Course title: Homological algebra (2 credits)
Course goals: Develop basic tools from homological algebra as used in algebra and algebraic geometry

The class will meet weekly for 80 minutes; 50 minutes will be a presentation by one of the students on the topic of the week, and 30 minutes will be a discussion of homework problems. Each student will be required to present at least twice in class. Prior to each presentation, the student must submit a written outline of their talk. Outlines and presentations will be graded on completeness, correctness and accuracy, and appropriateness for the audience.

Assessment overview: Completion of homework and participation in homework discussion (10%), presentation outlines (30%), presentations (60%).

Time/Location: Thursdays from 3:00-4:30pm in K9509

Course texts are:
[E] Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
[H] Hartshore, Algebraic Geometry

Prerequisites are:
** Math 817 and 818 recommended
** Instructor approval


Outline:
Jan 9 (Nathan): Introduction.  Free modules. Finitely generated modules over PIDS. (Graded) free resolutions. Hilbert functions and Hilbert polynomials. Hilbert's syzygy theorem. [E A3.1]. Complexes. Chain complexes for simplicial complexes. de Rham complex.

Jan 16 (Sharon): [E A3.2,A3.3] Projective modules. Free and projective resolutions. Minimal resolutions: [E 19.1] 473-474, 476 and Lemma 19.4, [E 20.1] up to statement of Theorem 20.2 (proof delayed until later).
Jan 23 (Negarin):[E A3.4] Injective modules and resolutions. [E A3.5] (basically review). [E A3.6] Maps of complexes up through the proof of Proposition A3.12.
Homework: Try several of exercises from [E A3.4.1].

Jan 30 (Chi Ki) Rest of [E A3.6]. Theorem 20.2 and its proof. [E A3.7]
Homework: Do exercises A3.8 and A3.9. Prove Proposition A3.15.

Feb 6 (Karolyn) Rest of [E A3.8]. Reminder of Hom and Tensor product ([E 2.2] up through bottom of pg 65). Derived functors [E A3.9].
Homework: Exercise 2.4, A3.15

Feb 13 (Pijush) Tor [E 3.10], and Exercise A3.18. Ext [E3.11]. For Theorem A3.18, recall global and projective dimension from pg 474. Exercise A3.26.
Homework: Exercises  A3.16, A3.17, A3.23, A3.24, A3.25
                               
Feb 27 (Negarin) [E A3.12] Mapping cones and double complexes. Regular sequences (pg 423). Koszul complexes: [E 17.2] up through proof of Corollary 17.5, [E 17.3] up through proof of Theorem 17.4.
Homework: Try several exercises from [E A3.12.1]


For the following four weeks, do some selection of exercise from A3.13.6

Mar 6 (Chi Ki) Spectral Sequences [E A3.13]. Mapping cones revisited [E A3.13.1]. Exact Couples [E A3.13.2]. 

Mar 13 (Karolyn) [E A3.13.3] Filtered Modules.

Mar 20 (Pijush) [E A3.13.4] Double complexes. Balanced Tor.

Mar 27 (Negarin and Karolyn) [E Corollaries 19.5-19.8] Proof of Hilbert's Syzygy theorem. [E A3.13.5] Exact sequence of terms of low degree. Grothendieck spectral sequence [Exercise A3.50a,b,c].

Apr 3 (Chi Ki and Pijush) Sheaf cohomology [H pg 206-208]. Higher direct images [H III.8 through Proposition 8.1]. Leray spectral sequence [Exercise A3.50d]. Local and global Ext [Start of H III.6]. Local to global Ext sequence (see e.g. Wikipedia). 

Read A3.14 on your own if you wish.