Timeline for K_2 of rings of algebraic integers
Current License: CC BY-SA 2.5
9 events
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| Jul 8, 2010 at 20:22 | comment | added | Jim Humphreys | Garland's theorem as originally stated appears in: A Finiteness Theorem for K2 of a Number Field, Ann. of Math. 94, No. 3 (1971), 534-548 (online via JSTOR). From the review on MathSciNet you can also find some but not all earlier/later related papers, including one by Tate in Invent. Math. and a Bourbaki talk by Bass. As other answers and comments here indicate, the subject hasn't reached a definitive state yet. | |
| Jul 8, 2010 at 17:20 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jun 24, 2010 at 15:57 | vote | accept | Andy Putman | ||
| Jun 24, 2010 at 10:35 | comment | added | JBorger | Good point. I agree. Nevertheless I thought Tate's theorem might be good to see for those not familiar with the subject. | |
| Jun 24, 2010 at 9:57 | comment | added | Minhyong Kim | James: I agree that expressing $K_2$ in terms of Galois cohomology should be something 'explicit' in principle, but our current working knowledge of Galois cohomology seems primitive enough to make the utility of it entirely unclear. Recall that finiteness of $H^2(G_S, \mathbb{Z}_l(2)$, for example, uses the finiteness of $K_2$, even though one might have expected the application to run in the other direction. | |
| Jun 24, 2010 at 9:51 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jun 24, 2010 at 9:26 | comment | added | JBorger | Oops. That should be $\mathbf{Z}_{\ell}(2)$ instead of $\mathbf{Z}_{\ell}$. | |
| Jun 24, 2010 at 7:30 | comment | added | JBorger | Also, Tate has an explicit description of $K_2(F)$, where $F$ is a global field, in terms of Galois cohomology: $K_2(F)\cong\bigoplus_{\ell}H^2(G_F,\mathbf{Z}_{\ell})_{\mathrm{tor}}$, where $\ell$ runs over all primes distinct from the characteristic of $F$. See his paper "Relations between K_2 and Galois Cohomology", Inv Math 1976. | |
| Jun 24, 2010 at 7:00 | history | answered | Minhyong Kim | CC BY-SA 2.5 |