Let $f : X \rightarrow Y$ be a proper map of schemes. How do I show that $f^* \vdash f_!$? That is, how do I show that proper direct image is left adjoint to inverse image?
Somewhere I need to use that $f$ is separated, i.e. $X \rightarrow X \times_Y X $ is closed, and that it is universally closed, i.e. that $X \times_Y Z \rightarrow Z$ is closed, and probably (but maybe not?) that it is of finite type. It seems like the proof would be long, so maybe a reference would be nice.
Evidently, one is supposed to show that $f_*$ has a right adjoint when $f$ is proper, and that the right adjoint is $f^*$? Maybe someone can outline the steps for me.
I messed up -- this is what should be the case. I would instead like a proof that $f_! $ is left adjoint to $f^!$.