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The Quarterly Journal of Mathematics Advance Access published online on April 10, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han003
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS

Octavio Mendoza {dagger}

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F. 04510, México

Corina Sáenz {ddagger}

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F. 04510, México

Changchang Xi§

School of Mathematical Sciences; Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, P.R. China

§ Corresponding author. E-mail: xicc{at}bnu.edu.cn

Received 21 August 2006; revised 7 January 2008
  Abstract

We introduce the so-called homological systems in a module category over a pre-ordered set, which generalize the notion of a stratifying system over a linearly ordered set, and study both the corresponding modules filtrated by the systems and algebras stratified by the systems. In particular, we recover the tilting theory for pre-standardly stratified algebras, and get a general formula for computing the Cartan determinants of pre-standardly stratified algebras in terms of standard modules and simple modules. Also, the finitistic dimension of a given algebra, and the relative homological dimensions of full subcategories of the modules related to a homological system, are discussed. As an application, we get a new bound for the finitistic dimension of a pre-standardly stratified algebra.

{dagger} E-mail: omendoza{at}matem.unam.mx

{ddagger} E-mail: ecsv{at}lya.fciencias.unam.mx


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