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We define a numerical invariant   • ˆÝ ޚs over Cohen-Macaulay local ring š , which is related to the presenting matrices ofthe ˆ -th syzygy module (with or without free summands). We show that   • ‚ Þ šsa   • œ¦ Þs and   • ‚Ý ޚsa   • œ¦Ý ޚs for a Cohen-Macaulay local ring š 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 ޚi ‹s 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 šcan be described as a column invariant when š is Gorenstein

We investigate properties of certain invariants of Noetherian local rings, including their behavior under flat local homomorphisms. We show that these invariants are bounded by the multiplicity for Cohen–Macaulay local rings with infinite residue fields, and they all agree with the multiplicity when such rings are hypersurfaces. We also show that these invariants are all equal to 2 for a non-regular Cohen–Macaulay local ring A if and only if A has a minimal multiplicity, provided its residue field is infinite.

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 Herzog's characterization of modules of finite projective and injective dimensions in characteristic p > 0. We also discuss other homological situations related to these restrictions, including previously known results on the existence of free summands in syzygy modules.