Geometrically Linked Ideals and Gorenstein Dimension

Abstract

This thesis consists of two chapters. Chapter one is devoted to the notion of geometric linkage in the context of modules. We generalize a theorem of Peskine and Szpiro about geometrically linked ideals in the context of modules. More precisely, we show that over an unmixed local ring R if a G-perfect R-module M is linked to an R- module| N by a quasi-Gorenstein ideal a and Ass_R(M)∩Ass_R( N) = ∅, then there exists a quasi-Gorenstein ideal b such that M⊕ _R N| is free over R/b. We show that if an R- moduleM is horizontally linked to an R-module N such that AssR( M)∩AssR(N) =∅, then Tor ^R₁ (M;N) = 0. Conversely, we prove that if R is Gorenstein, M is horizontally linked to N, and Tor^R1 (M;N ) = 0, then Ass_RM ∩ Ass_R N = ∅ provided AnnR(M) is linked to Ann_R(N). In this case, Ann_R(M) is geometrically linked to Ann_R(N). Also, we provide several examples of Gorenstein local rings and horizontally linked modules M and N such that Tor^R₁( M;N) = 0 but Ass_R(M) \ Ass_R( N) 6= ∅.

In chapter two,first by using geometrically linked ideals, we give a construction for inffinitely many non-isomorphic indecomposable (totally re exive) modules, each minimally generated by a given number of elements. Next, we show that if a Cohen-Macaulay non-Gorenstein local ring (R;m; k) admits a non-free totally re exive module Mof minimal multiplicity, then the Poincaré series of M is a factor of the Poincarée series of the residuefield k. As a consequence, we show that over such a ring, if cx_R( M) < ∞ then no syzygy of the residuefield has a non-zero direct summand offinite Gorenstein dimension.