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.
- What is the definition of an additive category?
- What does it mean in the definition to say that “composition of morphisms distributes over addition”?
- Do you see how the category of abelian groups is an additive category?
- Do you see how the category of finite-dimensional vector spaces over a field k is an additive category?
- Kernels of homomorphisms are defined by a universal property inside additive categories. What is that definition?
- 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.
- What are the three additional properties imposed on an additive category that makes it an abelian category?
- Verify that the category of abelian groups is an abelian category.
- Verify that the category of vector spaces over a fixed field k is an abelian category.
- 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?
- What is an image? a quotient? an exact sequence?