Module Degenerations and Finite Field Extensions

TL;DR

This paper explores how module degenerations and related properties change when considering modules over a base field and its finite extensions, extending geometric definitions to more general algebraic contexts.

Contribution

It generalizes the concept of module degeneration from geometric to algebraic definitions applicable over arbitrary fields, analyzing effects of finite field extensions.

Findings

Degeneration can be characterized via exact sequences over arbitrary fields.

Isomorphism classes vary with the base field and its extensions.

The hom-order of modules is affected by the choice of base field.

Abstract

Degeneration of modules is usually defined geometrically, but due to results of Zwara and Riedtmann we can also define it in terms of exact sequences. This definition also works over fields that are not algebraically closed. Let $k$ be a field, $K$ a finite extension of $k$ and $\Lambda$ a $k$-algebra. Then any $K\otimes_k\Lambda$-module is also a $\Lambda$-module. We study how the isomorphism classes, degeneration and hom-order differ depending on which algebra we are working over.