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Dissertation Defense--Casey Monday

Date:
-
Location:
945 Patterson Office Tower
Speaker(s) / Presenter(s):
Casey Monday, University of Kentucky

Title:  A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring

Abstract:  Serre classes of modules over a ring $R$ are important because they describe relationships between certain classes of modules and sets of ideals of $R$. In this talk we characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. When $R$ is complete local and noetherian, define $E$ as the injective envelope of the residue field of $R$. Then denote $M^\nu=Hom_R(M,E)$ as the dual of $M$. A module $M$ is reflexive if the natural evaluation map from $M$ to $M^{\nu\nu}$ is an isomorphism.  The main result provides a characterization of the Serre classes of reflexive modules over such a ring. This characterization depends on an ability to ``construct'' reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.

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