Lecture Monday Feb 1

This meeting will cover 2.3.C (page 79) through 2.4.3 (page 83).   The most important point will be to see how the category of presheaves of abelian groups on an topological space X forms an abelian category.

  1. 2.3.C.  How is the “sheaf Hom” defined?  What are the restriction maps? Why is it a sheaf?
  2. 2.3.C-Warning:  Can you find an example showing sheaf-hom does not commute with taking stalks?
  3. We will go over in detail the arguments showing that the category of presheaves of abelian groups on a space X forms  an abelian category.   Start with the defining properties of an additive category.  How are the sets of morphisms viewed as abelian groups?  What is the 0-object?  What is a product of two presheaves?
  4. What is the kernel and cokernel in this category?  How are restriction maps defined on these sheaves?
  5. Complete the verification that this category is abelian.
  6. 2.3.G. Specialize the definition of exactness to this abelian category.
  7. 2.3.H. Characterize exactness in terms of sections over each U.
  8. Exercise 2.3.I.
  9. Exercise 2.3.J.
  10. Exercise 2.4.A.
  11. Exercise 2.4.B.
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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.

 

Lecture Wednesday Jan 27

This meeting will cover Chapter 2 (Sheaves) up through 2.2.6. (plus examples 2.2.9 and 2.2.10).   The definition of sheaf is absolutely essential for what follows later in this course.

Read over the examples in 2.1.

  1. What is the definition of a presheaf on a topological space X?
  2. Do you see how differentiable functions on R^n can be made into a presheaf?
  3. Exercise 2.2.A. Aren’t we all category lovers?
  4. Do the terms “section” and “restriction” here carry their usual meanings?
  5. Work through the construction of the stalk at p of a presheaf.
  6. On a discrete topological space, how are the stalks related to sections?
  7. On the topological space X={p,q} with just three open sets {},{p},{p,q}, what is the stalk at q of a skyscraper sheaf at p?
  8. Do you see how the stalk is the colimit of a filtered index category?
  9. What two additional axioms make a presheaf a sheaf?
  10. Exercise 2.2.D.
  11. What is the constant presheaf on a topological space (Example 2.2.10.).
  12. How does the constant sheaf differ from the constant presheaf?

Lecture Monday Jan 25

Abelian Categories.  Study guide.

There is much more material in section 1.6 than can reasonably be discussed in one lecture.   At some point you should become familiar with everything in this section, because it is all very useful in algebraic geometry.  We will cover the early part of the section in class.

The definition of abelian category builds on the definition of additive category.

  1. What is the definition of an additive category?
  2. What does it mean in the definition to say that “composition of morphisms distributes over addition”?
  3. Do you see how the category of abelian groups is an additive category?
  4. Do you see how the category of finite-dimensional vector spaces over a field k is an additive category?
  5. Kernels of homomorphisms are defined by a universal property inside additive categories.  What is that definition?
  6. In the examples of (2) and (3) above, can you relate kernels and cokernels to notions of kernels and cokernels that you have seen before?

 

Study questions for abelian categories.  You main objective should be to develop intuition that allows you to see an abstract abelian category as being somehow similar to the  category of modules of a ring.

  1. What are the three additional properties imposed on an additive category that makes it an abelian category?
  2. Verify that the category of abelian groups is an abelian category.
  3. Verify that the category of vector spaces over a fixed field k is an abelian category.
  4. Think about the statement following the definition, “It is a nonobvious (and imprecisely stated) fact that every property you want to be true about kernel, cokernels, etc. follows from these three.”  What would you want to be true?
  5. What is an image? a quotient? an exact sequence?