# Math 6290 (Spring 2020):

Homological algebra

## Textbook

The textbook will be *An introduction to homological algebra*, by Charles A. Weibel.

## Lecture recordings

- April 24: Galois cohomology
- April 14: group cohomology
- April 10: derived categories
- April 6: derived functors
- April 3: triangulated categories
- March 30

## Homework

Homework problems will be assigned daily. We will discuss solutions in class on Wednesdays, with presenters chosen by a roll of the dice. Students who do not wish to participate in presentations may submit solutions instead. Exercises in **bold** will be discussed in class.

- For Wednesday, April 29: We will discuss homotopy limits and colimits and stable infinity categories, summarizing HTT, §§1.2.8, 1.2.12, 1.2.13 and HA, §§1.1.1, 1.1.2, 1.3.1, 1.3.2.
- For Wednesday, April 22: Do Exercises
**6.2.2**, 6.2.4, 6.4.1,**6.4.2**,**6.6.2**,**6.6.6**. - For Monday, April 18: Read Weibel, §§6.5, 6.6.
- For Wednesday, April 15: Read Weibel, §§6.2, 6.4.

Do Exercises**6.1.6**,**6.2.3**. - For Monday, April 13: Read Weibel, §6.1.

For Friday, April 17: Do Weibel, Exercises**10.7.1**,**10.7.2**, 6.1.1,**6.1.2**,**6.1.3**. - For Friday, April 10: Read Weibel, §§10.7, 10.8.
- For Wednesday, April 8: Do Weibel, Exercise
**10.4.2**. - Friday, April 3: class will meet on Zoom in Meeting Room 283-728-921.

For Wednesday, April 8: Do Weibel, Exercises**10.1.2**,**10.2.1**,**10.2.2**, 10.2.4, 10.2.6, 10.4.5, 10.4.6. - Wednesday, April 1: class will meet on Zoom in Meeting Room 283-728-921. Read Weibel, §10.2. Do Weibel, Exercise 10.3.1 (using this outline).
- Monday, March 30: class will meet on Zoom in Meeting Room 283-728-921. We will continue discussing the derived category.
- Friday, March 20: class will meet on Zoom in Meeting Room 283-728-921. Read Weibel, §10.3 and §10.4. Do Exercise 10.4.2, 10.4.5.
- Wednesday, March 18: class will meet on Zoom in Meeting Room 283-728-921. We will discuss Exercises 5.6.2, 1.4.5, and the exercise from class on Monday: show that if \( \varphi : A^\bullet \to B^\bullet \) is a quasi-isomorphism of bounded below cochain complexes and \( \alpha : A^\bullet \to I^\bullet \) is a morphism, where \( I^\bullet \) is a bounded below cochain complex of injectives, then there is a morphism of complexes \( \beta : B^\bullet \to I^\bullet \) such that \( \beta \varphi \) is homotopy equivalent to \( \alpha \).
- Monday, March 16: class will meet on Zoom in Meeting Room 283-728-921. Read Weibel, §10.1. Do Exercise 1.4.5.
- Friday, March 13: class will meet on Zoom in Meeting Room 283-728-921. Please read Weibel, §5.8 (skip 5.8.6 unless you already know about sheaves; refer back to §5.7 for the definition of a Cartan–Eilenberg resolution, but you don't need to read the whole section). Do Exercise 5.6.2.
- Monday, March 9:

(for Friday, March 13) read Weibel, §5.7. (for Wednesday, March 18) do Weibel, Exercises 5.6.2, 5.6.3, 5.6.4. - Friday, March 6:

(for Monday, March 9) read Weibel, §5.6.

(for Wednesday, March 11) do Weibel, Exercise**5.6.1**. - Wednesday, March 4:

(for Friday, March 6) read Weibel, §§5.2, 5.4.

