Lecture Friday January 29

This lecture will cover up through 2.3.D.  The main concepts to be covered: morphisms of presheaves, and ringed spaces.

Study problems.

  1. Exercise 2.2.G: show that the gluability axiom holds for sections of mu:Y->X.
  2. Go through the construction of the direct image of a sheaf on page 2.2.H. and check that the skyscraper is a pushforward.
  3. 2.2.I. Show that pushforward induces maps of stalks.
  4. If F is a sheaf, what is the difference between F(U) and F|_U?
  5. Pay very close attention to 2.2.13. This is absolutely essential.  This gives the definition of ringed space and O_X-modules.
  6. Do you see how to “promote” a topological space into a ringed space, and how to promote a differentiable manifold into a ringed space?  In general, we want to consider the structure sheaf as an essential part of the topological or geometrical setup.
  7. Can you give a nontrivial example of an O_X-module when O_X= sheaf of differentiable functions on a manifold X?
  8. What is a morphism of presheaves? of sheaves?
  9. Exercise 2.3.A. Describe the induced morphism of stalks.
  10. Show how to form categories of sheaves, categories of sheaves of abelian groups, etc.