Adjunction of pth Roots to Dedekind Domains and Injective Modules

Abstract

In the case when R is a Dedekind domain of positive prime characteristic p > 0, we give a parameterisation of the isomorphism classes of the injective indecomposable R ^∞-modules. Here R ^∞ is the smallest ring extension of R on which the Frobenius morphism acts bijectively. Further, in this case, every injective R ^∞-module is the injective hull of a direct sum of injective indecomposable R ^∞-modules.

Key Words:

• Dedekind domain
• Perfect closure
• Injective modules
• Characteristic p methods
• Frobenius morphism
• Tight closure
• Linear growth of primary decompositions

1991 Mathematics Subject Classification:

• 13A35
• 13C11
• 13G05

Acknowledgments

I would like to express my thanks to Prof. R. Y. Sharp, who suggested this topic and corrected many of the mistakes in the initial drafts. This work is supported by a fees-only studentship provided by the Engineering and Physical Sciences Research Council of the United Kingdom.