The direct summand conjecture and its derived variant

In the late 60s, Hochster conjectured that every regular ring is a direct summand, as a module, of any finite extension. Soon thereafter, Hochster himself proved his conjecture in equicharacteristic, and these ideas were crucial to the conception of the modern theory of F-singularities. In the mixed characteristic setting, Hochster’s conjecture was settled very recently by Yves Andre using perfectoid geometry. In my talk, I’ll discuss a proof of Hochster’s conjecture, and explain why related ideas also help establish a derived variant of the conjecture put forth by de Jong. One of my main goals in this talk 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.