Coherent Sheaves on Smooth Curves

algebraic-geometrycoherent-sheavesprojective-spacevector-bundles

Let $X$ be a locally noetherian integral non-singular scheme of dimension 1 (I'm really interested in the case $X=\mathbb{P}^1_K$) and $\mathcal{E}$ a coherent sheaf over $X$, I've already shown that $\mathcal{E}$ is either locally free (of finite rank) or has torsion (this use the theorem of the structure of finitely generated modules over PIDs on stalks). Let $\mathcal{T} \subseteq \mathcal{E}$ the torsion subsheaf, then $\mathcal{F} = \mathcal{E}/\mathcal{T}$ is locally free and we have a short exact sequence
$$
0 \longrightarrow
\mathcal{T} \longrightarrow
\mathcal{E} \longrightarrow
\mathcal{F} \longrightarrow
0
$$

My question is:

  1. I want this sequence to split, so every coherent sheaf over $\mathbb{P}^1_K$ would be a sum of a vector bundle and a torsion coherent sheaf. I know this sequence is split on stalks or even on affine opens, but it is not clear to me how to glue sections/retractions together since there is no canonical one.

  2. I saw somewhere that torsion sheaves over $X$ have finite support, but I don't know why. Does anyone have a hint or a reference to it?

Context: I already know Birkhoff–Grothendieck theorem about the classification of vector bundles over $\mathbb{P}^1_K$. Now I want a sort of classification to coherent sheaves. It sounds like every coherent sheaf is a finite sum of twists of $\mathcal{O}$ (lines bundles) and skyscraper sheaves over closed points.

Best Answer

  1. There are a few ways to argue that the sequence splits; here is one. The obstruction to splitting your sequence is an element of $\text{Ext}^1_{\mathcal{O}}(\mathcal{F}, \mathcal{T})$. Since $\mathcal{F}$ is locally free, we have $$ \text{Ext}^1_{\mathcal{O}}(\mathcal{F}, \mathcal{T}) \simeq \text{Ext}^1_{\mathcal{O}}(\mathcal{O}, \mathcal{T} \otimes \mathcal{F}^\vee) \simeq H^1(\mathbb{P}^1, \mathcal{T} \otimes \mathcal{F}^\vee). $$ Since $\mathcal{T} \otimes \mathcal{F}^\vee$ is supported at finitely many points, and hence on a closed subset of dimension 0, this cohomology group vanishes by Grothendieck vanishing. Intuitively, your question is about eliminating an obstruction to gluing some local sections, so it shouldn't be surprising that cohomology provides a resolution. Incidentally, this argument works on any nonsingular curve $X$.

  2. By quasi-compactness of $X$, we may assume that $X$ is an affine curve $\text{Spec} \, R$, for $R$ a Dedekind domain. Now the claim is that if $M$ is a finitely generated torsion module over $R$, then $\tilde{M}$ has finite support. By finite generation, we can find a single element $f \in R$, $f \neq 0$, such that $fm = 0$ for all $m \in M$. Now, $M_f = 0$. Geometrically, this means that $\tilde{M}\vert_{D(f)} = 0$. Since $D(f)$ is a non-empty Zariski open subset of $X$, $\tilde{M}$ must be supported on the complement, a proper Zariski closed subset of $X$. But a proper closed subset of an irreducible curve is finite.

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