String algebras over local rings: admissibility and biseriality

Abstract

For a path algebra over a noetherian local ground ring, the notion of an admissible ideal was defined by Raggi-C{á}rdenas and Salmer{ó}n. We provide sufficient conditions for admissibility and use them to study semiperfect module-finite algebras over local rings whose quotient by the radical is a product of copies of the residue field. We define string algebras over local ground rings and recover the notion introduced by Butler and Ringel when the ground ring is a field. We prove they are biserial in a sense of Kiričenko and Kostyukevich. We describe the syzygies of uniserial summands of the radical. We give examples of Bäckström orders that are string algebras over discrete valuation rings.

arXiv comment28 pages, comments welcome