Finding Maximal Cohen-Macaulay modules

Abstract: In this talk, we consider a problem that lies in the confluence of two topics. On one hand, we have maximal Cohen-Macaulay (MCM) modules - these are classical objects that have been studied extensively from algebraic and geometric viewpoints. There is a rich theory of MCM modules over Cohen-Macaulay (CM) rings and many beautiful connections to the singularities of the ring have been discovered. However, in the absence of the CM property in the ring, not as much is known - even the object's existence is largely unclear. On the other hand, we have a mixed characteristic phenomenon. In 1980, Paul Roberts showed that the integral closure of a regular local ring in an Abelian extension of its quotient field is CM, provided the characteristic of the residue field does not divide the degree of the extension. This fails in the "modular case" in mixed characteristic. We will look at some past results in the literature before considering the question of existence of MCMs in the modular case of Roberts's theorem.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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