Using Exterior Powers in Codimension Two Iwasawa Theory

Abstract

My research this summer was in the area of Iwasawa theory, a subfield of algebraic number theory which investigates infinite Galois extensions which are "the same as" (isomorphic to) the p-adic integers. There has been a recent burst of research output with the aim of exploring the consequences of the Iwasawa main conjectures in more exotic settings, as in higher codimension primes. My focus was on the algebraic aspects of the theory, which could be summarized as the module theory of 2-dimensional complete Noetherian regular local rings. A heavy emphasis is placed on homological techniques, which vaguely speaking prove things non-constructively by showing that they "cannot not be true". I follow contemporary development in this area in applying the exterior algebra functor to reduce these more exotic settings to those already well-studied, and then transporting results on the exterior algebra to the actual module of interest.