@Libli On M.SE someone asked for a reference of the Auslander "famous" result; see here. Could you help him?

Cross-posted: math.stackexchange.com/questions/4474761

This is a classical result of Serre.

@user782932 As I've mentioned in a comment on M.SE, this can be done by using Cohen factorization, a non-trivial result of Avramov, Foxby and Herzog. However, I suggested the OP to ask the same question here with the hope of a simpler approach.

@user782932 How can reduce the general case to the case $R\to S$ is onto? This is the main point here.

I wasn't able to locate that theorem in Serre's book. Did Serre really prove this?