Finite Cohen -Macaulay type

Let (R, [special characters omitted]) 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 x 1,…,xd in the maximal ideal [special characters omitted]. 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 [special characters omitted]-adic completion [special characters omitted] 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 hypersurfaces of mixed characteristic. Such rings are of the form R = V[[y]]/(f), where (V, π) is a complete discrete valuation ring of characteristic zero with algebraically closed residue field of prime characteristic p. Using the theory of Auslander–Reiten quivers, I prove that the obvious generalizations of the one-dimensional complete equicharacteristic hypersurfaces with finite CM type do indeed have finite CM type