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The homological conjectures, which date back to Peskine, Szpiro and Hochsterin the late sixties, make fundamental predictions about syzygies andintersection problems in commutative algebra. They were settled long ago in thepresence of a base field and led to tight closure theory, a powerful tool toinvestigate singularities in characteristic p. Recently, perfectoid techniquescoming 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 andweak functoriality of big Cohen-Macaulay algebras, which solve in turn thehomological conjectures in general. This also opens the way to the study ofsingularities 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 showshow such a study could build a bridge between singularity theory in char. p andin char. 0.

Singularities in mixed characteristic. The perfectoid approach