# 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.

• M. Hochster, Contracted ideals from integral extensions of regular rings: published version. This paper introduces the DSC and proves the equal-characteristic case.
• M. Hochster, Topics in the Homological Theory of Modules over Commutative Rings: not available for download. This short book is a series of lectures on the homological conjectures.
• B. Bhatt, On the direct summand conjecture and its derived variant: arXiv version. This will be one of our primary sources for the proof of DSC in the mixed-characteristic case. For the equal-characteristc case of the derived DSC, see B. Bhatt, Derived splinters in positive characteristic, published version.
• B. Bhatt, Almost direct summands, published version. This paper proves a special case of DSC using the original almost purity theorem of Faltings, providing a simple illustration of the use of almost commutative algebra in this setting.
• R. Heitmann and L. Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic: arXiv version. This gives an even more elementary proof of DSC than the one of Bhatt; it will be another one of our primary sources.
• C. Huneke and G. Lyubeznik, Absolute integral closure in positive characteristic: published version. This paper gives a short proof of the Hochster-Huneke theorem that for a complete excellent local ring of positive characteristic, the absolute integral closure is a big CM algebra.
• B. Bhatt, A. Caraiani, K.S. Kedlaya, J. Weinstein, Arizona Winter School 2017 lecture notes on perfectoid spaces: Arizona Winter School lecture notes. This is a recommended entry point into the literature on perfectoid rings and spaces. See also K.S. Kedlaya, New methods for (phi, Gamma)-modules, published version and errata.
• O. Gabber and L. Ramero, Almost Ring Theory: free download from UCSD. This is the standard reference for almost ring theory; we will only use material from Chapter 2. See also M. Olsson, On Faltings' method of almost étale extensions: published version; J. Tate, p-divisible groups, published version.
• D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, free download from UCSD. This will be our fallback reference for standard commutative algebra, particularly the Gorenstein and Cohen-Macaulay properties.

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.

Announcements:

• First lecture: Monday, April 3.
• The guest lecture on Friday, April 21 by Linquan Ma will double as an algebraic geometry seminar, and will be held in APM 6402.
• Holidays this term: Monday, May 29.

• Apr 3: overview (Hochster "Contracted ideals", through Example 1). For the relationship between pure morphisms and descent, see the Stacks Project.
• Apr 5: zero-dimensional Gorenstein rings (Eisenbud 21.1, 21.2); proof of DSC in equal positive characteristic (Hochster "Contracted ideals", Theorem 1, Theorem 2).
• Apr 7: history of the homological conjectures (guest lecture by Paul Roberts; Hochster "Topics").
• Apr 10: almost commutative algebra (Gabber-Ramero, Chapter 2).
• Apr 12: almost splittings (Bhatt, "Almost direct summands")
• Apr 14: redux of prior lectures; almost purity and almost splittings (see notes).
• Apr 17: the construction of André (Bhatt, "On the DSC...", section 2).
• Apr 19: Cohen-Macaulay algebras.
• Apr 21: homological conjectures and big Cohen-Macaulay algebras (guest lecture by Linquan Ma, in APM 6402).
• Apr 24: more on Andr&eactue;'s construction; almost purity revisited.
• Apr 26: inverse limit; Bhatt's theorem and its application to DSC (Bhatt, "On the DSC...", sections 4-5).
• Apr 28: proof of Bhatt's theorem (Bhatt, "On the DSC...", section 4).
• May 1: local cohomology (Eisenbud, appendix 4).
• May 3: local cohomology (KK away; guest lecture by Paul Roberts; Huneke-Lyubeznik).
• May 5: perfectoid fields and tilting (KK away; lecture by Zonglin Jiang).
• May 8: perfectoid rings and tilting (KK away; lecture by Jake Postema).
• May 10: overview of adic spaces (KK away; lecture by Iacopo Brivio).
• May 12: almost purity (KK away; lecture by Xin Tong).
• May 15: derived splinters (Bhatt, "Derived splinters").
• May 17: derived splinters continued (Bhatt, "Derived splinters", section 4).
• May 19: derived direct summand conjecture (Bhatt, "On the DSC...", section 6).
• May 22: Heitmann-Ma, section 2, lemma 3.2.
• May 24: Heitmann-Ma, lemma 3.4.
• May 26: Heitmann-Ma, lemma 3.5.
• May 29: NO LECTURE (Memorial Day).
• May 31: Absolute and relative Fargues-Fontaine curves (Kedlaya, AWS 3.1; lecture by Zonglin Jiang).
• Jun 2: Vector bundles on Riemann surfaces and slope formalism (Kedlaya, AWS 3.2-3.3; lecture by Daniel Smith).
• Jun 5: Harder-Narasimhan filtrations (Kedlaya, AWS 3.4; lecture by Peter Wear).
• Jun 7: Additional examples of slope formalism, slopes over a point (Kedlaya, AWS 3.5-3.6.12; lecture by Xin Tong).
• Jun 9: Slopes over a point, slopes in families (Kedlaya, 3.6.13-3.7; lecture by Jake Postema).