Krull--Gabriel dimension of Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type

TL;DR

This paper computes the Krull--Gabriel dimension for categories of maximal Cohen--Macaulay modules over hypersurfaces, revealing a dimension of 0 for finite and 2 for countable but not finite types, advancing understanding of their structure.

Contribution

It provides explicit calculations of the Krull--Gabriel dimension for Cohen--Macaulay modules over hypersurfaces with countable representation type, clarifying their categorical complexity.

Findings

Krull--Gabriel dimension is 0 for finite Cohen--Macaulay type.

Krull--Gabriel dimension is 2 for countable but not finite Cohen--Macaulay type.

The dimension distinguishes between finite and countable Cohen--Macaulay representation types.

Abstract

We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is $0$ if the hypersurface is of finite Cohen--Macaulay representation type and that is $2$ if the hypersurface is of countable but not finite Cohen--Macaulay representation type.