|Date: Wednesday, October 26, 2016
Location: 4096 East Hall (4:10 PM to 5:30 PM)
Title: The direct summand conjecture and its derived variant I
Abstract: In the late 60's, Hochster formulated the direct summand conjecture (DSC) in commutative algebra: a regular commutative ring splits off from any finite extension (as a module). A few years later, Hochster himself proved the DSC when the ring contains a field; these ideas had a significant impact on the ensuing development of tight closure and F-singularity theory.
In the mixed characteristic setting, the case of dimension <= 3 was settled by Heitmann in the 90's. The general case remained open until extremely recently (September 2016), when it was resolved beautifully by Andre using the theory of perfectoid spaces.
In these two talks, I'll discuss a proof of the DSC that is related to, but simplifies, Andre's proof. I will also explain why the ideas going into the new proof help establish a derived variant of the DSC put forth by de Jong; the latter roughly states that regular rings have rational singularities. One of my main goals in these talks is to explain why passing from a mixed characteristic ring to a perfectoid extension is a useable analog of the passage to the perfection (direct limit over Frobenius) in characteristic p.
The relevant background from non-archimedean and perfectoid geometry will be reviewed.
Speaker: Bhargav Bhatt