Timeline for $K_1(k[x]/(x^2))$ for a field $k$
Current License: CC BY-SA 4.0
8 events
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| Nov 17, 2022 at 13:08 | comment | added | user443060 | I wanted to post this as a separate question but in this thread if I get some hint it would be helpful. So $K_1(k[x]/(x^2))$ is properly understood. But if I extend it to $K_1((k[x]/(x^2))[y])$ will the result be same? Here I find the problem at $SK_1$. | |
| Nov 7, 2022 at 4:48 | comment | added | user443060 | Thank you very much. I will read through it definitely. | |
| Nov 6, 2022 at 18:32 | comment | added | F Zaldivar | For a commutative ring $A$, the determinant splits $K_1(A)\simeq A^{\times}\oplus (\text{SL}(A)/E(A)$, and the second summand is trivial when $\text{SL}(A)$ is generated by elementary matrices. This is the case when $A$ is a local ring, as in your example. I don't have Weibel's textbook, but probably this is discussed there since it is standard for an introduction to algebraic K-theory. | |
| Nov 6, 2022 at 17:58 | comment | added | Steven Landsburg | @Mohan: I don't think we even need Artin for this. | |
| Nov 6, 2022 at 17:40 | comment | added | Mohan | For any artin local ring, $K_1$ is just the group of units. | |
| Nov 6, 2022 at 17:35 | history | edited | YCor |
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| Nov 6, 2022 at 17:12 | history | edited | LSpice | CC BY-SA 4.0 |
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| Nov 6, 2022 at 17:09 | history | asked | user443060 | CC BY-SA 4.0 |