Local Rings of Finite Cohen–Macaulay Type

Abstract

Let (R, m) be a local Cohen–Macaulay ring whosem-adic completionR̂has an isolated singularity. We verify the following conjecture of F.-O. Schreyer:Rhas finite Cohen–Macaulay type if and only ifR̂has finite Cohen–Macaulay type. We also show that the hypersurfacek[[x₀,…,x_d]]/(f) has finite Cohen–Macaulay type if and only ifk^s[[x₀,…,x_d]]/(f) has finite Cohen–Macaulay type, wherek^sis the separable closure of the fieldk.