  The homological conjectures, which date back to Peskine, Szpiro and Hochster
in the late sixties, make fundamental predictions about syzygies and
intersection problems in commutative algebra. They were settled 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 us to get rid of any base field;
this solves the direct summand conjecture and establishes the existence and
weak functoriality of big Cohen-Macaulay algebras, which solve in turn the
homological conjectures in general. This also opens the way to the study of
singularities in mixed characteristic. We sketch a broad outline of this story,
taking lastly a glimpse at ongoing work by L. Ma and K. Schwede, which shows
how such a study could build a bridge between singularity theory in char. p and
in char. 0.
