Given a smooth proper scheme over the ring of integers of an algebraically closed $p$-adic field, we recently showed that the mod $p$ Betti numbers of the generic fibre are a lower bound for the de Rham Betti numbers of the special fibre. The main innovation was the construction of an integral $p$-adic cohomology theory that gets rid of certain factorials occurring as denominators in crystalline cohomology.

In my talk, I'll recall this story from the point of view of $p$-adic Hodge theory, and then explain a different perspective on this integral cohomology theory: it arises as a suitable graded piece of (an enrichment of) the Hochschild homology of the scheme relative to the sphere spectrum. In particular, I shall try to explain why working over the sphere spectrum instead of the integers is responsible for getting rid of the aforementioned denominators. (Everything reported is joint work with Matthew Morrow and Peter Scholze.)

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