Let $f: X \to Y$ be a morphism of schemes. Consider the graph morphism $(id,f): X \to X \times Y$. Denote by $\Gamma_f \subset X \times Y$ it's schematic image. Futhermore the projection $pr_X: X \times Y \to X$ induces the morphism $p:\Gamma \to X$
My question is under what conditions (preferably the most weak ones) on $f,X,Y$ the graph morphism $(id,f)$ and $pr$ are isomorphisms.
I know:
-$(id,f)$ is always an immersion and even a closed immersion (by a base change argument) if $Y$ is separable (can be seen by a base change argument)
-it's (set theoretically) injective since $pr . (id,f)= id$ for $pr: X\times Y \to X$.
-then there are always some boring conditions like when $f$ is already an isomorphism or similar
Does anybody know the most weakingst conditions which garantee that $(id,f)$ is an iso