Calculate Ext group

Question

Let $X$ be a projective Calabi-Yau threefold. Then I would like to compute $Ext^i(O_X,O_X)$ group, which makes me confusing. First suppose we take injective resolution of the $O_X$ on the right, then we are just calculating the cohomology group. Then $H^0(O_X)=H^3(O_X)^*=\mathbb{C}$, where the first equality is due to Serre duality and the second is just the fact of projective variety. On the other hand, if we take a locally free resolution of the first $O_X$, then the resolution will be $0\rightarrow O_X\rightarrow 0$ and take $Hom(-,O_X)$, we just get $Ext^0(O_X,O_X)=\mathbb{C}$ and others are all zero. I am sure I made a (quite stupid) mistake.

Answer 1

You can compute Ext sheaves using a locally free resolution of the first argument, but you can't compute Ext groups that way.