Professor: Mark Haiman
Office: 855 Evans
Spring Office hours: WF 11:30-12:30 or by appointment
Phone: (510) 642-4318
Prerequisites: Math 250A or equivalent; knowledge of some material from Math 250B also helpful. The fall semester (256A) is required for the spring semester (256B).
General: This course is a two-semester introduction to the foundations of algebraic geometry in the language of schemes, along with techniques, examples and applications.
The theory of schemes was developed by Grothendieck and his collaborators in the 1960's. The basics are set forth in Grothedieck's treatise Éléments de géométrie algébrique (EGA), with more advanced material presented in the various volumes of the Séminaire de Géométrie Algébrique (SGA).
In many ways EGA and parts of SGA still provide the best exposition of the subject, if we update the treatment of sheaf cohomology (EGA doesn't use derived categories) and add something about Grassman schemes (omitted in EGA). A problem for English speaking students is that EGA and SGA are only available in the original French. My plan is to use EGA as the course text by providing synopses in English of the sections we cover, stating theorems, definitions and sometimes outlines of proofs.
In the lectures I will discuss proofs of the most important theorems, and attempt to illustrate and motivate the concepts with examples, something that EGA does not do.
EGA: A. Grothendieck, Éléments de géométrie algébrique, I-IV, Publications Mathématiques de l'IHES, vols. 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967).
It seems no longer possible to obtain EGA in print, but it is available electronically in the collections of IHES Volumes hosted by Springer or NUMDAM.
SGA: All volumes except SGA 2 were published in the Springer Lecture Notes in Mathematics series. They are available electronically through the UC libraries. The easiest way to access them is from the list of references on Wikipedia. The entries there for the Springer volumes contain links with labels like "doi:10.1007/BFb0058656," which you click on the numeric string after "doi:". There are also entries with math arXiv links for typeset versions of SGA 1 and SGA 2.
Other references: You may find some of the following English language references helpful.
Homework and grading: Problems sets will be posted here from time to time. I will assign more problems than I actuallly expect you to do. Grading will be based on your doing a reasonable fraction of them, although you are of course welcome to try to do them all if you feel ambitious. Extra difficult problems are marked with an asterisk.
|Jan 22-31||EGA I, 2.5, 3.1-3.3, 5.1-5.2||Problem set 6||Feb 14|
|Feb 3-14||EGA I, 3.4-3.6, 4.3-4.5, 5.3-5.5; EGA IV, 1.1-1.2|
|Feb 19-28||EGA I, 1.6, 9.1-9.6; EGA II, 1.1-1.7||Problem set 7||Apr 25|
|Mar 3-April 9||EGA II, 2.1-2.9|
|April 11-23||EGA II, 3-5||Problem set 8||May 19|
|April 25-May 2||Notes on sheaf cohomology|
|Aug 30-Sept 9||Examples in class||Problem set 1 (9/20: corrected typos in 1 and 2)||Oct 4|
|Sept 11-23||Mumford "Red Book" I.2-I.5||Problem set 2||Oct 18|
|Sept 25-Oct 16||EGA 0, 1-3.4; EGA I, 1.1-1.3||Problem set 3||Nov 1|
|Oct 18-Nov 1||EGA 0, 3.5 & 4.1; EGA I, 1.7-1.8 & 2.1-2.4||Problem set 4||Nov 27|
|Nov 4-27||EGA I, 6.1-6.4 & 6.6; EGA IV, 10.1-10.4||Problem set 5||Dec 19|
|Dec 2-Dec 6||EGA I, 1.4 & 4.1-4.2|
Notes on Sheaf Cohomology
Syllabus: I've divided the topics below into foundations and applications. There are more topics on this syllabus, especially toward the end, than we are likely to cover.
Foundations, Part I
Foundations, Part II
Foundations, Part III