4

Let $C$ be a curve and $f:C \to \mathbb{P}^1$ a finite morphism. I know that the induced sheaf $\mathcal{A}= f_*(\mathcal{O}_C)$ is coherent.

I want to know why following statements are equivalent:

1) $\mathcal{A}= f_*(\mathcal{O}_C)$ is locally free; therefore therefore for every $u \in \mathbb{P}^1$ there exist an open affine neghboerhood $U= Spec(R) \subset \mathbb{P}^1$ such that $\Gamma(U, f_*{O}_C) \cong R^n$.

2) $C$ don't has any embedded components, where the embedded components are defined as $Emb(\mathcal{O}_C):= \{c \in C \ | \ m_c \in Ass_{\mathcal{O}_{C,c}}(\mathcal{O}_{C,c}) $and $c$ not generic$\} $

user267839
  • 7,293
  • 1
    A coherent sheaf on a smooth curve (in your case the projective line) is locally free if and only if it is torsion free. Can you see how to use that? – Mohan Jan 20 '18 at 02:40
  • One step is missing: Let wlog $c \in D_+(T_0) \cong k[T]$. $k[T]$ a domain, so $k[T]c$ also and we get - using your criterion and localisation property - that the stalk $\mathcal{O}{C,c} \cong (k[T]C)^n$ is a torionfree $k[T]_c$-module. This imply obviously $Ass{k[T]c}(\mathcal{O}{C,c})= {0 }$. But I dont't see any argument which allows me to controll $Ass_{\mathcal{O}{C,c}}(\mathcal{O}{C,c})$. Especially to show that if $m_c \in Ass_{\mathcal{O}{C,c}}(\mathcal{O}{C,c})$ then $m_c$ is a minimal prime (equivalently for beeing generic). – user267839 Jan 20 '18 at 15:19
  • Which implication do you find hard? I hope 1) implies 2) is clear. For 2) implying 1), if a closed point (this is what not generic means) is embedded, can you see why the direct image has torsion? – Mohan Jan 20 '18 at 16:09
  • Well, the implication 2) to 1) is clear: If $c$ not embedded, then $m_c$ can't be minimal in $\mathcal{O}{C,c}$; otherwise $c$ would be generic of it's apropriate open affin neighbourhood according affine correspondence between irreducible components and minimal prime ideals. So $Ann(x_c) = m_c \neq 0 $ for some $x_c \in \mathcal{O}{C,c}$ and $x_c$ is a torsion element. – user267839 Jan 20 '18 at 16:40
  • My problem is still 1) to 2): As I explained in my post above I just get by torsion freeness that $Ass_{R_c}(\mathcal{O}{C,c})= {0}$ for $\mathcal{O}{C,c}\cong R_c^n$ (by local freeness), but don't see how to cope with $m_c$ considered in $Ass_{\mathcal{O}{C,c}}(\mathcal{O}{C,c})$... – user267839 Jan 20 '18 at 16:40

0 Answers0