Hi,

I have a simple question about coherent sheaves and line bundles:

if I have a coherent sheaf $F$ on a good scheme $X$ and I know that $F_x \otimes k(x) = k(x)$ for alle points $x$ on $X$ (where $k(x)$ is the residue field of $x$ and the tensor product goes over the local ring $\mathcal O_{X,x}$), can I then say, that $F$ is already a line bundle?

Hi,

I have a simple question about coherent sheaves and line bundles:

if I have a coherent sheaf $F$ on a good scheme $X$ and I know that $F_x \otimes k(x) = k(x)$ for alle points $x$ on $X$ (where $k(x)$ is the residue field of $x$ and the tensor product goes over the local ring $\mathcal O_{X,x}$), can I then say, that $F$ is already a line bundle?

My strategy would be:

the stalk of $F$ is finitely generated over the local ring; by Nakayama you can choose a generating element $s_x$ of $F_x$ for every $x$, so you have $F_x \simeq \mathcal O_{X,x}$ and now use the statement that if a coherent sheaf is free in a point, then it is in a neighborhood. Is this the right way to see it?