Finite presentation and purity in categories $\sigma [M]$

Mike Prest ¹   ; Robert Wisbauer ²

¹ Department of Mathematics University of Manchester Manchester M13 9PL, UK
² Mathematisches Institut der Heinrich-Heine-Universität D-40225 Düsseldorf, Germany

Colloquium Mathematicum, Tome 99 (2004) no. 2, pp. 189-202

Résumé

For any module $M$ over an associative ring $R$, let $ \sigma [M] $ denote the smallest Grothendieck subcategory of ${\rm Mod}\hbox {-}R$ containing $M$. If $ \sigma [M]$ is locally finitely presented the notions of purity and pure injectivity are defined in $ \sigma [M]$. In this paper the relationship between these notions and the corresponding notions defined in ${\rm Mod}\hbox {-}R$ is investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an $M$ such that $ \sigma [M]$ does not contain any non-zero finitely presented objects.

Détail

DOI : 10.4064/cm99-2-4

Keywords: module associative ring sigma denote smallest grothendieck subcategory mod hbox containing sigma locally finitely presented notions purity pure injectivity defined sigma paper relationship between these notions corresponding notions defined mod hbox investigated connection between resulting ziegler spectra discussed example given sigma does contain non zero finitely presented objects