Homological conjectures and singularities in mixed characteristic

Ponente(s): Linquan Ma .

The homological conjectures have been a focus of research in commutative algebra since the 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 in 2016 when Andre proved the direct summand conjecture and the existence of big Cohen--Macaulay algebras, which lie in the heart of these conjectures. The main new ingredient is to use the theory of perfectoid algebras and spaces, and this leads to the further development of a mixed characteristic singularity theory. In this talk, we will give a survey on these results and methods, and mention some applications to birational geometry in mixed characteristic.