Linear relations over commutative rings

TL;DR

This paper studies the category of linear relations over arbitrary commutative rings, revealing its structure as a subcategory of Kronecker representations and generalizing key properties previously known only over fields.

Contribution

It identifies the category as a definable, faithful, hereditary torsion-free class and extends functorial filtrations results from fields to general commutative rings.

Findings

The category of linear relations over a commutative ring is a subcategory of Kronecker representations.

Generalization of covering and splitting properties from fields to arbitrary commutative rings.

Establishment of the category as a definable, faithful, hereditary torsion-free class.

Abstract

We consider the category of linear relations over an arbitrary commutative ring, and identify it as a subcategory of the category of Kronecker representations. We observe that this subcategory forms a definable, faithful and hereditary torsion-free class. We also generalise results used in the functorial filtrations method, known before only in case the ground ring is a field. In particular, our results strictly generalise what the so-called the `covering' and `splitting' properties from this method.