# Bivariant and extended Sylvester rank functions

## TL;DR

This paper extends Sylvester rank functions to all pairs of modules and maps over unital rings, providing new insights into their properties and relationships, including injectivity of pull-back maps and a novel proof of Schofield's result.

## Contribution

It introduces extensions of Sylvester rank functions to all module pairs and maps, and analyzes their behavior under ring epimorphisms, including new proofs of existing theorems.

## Key findings

- Extended Sylvester rank functions to all module pairs and maps
- Proved injectivity of the pull-back map for ring epimorphisms
- Provided a new proof of Schofield's characterization of the image of the pull-back map

## Abstract

For a unital ring R, a Sylvester rank function is a numerical invariant which can be described in 3 equivalent ways: on finitely presented left R-modules, or on rectangular matrices over R, or on maps between finitely generated projective left R-modules. We extend each Sylvester rank function to all pairs of left R-modules $M_1\subseteq M_2$, and to all maps between left R-modules satisfying suitable properties including continuity and additivity. As an application, we show that for any epimorphism $R\rightarrow S$ of unital rings, the pull-back map from the set of Sylvester rank functions of S to that of R is injective. We also give a new proof of Schofield's result describing the image of this map when S is the universal localization of R inverting a set of maps between finitely generated projective left R-modules.