Let $f : X \to Y$ be a morphism between two Noetherian schemes. Then $f_*$(respect to $R^1f_*$) sends coherent sheaves to coherent sheaves if and only if $f$ is universally closed (respect to separated) according to the answer of If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
Since $f$ is separated if and only if its diagonal morphism $\Delta_f$ is universally closed, I wonder if there exists any direct relation between $R^1f_*$ and $(\Delta_f)_*$. Also, can we generalize relation between two functors in the case of quasi separated & quasi compact morphism?
