Timeline for Why write GRR with the relative tangent sheaf?
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6 events
| when toggle format | what | by | license | comment | |
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| Feb 19 at 22:24 | comment | added | Tim | Only a comment instead of an answer, but in the introduction of SGA 6 they write that they will use the second formulation because it is "more useful for their needs". I'm not familiar with this part of SGA 6 so I can't say exactly why how this is true, but maybe an answer is to be found there :-) | |
| Apr 26, 2015 at 20:37 | comment | added | A Rock and a Hard Place | In fact the Baum-Fulton-MacPhearson Riemann-Roch theory seems to me to be an argument for writing the theorem in the first version, aka. as exhibiting a natural transformation: that is the version that generalizes to nonsingular varieties and even to separated schemes of finite type over a field. | |
| Apr 26, 2015 at 20:33 | comment | added | A Rock and a Hard Place | Well, there are ways of defining Todd classes even for singular varieties. One can take the Baum-Fulton-MacPhearson natural transformation $\tau\colon K_0\to A_{\mathbb Q}$ and define the homology Todd class $\tau_X(\mathscr O_X).$ For locally complete intersections in nonsingular varieties that reduces to $\operatorname{td}(\mathscr T_X)\cap [X],$ the Todd class of the virtual tangent bundle capped with the fundamental class. Provided, GRR in the form above fails to hold for these new Todd classes, but still. | |
| Apr 21, 2015 at 6:26 | comment | added | Donu Arapura | The second version is also more general. If $X$ or $Y$ are singular, then the terms in first statement are undefined, but $\mathcal{T}_f$ may still be defined, e.g. when $f$ is smooth and proper. | |
| Apr 20, 2015 at 20:36 | comment | added | Will Sawin | One reason is just because people may want to compute $ch(f_!a)$ and the second formulation tells you how to do that. | |
| Apr 20, 2015 at 20:17 | history | asked | A Rock and a Hard Place | CC BY-SA 3.0 |