Wednesday March 30, 2016

Z-valued points

Section 6.3.7 introduces a new notion of point (the Z-valued points).  Vakil notes that the “terminology ‘Z-valued point’ is unfortunate” because a scheme already has a set of points and Z-valued points are not points on the scheme.  Remember: algebraic geometry is a world where functions are not functions and points are not points.

  • 6.3.L.a A morphism induces a map of Z-valued points.
  • 6.3.8. Check the statement that A-valued points of an affine scheme are given by solutions to equations f1(x1,…,xn) = … = fr(x1,…,xn) = 0 in A.
  • 6.3.M. Points that take values in a local ring.
  • 6.3.N. Morphisms into projective space.

Read to the end of 6.3.

Maps of projective schemes

  • Exercise 6.4.A going from morphisms of graded rings to morphisms of schemes.
  • Work through example 6.4.1.
  • Work through example 6.4.2.
  • Exercise 6.4.C.  Different maps of graded rings can give the same morphism of schemes.
  • Exercise 6.4.D.