Math 6290 (Fall 2023):
Homological Algebra

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Homework

  1. For Monday, 4 December: Here are some more problems. Next week: we will complete the proof that soft sheaves are acyclic.
  2. For Wednesday, 29 November: Here are some problems to think about. The next topic is the acyclicity of flasque, fine, and soft sheaves, in that order. We will use these calculations to relate sheaf cohomology to singular cohomology and de Rham cohomology.
  3. For Monday, 27 November: Here are some problems to think about. The first goal when we return is to do some calculations and relate sheaf cohomology to other cohomology theories. Mainly this will come from §§2.4, 2.5, and 2.8. When you encounter D+ in the book, you may read it as the category of bounded below cochain complexes of injective sheaves.
  4. For Friday, 17 November: Here are some exercises about sheaves. The goal for Friday is to show that sheaves of abelian groups form an abelian category.
  5. For Monday, 13 November: We are starting to study sheaves. Please start reading Chapter II of Kashwiara–Schapira. I will try to get through §§2.1–3 this week. For a gentler introduction, you might try Tu's notes.
  6. For Friday, 10 November: We will do the Grothendieck spectral sequence on Friday (§5.8). You may want to review the horseshoe lemma (2.2.8) and read the construction of Cartan–Eilenberg resolutions (§5.7). Here are some problems that review what we've done with spectral sequences.
  7. For Wednesday, 1 November: You may want to look at section 3.6 on universal coefficient theorems.
  8. For Monday, 30 October: Please start reading Chapter 5 of Weibel.
  9. For Friday, 27 October: Here are some exercises. On Friday: Serre's theorem on global dimension (Theorem 4.4.16 from Weibel, or 19.2 from Matsumura)
  10. For Wednesday, 25 October: One more problem. I also added a problem I had forgotten to include on Monday's list. The goals are still Theorems 4.4.15 and 4.4.16.
  11. For Monday, 23 October: Here are some problems to think about. Goals for next week: Theorems 4.4.15 and 4.4.16.
  12. For Wednesday, 18 October: We are reviewing some commutative algebra. I am partly following chapters 5, 6, and 7 of Matsumura. The main result for Wednesday will be the Hauptidealsatz, for which I will mostly follow Vakil, \S11.5. Here are a few problems to think about.
  13. For Friday, 11 October: Here are some problems to think about. We will skip sections 3.5 and 3.6 for now and start Chapter 4 on Friday. For now, we will mostly discuss §§4.1, 4.3, and 4.4. The first goals are Corollary 4.3.8 and Theorem 4.4.15.
  14. For Monday, 9 October: Here are some problems to think about. The next topic is Yoneda extensions, which is §3.4 in the book.
  15. For Friday, 6 October: Here are some problems to think about. The first topic on Friday will be lemma 2.6.14 and theorem 2.6.15. Please continue reading through chapter 3. I won't have a lot to say in class about 3.2 and 3.3, but you should read them. Section 3.4 is about Yoneda extensions (one of my favorite topics), so we will spend a while there.
  16. For Monday, 2 October: Please start reading §2.6 (the important part for us right now is the bit about exactness of filtered colimits at the end on pp. 56–58) and §3.1. Here are some exercises to think about.
  17. For Wednesday, 27 September: We will discuss two proofs that R-mod has enough injectives (the proof in the textbook is on pp. 39–40). Reading: §2.1 and §2.7. We will need to show that Tor and Ext are independent of which variable is used to derive them. We will have two proofs of this: one similar to the textbook and one based on §2.1. No new exercises for today.
  18. For Monday, 25 September: We have already started discussing §§2.4–2.5, so you should read these. We will talk about injective modules and right derived functors on Monday. We will talk about the proof from Tohoku for the existence of enough injectives. Here are some problems to think about.For Friday, 22 September: We will continue covering §§2.2–2.3. It will be very important to know the definition of a projective object on Friday. Here are some problems to think about.
  19. For Wednesday, 20 September: Read §§2.2–2.3. Here are some questions to think about.
  20. For Monday, 18 September: I promise to answer the question about kernels and cokernels in the chain homotopy category.
  21. For Friday, 15 September: Make sure you are comfortable with the key definitions: chain homotopy, quasiïsomorphism, cone. Here are some more exercises to think about. We will continue the discussion of cones on Friday.
  22. For Wednesday, 13 September: Exercises 1.4.1–5 are all good to think about. The last part of 1.4.5 is quite subtle. We will discuss §1.5 on Wednesday.
  23. For Monday, 11 September: Here are some questions to think about. We should finally get to §1.4 on Monday. This important definitions will be 1.1.2 and 1.4.4. The plan is to talk a little more about the combinatorial n-sphere example and do some examples of chain homotopy equivalences.
  24. For Friday, 8 September: Here are some questions to think about. Read §§1.4 and start §1.5. Homework is due.
  25. For Wednesday, 6 September: Here are some questions to think about. Continue reading chapter 1 of Weibel's book. We will discuss material from §§1.1–4 in class. Here is a Canvas dropbox for your first assignment.
  26. For Friday, 1 September: Here are some questions to think about. Continue reading chapter 1 of Weibel's book. We will probably talk about material from 1.1 and 1.2 on Friday.
  27. For Wednesday, 30 August: Here are some questions to think about. Start reading chapter 1 of Weibel's book. The important terms for Wednesday are all on p. 6.