(for Wednesday, March 11) do Weibel, Exercises**3.5.1**,**5.2.3**, 5.4.2,**5.4.4**. Do Weibel, Exercises 3.1.1 again, this time using a spectral sequence. - Friday, February 28:

(for Monday, March 2) read Weibel, §§5.2, 5.4.

(for Wednesday, March 4)**prove that an \( R \)-module \( M \) is finitely presented if and only if \( \varinjlim \operatorname{Hom}(M, N_i) \to \operatorname{Hom}(M, \varinjlim N_i) \) for every filtered system of \( R \)-modules \( N_i \)**. Prove Lazard's theorem: an \( R \)-module is flat if and only if it is a filtered colimit of free \( R \)-modules.

- Wednesday, February 26:

(for Wednesday, March 4) do Weibel, Exercises**3.5.1**,**5.2.1**,**5.2.2**. - Monday, February 24:

(for Friday, February 28) read Weibel, §§2.6, 5.1.

(for Wednesday, March 4) Recall that if \( M \) is a sequence of abelian groups \( M(0) \leftarrow M(1) \leftarrow \cdots \), we defined \( \lim^{(1)} M \) as the set of equivalence classes of \( M \)-torsors. Show that \( \lim^{(1)} M = 0 \) if \( M \) is injective (Hint: use the Mittag-Leffler condition). Show that \( \lim^{(0)} = \lim \) and \( \lim^{(1)} \) are a universal \( \delta \)-functor in degrees \( \leq 1 \). Conclude that \( \mathrm R^1 \lim M = \lim^{(1)} M \).

do Weibel, Exercises 5.1.1, 5.1.2, 5.1.3.

- Friday, February 21:

(for Wednesday, February 26) Do Weibel, Exercise**3.5.3**.

**Prove that \( \hat{\mathbf Z}_p[p^{-1}] \simeq \varinjlim_m \varprojlim_n p^{-m} \mathbf Z / p^n \mathbf Z\simeq \varprojlim_n \varinjlim_n p^{-m} \mathbf Z / p^{-n} \mathbf Z \).**

Calculate \( \varprojlim \mathbf Q / p^n \mathbf Z \)*correctly*. I did this incorrectly in class (thanks to Shen and Sarah for correcting me!). The correct answer is that there is a split exact sequence: \( 0 \to \hat{\mathbf Q}_p \to \varprojlim (\mathbf Q / p^n \mathbf Z) \to \mathbf Q / \mathbf Z[p^{-1}] \to 0 \). Hint: Show that \( R^1 \lim \mathbf Q / \mathbf Z[p^{-1}] = 0 \).

- Wednesday, February 19:

Read Weibel, §5.1. Fill out the survey on topics for the latter half of the semester. - Monday, February 17:

(for Friday, February 21) read Weibel, §§2.6, 3.5.

(to discuss on Wednesday, February 26) Do Weibel, Exercises 2.6.4, 2.6.5,**3.5.1**,**3.5.2**,**3.5.5**. - Friday, February 14:

(for Monday, February 17) read Weibel, §3.4.

(to discuss on Wednesday, February 19) Prove that \( R^n \operatorname{Hom}(N,-)(M) \) is a universal \( \delta \)-functor of the variable \( N \) and conclude that \( R^n \operatorname{Hom}(N,-)(M) = R^n \operatorname{Hom}(-,M)(N) \) for all \( n \).

In class we defined \( \underline{\operatorname{Ext}}^1(M,N) \) to be the set of isomorphism classes of extensions of \( M \) by \( N \). Define \( \underline{\operatorname{Ext}}^0(M,N) = \operatorname{Hom}(M,N) \) and**show that \( \underline{\operatorname{Ext}}^\bullet(M,N) \) is an effaceable \( \delta \)-functor in degrees \( \leq 1 \). Conclude that \( \underline{\operatorname{Ext}}^1(M,N) = \operatorname{Ext}^1(M,N) \).**

- Wednesday, February 12:

(for Friday, February 14) read Weibel, §§3.2, 3.3.

