Math 953: Problem set 8, due April 4

  • If X is a noetherian scheme, show that the topology on X is noetherian.
  • We say a morphism f: X -> Y of schemes is quasi-finite if f has finite fibers. Grothendieck's version of Zariski's Main Theorem is that: a morphism is finite if and only if it is quasi-finite and proper. (Proper morphisms are defined in section II.4, but in any case, proper morphisms are closed, meaning that the image of any closed set is closed). Give an example of a morphism f which is quasi-finite, surjective and closed, but which is not finite. [Hint: Consider a finite morphism X -> Y, where X = Y = A1, almost any morphism nonconstant nonisomorphism will do, and remove a point from X, just be sure the point is not the only one in its fiber.]
  • Do Problem 3.2 on p. 91 of Hartshorne.