Off-campus UNL users: To download campus access dissertations, please use the following link to log into our proxy server with your NU ID and password. When you are done browsing please remember to return to this page and log out.
Non-UNL users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Geometrically Linked Ideals and Gorenstein Dimension
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 AssR(M)∩AssR( 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 TorR1 (M;N ) = 0, then AssRM ∩ AssR N = ∅ provided AnnR(M) is linked to AnnR(N). In this case, AnnR(M) is geometrically linked to AnnR(N). Also, we provide several examples of Gorenstein local rings and horizontally linked modules M and N such that TorR1( M;N) = 0 but AssR(M) \ AssR( 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 cxR( M) < ∞ then no syzygy of the residuefield has a non-zero direct summand offinite Gorenstein dimension.^
Gheibi, Mohsen, "Geometrically Linked Ideals and Gorenstein Dimension" (2018). ETD collection for University of Nebraska - Lincoln. AAI10827949.