Compactness of the Alexandrov topology of maximal Cohen-Macaulay modules

Let $R$ be a Cohen-Macaulay local ring. In this paper, we first describe the radicals of annihilators of stable categories of maximal Cohen-Macaulay $R$-modules. We then prove that the Alexandrov topology of the stable category of maximal Cohen-Macaulay $R$-modules is compact provided that the completion of $R$ has an isolated singularity. Finally, we consider the case of a hypersurface of countable CM-representation type.