Finding Maximal Cohen-Macaulay and Reflexive Modules

Abstract

In this work we study two classical objects in algebra - maximal Cohen-Macaulay and reflexive modules. We show the existence of a small Cohen-Macaulay module or algebra for a newclass of rings in mixed characteristic. In particular, we show the existence of a birational small Cohen-Macaulay module over general biradical extensions of an unramified regular local ring of mixed charateristic and then use it to show the existence of a small Cohen-Macaulay module (algebra) under certain circumstances for general radical towers. This builds towards understanding generically Abelian extensions of an unramified regular local ring in mixed characteristic vis-à-vis Roberts (1980). We then study the class of reflexive modules over curve singularities through the lens of I-Ulrich modules and provide applications to finite type results and strongly reflexive extensions. This is a contribution towards understanding reflexivity in the one dimensional non-Gorenstein case - the one-dimensional case is key to understanding reflexivity in higher dimensions over "nice" rings.