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November 17 (Sat), 2018 13:35--14:35, 16:45--17:45 Graduate School of Mathematical Sciences The University of Tokyo, Tokyo, Japan |
Abstract
The homological conjectures, which date back to Peskine, Szpiro and Hochster
at the end of the 60fs, make fundamental predictions about syzygies and
intersection problems in commutative algebra. They were established long ago
in the presence of a base field, and led to tight closure theory, a powerful
tool to investigate singularities in characteristic $p$.
Recently, perfectoid techniques coming from $p$-adic Hodge theory have allowed
to get rid of any base field. We shall report on our proof of the direct summand
conjecture and the existence and weak functoriality of big Cohen-Macaulay algebras,
which solve the homological conjectures in general.
This also opens the way to the study of singularities in mixed characteristic,
and we shall outline ongoing work by Ma and Schwede which shows how such a study
can even build a bridge between singularity theory in char. $p$ and in char. $0$.