Math 206A - Topics in Algebraic Geometry

Course description: This is a graduate topics course on the topic of homological conjectures in commutative algebra, many of which have been resolved very recently using tools from p-adic Hodge theory (perfectoid rings and spaces). Our primary goal will be to understand the proof of the direct summand conjecture: if R is a regular local ring, then any finite injective morphism R -> S of rings splits in the category of R-modules. This is trivial in the case where R is of equal characteristic 0; the general conjecture is due to Hochster, who proved the case of equal characteristic p>0 in 1973. The remaining case, in which R is of mixed characteristic, was settled by Yves André in 2016; the proof was then simplified by Bhargav Bhatt, who also established the splitting in the derived category of R-modules (the derived direct summand conjecture). We will follow the approach of Bhatt, supplemented as needed with background lectures on commutative algebra and/or perfectoid rings. As time permits, we will also discuss some related results.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours: by appointment.

Lectures: MWF 12-1, in APM 7421.

Textbook: No required text. Readings will be from a variety of sources, including those listed below. In addition to the links below, these files can also be found in the CoCalc (formerly SageMathCloud) project associated to this course; email me for access.

Prerequisites: Math 200C or equivalent (with instructor's permission). Familiary with algebraic geometry at the level of the Math 203 series is also strongly recommended. In addition, it will be helpful at certain points to have attended the reading seminar on perfectoid spaces during the preceding two quarters and/or the 2017 Arizona Winter School, but we will summarize the key points as they arise.

Homework: None. However, students attending the course will be expected to give some lectures and/or help with the assembly of lecture notes. This is currently taking place on CoCalc; email me for access.

Final exam: None.

Grading: Let me know if you need a meaningful grade.


Topics: (see also the running lecture notes)