What theorem of Serre's is being referenced?

Question

I am reading Joseph Bernstein's notes on D-modules which are available online here

In section 1 (page 1 of the pdf) Bernstein writes "By Serre's theorem this condition is local."

I was wondering to what theorem Bernstein is referring and wondering where I could find a statement and proof of this theorem.

Answer 1

I believe the first theorem in article 1.2 here