- I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities are planar).
The Chern class can be, of course, extracted from an exact sequence relating $F$ to some locally free sheaves. But I would like some more direct definition, like the one given by Hartshorne (Generalized divisors on Gorenstein curves and a theorem of Noether. J. Math. Kyoto Univ. 26 (1986), no. 3, 375–386).
At least for the first Chern class.
1.5 Even if one wants to define $c_1(F)$ from some resolution: torsion free sheaves on singular curves sometimes have no finite locally free resolutions. What would you do in this case?
- I'm looking for the Riemann-Roch for torsion-free sheaves on a singular curve (can assume the singularities to be planar). For example Hartshorne in the paper above does it for rank one.
Of course, if the only definition of the first Chern class is from the exact sequence, then Riemann-Roch is tautological (an alternative way to define $c_1(F)$). So this question is meaningful modulo the first question.
Somehow I do not find all this in classical textbooks.
Thanks to everybody!!!!
Best Answer
In the affine case, there is a sweet way to define the first Chern class as follows:
Let $R$ by the coordinate ring and $M$ the $R$-module correspond to our sheaf. As $M$ is torsion-free, one can embed $M$ in to a free module: $ 0\to M \to F \to N \to 0$ ( you need $M$ to be of constant rank, and that rank would be the rank of $F$). In this $N$ would be torsion, so the support is finite. Take the cycle $c(N) = \sum length(N_p)[p]$ where $p$ runs over the support of $N$. Then define $c(M)= -c(N)$.
In general, one could get codimension 1 cycles by picking them from any prime filtration of $M$(you needs to show that what you get from 2 different filtrations are rationally equivalent). This paper contains a treatment of that result.