By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?


**EDIT** By torsion sheaf I mean the stalk at the generic point is torsion. The curve is not necessarily reduced.



Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve
Just making the above comment an answer.


Take $C:=\text{Proj}\ \big(k[x,y,z]/z^2\big)$. Then $C\_{\text{red}}={\mathbb P}\_k^1$ and the coherent sheaf $F:={\mathcal O}\_{{\mathbb P}\_k^1}(-2)$ over ${\mathbb P}\_k^1$ is known to satisfy
$H^1\big({\mathbb P}\_k^1,{\mathcal O}\_{{\mathbb P}\_k^1}(-2)\big)=k$. Since $F$ can be seen as a coherent sheaf over $C$, and it has torsion at a generic point that comes from the nilpotent $z/x$, you get a desired example due to $H^1(C,F)=H^1(C\_{red},F)=k\ne0$. 


