Degenerations of graded Cohen-Macaulay modules

TL;DR

This paper introduces a new concept of degenerations for graded modules and explores the relationships between various partial orders in the context of graded Cohen-Macaulay modules, especially under finite representation type conditions.

Contribution

It defines degenerations of graded modules and establishes the equivalence of several partial orders in the setting of graded Cohen-Macaulay modules for specific classes of rings.

Findings

Partial orders coincide on graded Cohen-Macaulay modules under finite representation type.

Introduces graded analogies of hom, degeneration, and extension orders.

Provides a framework for understanding degenerations in graded module categories.

Abstract

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are identical on the graded Cohen-Macaulay modules if a graded ring is of graded finite representation type and representation directed.