# Math 6290 (Spring 2020):

Homological algebra

## Textbook

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

## 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.

- 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).