Ascent of finite Cohen-Macaulay type

Let (R,m) be a (commutative Noetherian) local ring of Krull dimension d. A non-zero R-module M is maximal Cohen–Macaulay (MCM) provided it is finitely generated and there exists an M-regular sequence x1,..., xd in the maximal ideal m. In particular, R is a Cohen–Macaulay (CM) ring if R is a MCM module over itself. The ring R is said to have finite Cohen–Macaulay type (or finite CM type) if there are, up to isomorphism, only finitely many indecomposable MCM R-modules. The first part of this dissertation investigates the non-complete CM local rings of finite CM type. In this direction, the main focus has been on a conjecture of Schreyer [33], which states that a local ring R has finite CM type if and only if the m-adic completion R ̂ has finite CM type. In Chapter 1, which contains joint work with R. Wiegand, I prove that finite CM type ascends to the completion for excellent CM local rings. In Chapter 2, I prove ascent of finite CM type when R is a CM local ring with a Gorenstein module, and such that the Henselization Rh is excellent. The second part of this dissertation considers one-dimensional complete hyper-surfaces of mixed characteristic. Such rings are of the form R = V [[y]]/(f), wher