Math 685: Directed Studies

Course description: Offered to enable students to undertake and complete, with credit, limited investigations not within their thesis research and not covered by any other courses in the curriculum.

Spring and Summer 2017 topic:
A Course in Homological Algebra, Hilton and Stammbach

SUGGESTED EXERCISES:

Chapter I Modules
2.1, 2.2
3.7
4.1, 4.2
5.4

Chapter II Categories and Functors
1.1, 1.2, 1.4
2.1, 2.2, 2.5 (and possibly 2.6)

Chapter III Extensions of Modules
1.1, 1.2, 1.5, 1.6
2.4, 2.5 (this is the Baer sum; it might help to look up on the web other ways to view it), 2.6, 2.8
3.3
4.1, 4.4, 4.6
8.1, 8.3

Chapter IV Derived Functors
1.1, 1.2
2.2
3.1
4.1, 4.2 (Hint: If φ is a homotopy equivalence, show directly that φ induces an isomorphism on homology. For the converse, apply Theorem IV.2.1 to obtain a long exact sequence relating the homology of the three complexes. Show that the connecting homomorphism is precisely the map induced by φ. It follows that φ induces an isomorphism on homology if and only if E(φ) is exact. By exercise 4.1, if E(φ) is exact, then 1 is homotopic to 0. Writing a homotopy as a 2x2 matrix, it follows that φ is a homotopy equivalence.)

5.7 (For some examples, recall Chapter III, exercises 4.1 and 4.4. Find Phom of some pairs of Z-modules.)

7.1, 7.2, 7.4 (For some examples, recall Chapter III, exercises 4.1 and 4.4. Find the first few terms of the long exact sequence in this exercise for some examples of Z-modules.)