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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.13624 |
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| _version_ | 1866916329618931712 |
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| author | Banerjee, Sourayan Kuber, Amit |
| author_facet | Banerjee, Sourayan Kuber, Amit |
| contents | Motivated by Krajiček and Scanlon's definition of the Grothendieck ring $K_0(M)$ of a first-order structure $M$, we introduce the definition of $K$-groups $K_n(M)$ for $n\geq0$ via Quillen's $S^{-1}S$ construction. We provide a recipe for the computation of $K_1(M_R)$, where $M_R$ is a free module over a PID $R$, subject to the knowledge of the abelianizations of the general linear groups $GL_n(R)$. As a consequence, we provide explicit computations of $K_1(M_R)$ when $R$ belongs to a large class of Euclidean domains that includes fields with at least $3$ elements and polynomial rings over fields with characteristic $0$. We also show that the algebraic $K_1$ of a PID $R$ embeds into $K_1(R_R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Model-theoretic $K_1$ of free modules over PIDs Banerjee, Sourayan Kuber, Amit Logic K-Theory and Homology 03C60, 19B99, 03C07, 19D23, 19B14 Motivated by Krajiček and Scanlon's definition of the Grothendieck ring $K_0(M)$ of a first-order structure $M$, we introduce the definition of $K$-groups $K_n(M)$ for $n\geq0$ via Quillen's $S^{-1}S$ construction. We provide a recipe for the computation of $K_1(M_R)$, where $M_R$ is a free module over a PID $R$, subject to the knowledge of the abelianizations of the general linear groups $GL_n(R)$. As a consequence, we provide explicit computations of $K_1(M_R)$ when $R$ belongs to a large class of Euclidean domains that includes fields with at least $3$ elements and polynomial rings over fields with characteristic $0$. We also show that the algebraic $K_1$ of a PID $R$ embeds into $K_1(R_R)$. |
| title | Model-theoretic $K_1$ of free modules over PIDs |
| topic | Logic K-Theory and Homology 03C60, 19B99, 03C07, 19D23, 19B14 |
| url | https://arxiv.org/abs/2407.13624 |