.

Reviewer: I.Assem

##### MSC:

16P10 | Finite rings and finite-dimensional associative algebras |

16Gxx | Representation theory of associative rings and algebras |

16E10 | Homological dimension in associative algebras |

16Dxx | Modules, bimodules and ideals in associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |