Let S be the affine line over the complex numbers. Consider the big fppf site (Sch/S)_{fppf} of S. By a theorem of Deligne this site has enough points. How can we describe these points?

Here is one way to construct points. Write S = Spec(C[x]) and suppose that B is a local C[x]-algebra such that any faithfully flat, finitely presented ring map B —> C has a section. Then the functor which associates to an fppf sheaf F the value F(Spec(B)) is a stalk functor, hence determines a point. In fact, I think all points of (Sch/S)_{fppf} are of this form.

Actually, if B is henselian, then it suffices if finite free ring maps B —> C have a section; this uses the material discussed here. If B is a henselian domain, it suffices if its fraction field is algebraically closed. A specific example is the ring B = ∪ C[[x]][x^(1/n)].

Anyway, I was hoping to use this description to say something about question 4 of this post on exactness of pushfoward along closed immersions for the fppf topology. I still don’t know the answer to that question. Do you?