$Ext_{\mathbb{P}^1}^1(F, \mathcal{O}_{\mathbb{P}^1})=H^1(\mathbb{P}^1, F^* \otimes \mathcal{O}_{\mathbb{P}^1})$

Question

My question refers to a step in the proof of THM 2.1.1 (Grotherdieck) from excerpt from "Vector Bundles on Complex Projective Spaces" by Christian Okonek, Michael Schneider, Heinz Spindler (page 12):

Why does hold

$$Ext_{\mathbb{P}^1}^1(F, \mathcal{O}_{\mathbb{P}^1})=H^1(\mathbb{P}^1, F^* \otimes \mathcal{O}_{\mathbb{P}^1})$$

In my former thread Hom Tensor Adjunction for Ext Groups

I tried to proof a statement that had been instantly provided wished equality, but the statement was wrong, so I wasn't able to apply it.

How to show the statement above otherwise?

Answer 1

For any locally free sheaf $E$ and any quasi-coherent sheaf $F$ over some scheme $X$, we have $Hom_X(E,F)\cong Hom_X(\mathcal{O}_X,E^\wedge\otimes F)$ where $E^\wedge$ denotes the dual sheaf of $E$. Taking derived functors of both sides and noting that $Hom(\mathcal{O}_X,-)=\Gamma(X,-)$, we obtain the result.