Timeline for Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 23, 2018 at 2:46 | vote | accept | user40276 | ||
| Nov 19, 2018 at 11:30 | answer | added | D.-C. Cisinski | timeline score: 16 | |
| Nov 18, 2018 at 20:47 | comment | added | user40276 | @MarcHoyois Hmm...Right. I thought at first that restricting to flat closed immersions wouldn't be problematic, but that was totally silly as no compactification will be of this form if, for instance, $\overline{X}$ is Noetherian (maybe if the ideal defining $\overline{X} \setminus X$ is not finitely generated, the immersion will be flat; but that's too weird). Any ideas on how to remedy this problem? What about maybe applying the Waldhausen construction to bounded complexes of coherent sheaves with cohomology having support on $\overline{X} \setminus X$? | |
| Nov 16, 2018 at 3:00 | comment | added | Marc Hoyois | How do you define $G_c$? $G$-theory does not have pullbacks along closed immersions. | |
| Nov 15, 2018 at 21:43 | history | edited | user40276 | CC BY-SA 4.0 |
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| Nov 13, 2018 at 18:53 | comment | added | user40276 | @Gasterbiter Thanks for the comment. That definition still weird to me, though. I guess its the just the best that one can do when something fails to be smooth (be it $\overline{X}$ or $\overline{X}\setminus X$). In any case, I suppose that by dévissage the naive definition is safe enough for $G$. | |
| Nov 10, 2018 at 6:25 | comment | added | user123627 | @user40276 check out section 4.1 in arxiv.org/pdf/1211.1813.pdf | |
| Nov 9, 2018 at 21:05 | history | edited | user40276 | CC BY-SA 4.0 |
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| Nov 9, 2018 at 21:04 | comment | added | user40276 | @MarcHoyois Thanks for the comment! It's somehow surprising my $K_c$ is ill defined. What exactly will fail? If I recall correctly, Gillet seems to define the $K$-theory of a pair in the analogous way when proving higher GRR, but maybe I'm overseeing something... By the way, do you know some reference where such $K_c (X)$ that you mentioned is studied? | |
| Nov 9, 2018 at 15:34 | comment | added | Marc Hoyois | G-theory is like Borel-Moore homology. The simplest reason is its functoriality: it is covariant for proper maps and contravariant for etale maps, like BM homology. Your $K_c$ is not well-defined (now that we know K-theory satisfies pro-cdh descent, we can define $K_c(X)$, but this involves taking the limit over all nilpotent thickenings of $\bar X-X$). | |
| Nov 9, 2018 at 10:54 | history | edited | user40276 | CC BY-SA 4.0 |
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| Nov 8, 2018 at 20:27 | history | asked | user40276 | CC BY-SA 4.0 |