Dominance of Cohen–Macaulay local rings of finite CM type

Prove that every Cohen–Macaulay local ring of finite Cohen–Macaulay type is dominant, meaning that its residue field belongs to every nonzero thick subcategory of its singularity category.

Background

The paper studies dominant local rings, a class defined by the condition that the residue field belongs to every nonzero thick subcategory of the singularity category. Dominant local rings have strong homological consequences, including the Tor-friendly property.

Takahashi proposed that all Cohen–Macaulay local rings of finite Cohen–Macaulay type should be dominant. The paper establishes this conjecture under several additional numerical hypotheses, including bounded multiplicity, bounded codimension, and certain relationships between multiplicity and codimension, but does not prove the conjecture in full generality.