In my thesis work, the main theorem was showing that the cohomology annihilator of a one dimensional Gorenstein local ring (with some extra assumptions) is given by the stable annihilator of its integral closure. This raises the question: Given any Gorenstein local ring, is it possible to find a single object M in the stable category of maximal Cohen-Macaulay modules such that whenever a ring element annihilates the stable endomorphism ring of M, that ring element annihilates all Hom-sets in the entire category. We introduce an Alexandrov topology on the stable category of maximal Cohen-Macaulay modules so that the answer to the above question is affirmative if and only if the space is compact. We look at some continuous functions and also the finite Cohen-Macaulay type.
We started to work on this project with three undergraduate students from Istanbul who attended the Mathematics Research Program where I gave three lectures on cohomology annihilators.