Back to Algebraic Geometry Seminar: Spring 2019
Algebraic Geometry Seminar
Date: Tuesday, Mar 5, 2019
Time: 3:30PM - 4:30PM
Location: LCB 222
Linquan Ma
Purdue University
Title |
Homological conjectures, perfectoid spaces, and singularities in mixed characteristic |
Abstract |
The homological conjectures have been a focus of research in commutative algebra since 1960s. They concern a number of interrelated conjectures concerning homological properties of commutative rings to their internal ring structures. These conjectures had largely been resolved for rings that contain a field, but several remained open in mixed characteristic–until recently Yves Andre proved Hochster’s direct summand conjecture and the existence of big Cohen-Macaulay algebras, which lie in the heart of these homological conjectures. The main new ingredient in the solution is to systematically use the theory of perfectoid spaces, which leads to further developments in the study of mixed characteristic singularities. For example, using integral perfectoid big Cohen-Macaulay algebras, one can define the mixed characteristic analog of rational/F-rational and log terminal/F-regular singularities, and they have applications to study singularities when the characteristic varies (based on recent joint work with Karl Schwede). In this talk, we will give a survey on these results and methods. |
Algebraic Geometry