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?<\/p>\n
Here is one way to construct points. Write S = Spec(C<\/strong>[x]) and suppose that B is a local C<\/strong>[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.<\/p>\n Actually, if B is henselian, then it suffices if finite free ring maps B —> C have a section; this uses the material discussed here<\/a>. If B is a henselian domain, it suffices if its fraction field is algebraically closed. A specific example is the ring B = \u222a C<\/strong>[[x]][x^(1\/n)].<\/p>\n Anyway, I was hoping to use this description to say something about question 4 of this post<\/a> on exactness of pushfoward along closed immersions for the fppf topology. I still don’t know the answer to that question. Do you?<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading