Abstract
AbstractWe define a numerical invariant Ý Þs over Cohen-Macaulay local ring , which is related to the presenting matrices ofthe -th syzygy module (with or without free summands). We show that Þ sa ¦ Þs and Ý Þsa ¦Ý Þs for a Cohen-Macaulay local ring of dimension . Key words : Row Invariants, Cohen-Macaulay Local Ring, Maximal Cohen-Macaulay Module, Free Summands 1. Introduction 1 Throughout this paper, we assume that Þis is a Noetherian local ring, and all modules are unitary. It is proved [1] 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. This fact provides not only a new way to understand some previously known results in commutative ring theory (see for instance Corollary 2.8 [1] , or Proposition 2.2 [1] ), but also new interesting invariants of local rings. These invariants have turned out to be quite useful; for example, the Auslander index of can be described as a column invariant when is Gorenstein
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