QUASI-HEREDITARY ALGEBRAS ASSOCIATED WITH UPPER TRIANGULAR BIMODULES

Abstract

Let A be an aggregate with a finite spectroid S and B a bimodule over A. If B is upper triangular, it is shown by Brüstle and Hille that the category mat B of matrices over B is equivalent to the Δ-good module category over a quasi-hereditary algebra which is the opposite of the endomorphism algebra of a projective generator in mat B. In the present paper we provide an explicit construction of indecomposable projectives and injectives in mat B by defining left and right radicals of B. In particular, we obtain a description of the characteristic tilting module over the quasi-hereditary algebra associated with B. Moreover, the Ringel dual of this quasi-hereditary algebra is the opposite of the quasi-hereditary algebra associated with a bimodule over A ^op.

ACKNOWLEDGMENT

The author is grateful to Thomas Bruüstle for helpful discussions. Supported by the National Natural Science Foundation of China (Grant No. 19971009).