9

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne:

Let $X$ be a nonsingular projective curve over an algebraically closed field $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$ with rank $r$ (at the generic point). Part 11(c) says there is a divisor $D$ and a short exact sequence:

$0 \rightarrow O_X(D)^{\oplus r} \rightarrow \mathcal{F} \rightarrow \mathcal{J} \rightarrow 0$

where $\mathcal{J}$ is a torsion sheaf. But $X$ is defined over a field, doesn't that imply that $\mathcal{J} = 0$? (Since $H^0(X,\mathcal{J}) = \bigoplus_{P \in X} \mathcal{J}_P$, and $H^0(X,\mathcal{J}) = 0$ being a torsion $k$-module, hence all the stalks of $\mathcal{J}$ are $0$).

That would mean all coherent sheaves on $X$ of rank $r$ are locally free, I don't quite believe that.

(Side note: What I was trying to think about originally is whether such a sheaf $\mathcal{F}$ has global sections or not by considering the degree of the divisor $D$. Any comment on that would also be quite helpful).

Nadim Rustom
  • 839
  • "Chapter 7": What Hartshorne is this? – aegbert Jul 17 '14 at 14:33
  • Sorry, it's chapter 2, of Algebraic Geometry. I fixed the typo. – Nadim Rustom Jul 17 '14 at 14:41
  • 4
    Being a torsion sheaf means that the stalk at the generic point is zero. I don't think this implies $H^0(X,\mathcal{J})$ is a torsion $k$-module. – RghtHndSd Jul 17 '14 at 15:19
  • Taking $\mathcal{J}$ to be the skyscraper sheaf at a closed point will give you a torsion sheaf such that $H^0(X,\mathcal{J})$ is not a torsion $k$-module. So I should have written "This doesn't imply" in my comment above. – RghtHndSd Jul 17 '14 at 15:22
  • Thanks, it seems I misunderstood the definition of "torsion sheaf". But does what you said apply to the sheaf $\mathcal{J}$ that appears in the short exact sequence above? I expect that the sheaf $\mathcal{J}$ is the same as the torsion subsheaf of $\mathcal{F}$, which associates to an open affine $U$ the torsion submodule of $\mathcal{F}(U)$. In this case the stalks at any point would be torsion? – Nadim Rustom Jul 17 '14 at 15:52

1 Answers1

7

Torsion means torsion over $\mathcal O_X$. For a coherent sheaf on a curve, it is equivalent to having support a finite number of closed points.

user160609
  • 1,747