Monday March 28, 2016

This meeting will continue with morphisms of schemes from Chapter 6.

Read carefully the proof of 6.3.2 (the key proposition: every morphism of locally ringed spaces between affine schemes is induced by a ring homomorphism).  Someone will be asked to present the proof in class.  We will break it into several steps. In particular,

  • The map of points is determined by its map on global sections.
  • The map of sections is determined by the map of global sections.

 

We will continue with various exercises:

  • 6.3.D.  The category of rings and the opposite category of affine schemes are equivalent.
  • 6.3.C.  A morphism of schemes is a morphism of ringed spaces that is locally a morphism of affine schemes.

Read section 6.3.4 and the definition of S-schemes.   We have already seen a different definition of A-schemes.  Show that this new definition is compatible with the old (exercise 6.3.G).

  • 6.3.E.  Practice with morphisms.
  • 6.3.F. Morphisms X -> Spec A.
  • 6.3.H.
  • 6.3.I.
  • 6.3.J.
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