Partial bases and homological stability of GL_n(R) revisited

2024/05/16 by Calista Bernard, Jeremy Miller, Bernard, Calista +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #11E57 #18N70 #19B14 #20J06 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Amino Acid Enzymes and Metabolism #FOS: Mathematics #Finite Group Theory Research #K-Theory and Homology (math.KT) #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2405.09998

openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a unital ring satisfying the invariant basis number property, that every stably free R-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes every ring of stable rank 1 (e.g. any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain O_S of arithmetic type where |S| > 1 and S contains at least one non-complex place. Extending recent work of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt, we prove that the sequence of general linear groups GL_n(R) satisfies slope-1 homological stability with ℤ[1/2]-coefficients.

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