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# 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*- module

*M*is horizontally linked to an

*R*-module

*N*such that Ass

*R*(

*M*)∩Ass

*R*(

*N*) =∅, then Tor

^{ R}

_{1}(

*M;N*) = 0. Conversely, we prove that if

*R*is Gorenstein,

*M*is horizontally linked to

*N*, and Tor

^{R}

^{1}(

*M;N*) = 0, then Ass

_{R}*M*∩ 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

^{R}

_{1}(

*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

*M*of 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.^

#### Subject Area

Mathematics

#### Recommended Citation

Gheibi, Mohsen, "Geometrically Linked Ideals and Gorenstein Dimension" (2018). *ETD collection for University of Nebraska - Lincoln*. AAI10827949.

http://digitalcommons.unl.edu/dissertations/AAI10827949