*Note Time*

Online Talk

The "crystalline miracle" states that the de Rham cohomology of a smooth p-adic formal scheme X is functorially determined by its mod p reduction X_{p=0}: namely, it is isomorphic to the crystalline cohomology of X_{p=0}. Now, more refined cohomology theories of p-adic formal schemes are intensely studied: for example, prismatic cohomology, syntomic cohomology, and topological cyclic homology. Jointly with Jeremy Hahn and Ishan Levy, we ask: how much of these invariants does the mod p^n reduction X_{p^n = 0} of a p-adic formal scheme X see? We find that, as n increases, we can recover more and more of the v_1-adic tower of the mod p reduction of these invariants, where v_1 is a certain class coming from chromatic homotopy theory.

As an application, we are able to give a complete and explicit computation of the mod (p,v_1) algebraic K-theory of Z/p^n. Another application states that given a p-adic formal scheme X smooth of dimension d over Z_p, the mod p etale cohomology of the generic fiber may be recovered (as a Galois representation) from X_{p^{b(d)} = 0}, where b(d) is a certain function of d. For example, if d \leq p-2 then X_{p^2 = 0} is enough, while if d \leq p^2-p-1 then X_{p^3 = 0} is enough.