Some Restrictions on the Maps in Minimal Resolutions

We show that there are certain restrictions on the entries of the maps in the minimal free resolutions of finitely generated modules of infinite projective dimension over Noetherian local rings. These restrictions provide a way to slightly improve Herzogʹs characterization of modules of finite projective and injective dimensions in characteristicp > 0. We also discuss other homological situations related to these restrictions, including previously known results on the existence of free summands in syzygy modules.