(for Wednesday, February 19) do Weibel, Exercises 3.2.1,**3.2.2**, 3.2.3,**3.2.4**(prove more generally that in an abelian category with enough injectives, \( A \to B \to C \) is exact if and only if \( \operatorname{Hom}(C,I) \to \operatorname{Hom}(B,I) \to \operatorname{Hom}(A,I) \) is exact for all injectives \( I \) ),**3.3.1**,**3.3.2**

- Friday, February 7:

(for Monday, February 10) read Weibel, §§2.7, 3.1 (you may want to read §3.1 first; it is a little more concrete); we will get the results of §2.7 a different way in class on Monday.

(to discuss on Wednesday, February 12) Weibel, Exercises**2.5.1**,**2.5.2**(careful: these exercises are not equivalent by reversing arrows, at least not trivially),**3.1.1**,**3.1.2**,**3.1.3**

- Wednesday, February 5:

(for Friday, February 7) read Weibel, §2.6, up to page 53 (stop before Application 2.6.5). - Monday, February 3:

(for Friday, February 7) read Weibel, §§2.1, 2.4, 2.5.

(to discuss on Wednesday, February 12) Weibel, Exercises 2.1.2, 2.4.2,**2.4.3**, 2.4.5. - Friday, January 31:

(for Monday, February 3): read Weibel, §§2.1, 2.4.

(to discuss on Wednesday, February 5) Weibel, Exercises**1.4.5**, 2.3.1. Prove that**an abelian group is injective if and only if it is divisible**and that**an abelian group is projective if and only if it is free**.

- Wednesday, January 29:

(to discuss on Wednesday, February 5) Weibel, Exercises 2.2.1,**2.3.2**. - Monday, January 27:

(for Friday, January 31) Read Weibel, §2.2–2.3

(to discuss on Wednesday, February 5) Weibel, Exercises 1.5.3, 1.5.6,**1.5.8**.

- Friday, January 24 (problems to be discussed on Wednesday, January 29): Read Weibel, §1.5. Let \( \operatorname{Hom}_\bullet(A_\bullet, B_\bullet) \) be defined as in class. Show that it is a complex and that \( Z_0 \operatorname{Hom}_\bullet(A_\bullet, B_\bullet) = \operatorname{Hom}(A_\bullet, B_\bullet) \). Do exercises
**1.4.1**, 1.4.2, 1.4.3,**1.4.4**,**1.4.5**. - Wednesday, January 22 (problems to be discussed on Wednesday, January 29): Read Weibel, §1.4. Do exercises 1.2.5,
**1.3.1**, 1.3.2,**1.3.5**. - Friday, January 17 (to be discussed on Wednesday, January 22): Read Weibel, §1.3. Do Weibel, Ex. 1.2.7,
**1.2.8**, 1.3.1. Please let me know before class if you do not want to participate in the presentations in class on Wednesday. - Wednesday, January 15 (to be discussed on Wednesday, January 22): Read Weibel, §1.1 – 1.2.
**Complete the construction of the abelian group structure on \( \operatorname{Hom}_{\scr A}(X,Y) \) when \( X \) and \( Y \) are objects of an abelian category \( \scr A \)**: verify that the addition law is associative and commutative with \( 0 \) as identity and that composition is bi-additive. Do Weibel, Ex. 1.1.1, 1.1.5,**1.1.6**, 1.1.7. - Monday, January 13 (to be discussed on Wednesday, January 22): Read §A.4 through Example A.4.4 (pp. 424 — 426). Verify that, for any not-necessarily-commutative ring \( R \), the category of left \( R \)-modules is an abelian category. You may use the definition of an abelian category given in Weibel or the definition given in class (but please be clear about which one you are using!).

## Office hours

## Grading

To get an A in this course, you should participate actively in class, and present homework solutions when called upon to do so (or submit written homework solutions).