MATH 607
Homological Algebra, Fall 2022

Lectures:

TR 12-1:20, 210 University Hall

Office hours:

Wed. 2-2:50 UNIV 208; Thur. 1:30-2:20 UNIV 301, or by appointment

My office:

207 University, phone 3465635, e-mail apolish@uoregon.edu

Textbooks:

We will use the following:

  • "Introduction to homological algebra" by Weibel
  • "Sheaf theory" by Tennison

    • You are also welcome to read the beginning of Grothendieck's paper on homological algebra

    Pre-requisites:

    Math 647-649

    Minimal plan:

    Weibel, ch. 1-3, selected stuff from Tennison

    Homework

    There will be no exams in this course, so the grade will be based solely on your homework scores. Homework should be turned in by email, as pdf file, on the day when it's due. You are encouraged to collaborate on homeworks, however, writing up the solutions should be an individual work.

    Fall homework assignments:

    Abbreviations: "W" is Weibel, "T" is Tennison
    The problems marked as "WU" have to be written up and submitted for grading

    Assignment #1 (due October 6):

  • Exercise (WU): Prove that for any morphisms A-> B-> C in an abelian category, if the induced morphism coker(A->B)->coker(A->C) is surjective (epi) then the morphism B->C is surjective. You can use axioms of abelian category and their corollaries established in class.
  • Exercises from W: 1.1.3, 1.2.5 (WU, assume A is the category of R-modules), 1.2.6 (WU), 1.3.1, 1.3.3 (5-lemma), 1.3.5 (WU), 1.3.6 (WU), 1.4.5, 1.5.2 (WU), 1.5.6.

    • Assignment #2 (due October 20):

  • Exercises from W: 1.5.10, 2.1.1, 2.2.1 (WU), 2.2.2, 2.2.3, 2.3.3, 2.3.5, 2.4.2 (WU), 2.4.3 (WU), 3.1.2 (WU), 3.1.3 (WU). 3.2.2 (WU).

    • Assignment #3 (due November 10):

  • Exercises from T: Exercises to ch.3 (p.68): #2 (WU), #3c,d, #4 (WU), #6, #10 (WU). Additional exercise (WU): (a) Let X be the topological space consisting of two points p and u, such that the only open subsets are {u}, X and the empty set. Prove that every presheaf of sets on X is a sheaf. Describe projective objects in the category of sheaves of k-vector spaces over X. Describe explicitly the pull-back and push-forward functors between categories of sheaves of k-vector spaces for the embeddings p --> X and u --> X. (b) Now let X be the topological spaces consisting of three points p_1,p_2 and u, such that all nonempty proper open subsets are {u}, {p_1,u} and {p_2,u}. Give an example of a presheaf over X which is not a sheaf. Describe explicitly the categories of presheaves and sheaves of sets over X and the functor of sheafification.