It depends on what kind of scheme $S$ is. If $S$ is reduced of finite type over a field, then the set of closed points is dense in $S$, so all morphisms are determined by their behaviour on the closed points. On the other hand if $S$ is a local scheme, then it has a single closed point, so you can't say much about automorphisms just from the closed points.

It depends on what kind of scheme $S$ is. If $S$ is (reduced) of finite type over a field, then the set of closed points is dense in $S$, so all morphisms are determined by their behaviour on the closed points. On the other hand if $S$ is a local scheme, then it has a single closed point, so you can't say much about automorphisms just from the closed points.

It depends on what kind of scheme $S$ is. If $S$ is of finite type over a field, then the set of closed points is dense in $S$, so all morphisms are determined by their behaviour on the closed points. On the other hand if $S$ is a local scheme, then it has a single closed point, so you can't say much about automorphisms just from the closed points.

It depends on what kind of scheme $S$ is. If $S$ is reduced of finite type over a field, then the set of closed points is dense in $S$, so all morphisms are determined by their behaviour on the closed points. On the other hand if $S$ is a local scheme, then it has a single closed point, so you can't say much about automorphisms just from the closed points.