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arXiv · 1901.07158

Bivariant and extended Sylvester rank functions

Hanfeng Li
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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.

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Hanfeng Li. 2019-01-22. Bivariant and extended Sylvester rank functions. https://doi.org/10.1112/jlms.12372

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