In this text on page 15, remark 3.17 they note (for a projective variety $X_F$ over $F$, $F$ being a function field over $k$) that
$$\operatorname{Ext}^1(\Omega_{X_F/F}^1; \mathcal{O}_{X_F})$$ is canonically isomorphic to [the $F$ finite dimensional vector space] $$H^1(X_F; f^*(\Omega_{F/k}^1) \otimes (\Omega_{X/F}^1)^{\vee}).$$
Unfortunately, I have no idea why. Could anyone point me to the right lemma's in some book (probably Hartshorne) or show me why this follows?
EDIT: I also question if this was really the remark he wanted to make, because a few lines above this he is looking at $$\operatorname{Ext}^1(\Omega_{X_F/F}^1; f^*(\Omega_{F/k}^1))$$ which makes more sense to me.