Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of certain real affine algebras
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909724239200256 |
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| author | Banerjee, Sourjya |
| author_facet | Banerjee, Sourjya |
| contents | Let $R$ be a reduced real affine algebra of (Krull) dimension $d \ge 2$ such that either $R$ has no real maximal ideals, or the intersection of all real maximal ideals in $R$ has height at least one. In this article, we prove the following: (1) the $d$-th Euler class group $\text{E}^d(R)$, defined by Bhatwadekar-R.~Sridharan, is canonically isomorphic to the Levine-Weibel Chow group of zero cycles $\text{CH}_0(\text{Spec}(R))$; (2) the universal Mennicke symbol $\text{MS}_{d+1}(R)$ is canonically isomorphic to the universal weak Mennicke symbol $\text{WMS}_{d+1}(R)$; and (3) additionally, if $R$ is a regular domain, then the Whitehead group $\mathrm{SK_1}(R)$ is canonically isomorphic to $\frac{\text{SL}_{d+1}(R)}{\text{E}_{d+1}(R)}$. As an application, we investigate some Eisenbud-Evans type theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14384 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of certain real affine algebras Banerjee, Sourjya Commutative Algebra 14C25, 13C10, 19B10, 19B14 Let $R$ be a reduced real affine algebra of (Krull) dimension $d \ge 2$ such that either $R$ has no real maximal ideals, or the intersection of all real maximal ideals in $R$ has height at least one. In this article, we prove the following: (1) the $d$-th Euler class group $\text{E}^d(R)$, defined by Bhatwadekar-R.~Sridharan, is canonically isomorphic to the Levine-Weibel Chow group of zero cycles $\text{CH}_0(\text{Spec}(R))$; (2) the universal Mennicke symbol $\text{MS}_{d+1}(R)$ is canonically isomorphic to the universal weak Mennicke symbol $\text{WMS}_{d+1}(R)$; and (3) additionally, if $R$ is a regular domain, then the Whitehead group $\mathrm{SK_1}(R)$ is canonically isomorphic to $\frac{\text{SL}_{d+1}(R)}{\text{E}_{d+1}(R)}$. As an application, we investigate some Eisenbud-Evans type theorems. |
| title | Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of certain real affine algebras |
| topic | Commutative Algebra 14C25, 13C10, 19B10, 19B14 |
| url | https://arxiv.org/abs/2111.14384 |