Let $X$ be a scheme and $\mathcal{F}$ be a sheaf on $X$ which is torsion $\mathcal{O}_X-$module (i.e., every local section is annihilated by an element of the ring $\mathcal{O}_X(U)$) or nilpotent (i.e., a power of any local section is zero). When can we say that $H^i(\mathcal{F})$ or $H^i(Hom_X(\mathcal{F},\mathcal{O}_X))$ vanishes for $i>0$. For example if $X$ is a curve then does the first cohomology groups given above vanish?

Let us take for example $X$ is a non-reduced curve in $\mathbb{P}^3$ ($X_{red}$ not smooth). Can we say that $H^i(Hom_X(I_X/I_X^2,I_{X_{red}}/I_X))$ equal to zero for $i$ equal to $0$ or $1$? (Here $I_X, I_{X_{red}}$ denotes respectively the ideal sheaves of $X, X_{red}$ in $\mathbb{P}^3$.)

torsiondoes not include the notion ofnilpotentfor the simple reason that $\mathcal F$ may not have a multiplication. Perhaps you want to look at $\mathcal O_X$-algebras? Torsion and nilpotent are still two different notions, but perhaps you can figure out what it is that you are asking. Furthermore, in your particular example you are asking if the $0^{\mathrm{th}}$ or $1^{\mathrm{st}}$ cohomology of a sheaf on a curve vanishes. It is unlikely that anything like that could happen. If it is torsion... (cont'd) – Sándor Kovács Dec 3 '12 at 18:14