Timeline for Chern classes of torsion free sheaves
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jan 10, 2020 at 16:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 11, 2019 at 15:28 | answer | added | Tim | timeline score: 1 | |
| Jul 7, 2018 at 20:45 | comment | added | User43029 | Thank you very much Jason Starr, that is exactly what I was looking for. | |
| Jul 6, 2018 at 19:49 | comment | added | Jason Starr | Similarly, for a smooth, projective $k$-scheme $X$, the natural map from the K-group of locally free $\mathcal{O}_X$_modules to the K-group of coherent $\mathcal{O}_X$-modules is an isomorphism of modules. Thus, as for projective space, $\chi$ is uniquely determined by its values on locally free $\mathcal{O}_X$-modules. All of this is described in the beautiful notes of Manin, "Lectures on the K-functor in algebraic geometry". | |
| Jul 6, 2018 at 19:47 | comment | added | Jason Starr | "I am looking for the formal definition of Chern classes for torsion free sheaves, at least on the projective spaces." The K-group of projective space $\mathbb{P}^n$ is generated as a free Abelian group by the classes of $\mathcal{O}(-d)$, for $d=1,\dots,n$. Thus, for every Abelian group $A$ and for every formula $\chi$ defined on coherent sheaves that is additive for short exact sequences, i.e., for $A= 1 + t\mathbb{Z}[[t]]$ under multiplication and for $\chi$ equal to the total Chern class, the formula is uniquely determined by its values on $\mathcal{O}(-d)$. | |
| Jul 6, 2018 at 18:46 | history | asked | User43029 | CC BY-SA 4.0 |