Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited
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| Format: | Preprint |
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2024
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| _version_ | 1866916249275990016 |
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| author | Bernard, Calista Miller, Jeremy Sroka, Robin J. |
| author_facet | Bernard, Calista Miller, Jeremy Sroka, Robin J. |
| contents | 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 $\mathcal{O}_S$ 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 $\operatorname{GL}_n(R)$ satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited Bernard, Calista Miller, Jeremy Sroka, Robin J. Algebraic Topology K-Theory and Homology Number Theory Representation Theory 11E57, 18N70, 19B14, 20J06 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 $\mathcal{O}_S$ 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 $\operatorname{GL}_n(R)$ satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients. |
| title | Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited |
| topic | Algebraic Topology K-Theory and Homology Number Theory Representation Theory 11E57, 18N70, 19B14, 20J06 |
| url | https://arxiv.org/abs/2405.09998 |