## 18.705 (Fall 2014): COMMUTATIVE ALGEBRA

Please see the Course Info fordetails about the course (MUST READ!).

The following assignments refer to Lecture Notes written by AllenAltman and Steven Kleiman. These notes are available here in threeformats: the first implements hyperreferencing in xdvi; the second, inacroread; and the third saves trees when printed.

LECTURE NOTES (current 13/09/01): 13Ed.dvi and 13Ed.pdf and 13Ed-2up.pdf.

The notes are also available at the URL

in digital and print forms. The digital form is free of charge; the print form is a perfect-bound paperback, and costs \$19.95.

Caution, the notes are in beta form. Comments, however trivial, arealways welcome. Please email them to the instructor(liuyf@math.mit.edu) and to Prof. Kleiman(Kleiman@math.mit.edu).

Assigned problems are indicated below, after the date they're due.The numbers between parentheses refer to subsections, which areexercises, in the Lecture Notes.

SYLLABUS (Temporary):

Final Exam: 9:00--12:00 Dec 16, 2014 (Tuesday), at E17--122

1. R.09/04 --- Rings and ideals --- Write up for 09/15: (1.7), (1.10), (1.14), (1.15), (1.17).
2. T.09/09 --- Prime ideals --- Write up for 09/15: (2.11), (2.16), (2.18), (2.22), (2.23).
3. R.09/11 --- Radicals --- Write up for 09/15: (3.9), (3.13), (3.17), (3.18), (3.32), (3.39).
4. T.09/16 --- Modules --- Write up for 09/22: (4.12), (4.17), (4.18), (4.19), (4.20).
5. R.09/18 --- Homological algebra I --- Write up for 09/22: (5.11), (5.15), (5.16), (5.24), (5.26), (5.29).
6. T.09/23 --- Homological algebra II --- Write up for 09/29: (6.3), (6.9), (6.16), (6.17).
7. T.09/25 --- Homological algebra III --- Write up for 09/29: (7.2), (7.9), (7.15), (7.17), (7.20).
8. T.10/01 --- Tensor products --- Write up for 10/06: (8.7), (8.9), (8.16), (8.24), (8.25), (8.26).
9. R.10/03 --- Flatness --- Write up for 10/06: (9.4), (9.15), (9.17), (9.18), (9.25), (9.28).
10. T.10/07 --- Finitely generated modules --- Write up for 10/14: (10.6), (10.8), (10.16), (10.22), (10.31), (10.35).
11. R.10/09 --- Localization of rings --- Write up for 10/14: (11.4), (11.7), (11.10), (11.18), (11.24), (11.32).
12. T.10/14 --- Localization of modules --- Write up for 10/20: (12.6), (12.8), (12.14), (12.19), (12.28).
13. R.10/16 --- Support --- Write up for 10/20: (13.3), (13.7), (13.10), (13.18), (13.25), (13.40), (13.41).
14. T.10/21 --- Integral extensions --- Write up for 11/03: (14.4), (14.12), (14.14), (14.15), (14.17).
15. R.10/23 --- Midterm exam, in class (covering through 10/14)
16. T.10/28 --- Noether normalization --- Write up for 11/03: (15.2), (15.3), (15.8), (15.11), (15.12).
17. R.10/30 --- Noetherian rings --- Write up for 11/03: (16.9), (16.18), (16.20), (16.29), (16.30).
18. T.11/04 --- Associated primes --- Write up for 11/17: (17.6), (17.7), (17.11), (17.16), (17.22), (17.26).
19. R.11/06 --- Primary decomposition --- Write up for 11/17: (18.7), (18.8), (18.17), (18.18), (18.22), (18.26).
20. T.11/11 --- Veterans Day, no class
21. R.11/13 --- No class
22. T.11/18 --- Length --- Write up for 11/24: (19.4), (19.5), (19.10), (19.12), (19.13), (19.16).
23. R.11/20 --- Hilbert functions --- Write up for 11/24: (20.5), (20.6), (20.9), (20.10), (20.19).
24. T.11/25 --- Dimension --- Write up for 12/05: (21.6), (21.9), (21.12), (21.15), (21.18), (21.19).
25. R.11/27 --- Thanksgiving, no class
26. T.12/02 --- Completion --- Write up for 12/05: (22.4), (22.6), (22.11), (22.14), (22.19), (22.21).
27. R.12/04 --- Discrete valuation rings --- Write up, optional: (23.6), (23.9), (23.12), (23.13), (23.17), (23.22).
28. T.12/09 --- Dedekind domains --- Write up, optional: (24.5), (24.6), (24.8), (24.12), (24.13).