Math 250B Syllabus
| Week of | Mon | Wed | Fri | Topics | Reading |
| 1/18 | Holiday | Categories. Yoneda's theorem. Universal solutions of equations. Ideals, prime and maximal |
See Lang, section 7 of Chapter I, and perhaps some of MacLane's Categories for the Working Mathematician | ||
| 1/25 | Categorical limits and adjoint functors. Free objects. Localization. The spectrum of a ring as a topological space. Linear algebra over rings. |
See Chapter IX of MacLane, and Chapter I.2 and Appendix V of Eisenbud, | |||
| 2/01 | Homomorphisms and exact sequences. Abelian categories. | See Maclane Chapter VIII. | |||
| 2/08 | Tensor products. Flat, projective, and injective modules. Finite type and finite presentation. Lazard's theorem. | See Chapter I.2 and I.6 of Eisenbud, and Appendix VI of Eisenbud | |||
| 2/15 | Holiday | Injective resolutions. Ext and Tor. Nakayama's lemma. |
See section 4.1 of Eisenbud, and Exercise 6.2 p 172 of Eisenbud. | ||
| 2/22 | Noetherian rings and modules.Hilbert basis theorem, Associated primes. | See Chapter X of Lang | |||
| 3/01 | Midterm. Primary decomposition. More on flatness, including the local criterion. | See the notes on the local criterion. | |||
| 3/08 | Tensor products of rings. Lifting prime ideals: Finite extensions of rings, and flat extensions of rings. | ||||
| 3/15 | Noether normalization and the Nullstellensatz. Openness of the spectral image of a flat and finite type morphism (Noetherian case). | See the notes on constructible, open, and closed sets | |||
| 3/22 | Spring break | ||||
| 3/29 | Filtrations, gradations, completions. The Artin-Rees lemma. Flatness, including flatness of completions. | ||||
| 4/05 | Dimension of local rings. Krull dimension, Hilbert polynomials, | See Lang X.5, X.6 and Eisenbud Chapters 8, 9, and 12 | |||
| 4/12 | Principal ideal theorem, Catenary rings, Regular local rings. | ||||
| 4/19 | Depth and Cohen Macalay rings. Homological methods. | See Eisenbud Chapter 18, and Serre Chapter IV | |||
| 4/26 | Depth and projective dimension. Regular local rings II. Koszul complexes. | ||||
| 5/03 | Reading Week |
The above schedule is just a rough guide and subject to change as the course progresses.