Twisted homology stability of O_n for valuation rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911609096503296 |
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| author | Harr, Oscar |
| author_facet | Harr, Oscar |
| contents | In this article, we extend an argument of Vogtmann in order to show homology stability of the Euclidean orthogonal group $O_n(A)$ when $A$ is a valuation ring subject to arithmetic conditions on either its residue or its quotient field. In particular, it is shown that if $A$ is a henselian valuation ring, then the groups $O_n(A)$ exhibit homology stability if the residue field of $A$ has finite Pythagoras number. Our results include those of Vogtmann, and hold with various twisted coefficients. Using these results, we give analogues for fields $F\neq\mathbb R$ of some computations that appear in the study of scissor congruences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03213 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Twisted homology stability of O_n for valuation rings Harr, Oscar Algebraic Topology Representation Theory In this article, we extend an argument of Vogtmann in order to show homology stability of the Euclidean orthogonal group $O_n(A)$ when $A$ is a valuation ring subject to arithmetic conditions on either its residue or its quotient field. In particular, it is shown that if $A$ is a henselian valuation ring, then the groups $O_n(A)$ exhibit homology stability if the residue field of $A$ has finite Pythagoras number. Our results include those of Vogtmann, and hold with various twisted coefficients. Using these results, we give analogues for fields $F\neq\mathbb R$ of some computations that appear in the study of scissor congruences. |
| title | Twisted homology stability of O_n for valuation rings |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2212.03213 |