Workshop

We will explain the the definition of a perfectoid field and the tilting construction. We will then state the Fontaine-Wintenberger theorem and prove it in the algebraically closed case.

This talk will describe the basic concepts of Almost Mathematics, give some background of the connections between Almost Mathematics and the Homological Conjectures, and present a proof of the Almost Purity Theorem in positive characteristic.

The talk concerns morphisms between perfect complexes over commutative noetherian rings.  The central result is a criterion for the tensor-nilpotence of such morphisms, in terms of numerical invariants of complexes known as levels. The proof uses the existence of big Cohen-Macaulay modules. Applications to local rings include a strengthening of the Improved New Intersection Theorem, and short direct proofs of several results equivalent to it.  The results come from recent joint work with Iyengar and Neeman; see https://arxiv.org/abs/1711.04052

Inspired by the solution of the direct summand conjecture, we introduce perfectoid multiplier/test ideals in mixed characteristic. As an application, we prove the uniform bound on the growth of symbolic powers in regular local rings of mixed characteristic analogous to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal characteristic. This is joint work with Karl Schwede

Suppose that R is a local ring of mixed characteristic. Using recent breakthrough results of André on the existence of big Cohen-Macaulay algebras, we defined a mixed characteristic analog of the multiplier ideal / test ideal and show it satisfies many of the same formal properties as its equal characteristic brethren. Using the same ideas, we show that if R is mixed characteristic and local and R/pR has F-rational or F-regular singularities, then R itself has analgous singularities in mixed characteristic. This is joint work with Linquan Ma