For a p-adic ring R, Bhatt--Morrow--Scholze define p-complete complexes Z_p(i)(R), either in terms of the homotopy sheaves of p-adic algebraic K-theory or equivalently as filtered Frobenius eigenspaces of prismatic cohomology (an integral refinement of earlier constructions of Fontaine-Messing and Kato).

I will discuss a description of the Z_p(i) on a class of p-adic rings in terms of p-adic vanishing cycles, recovering constructions of Geisser--Schneider--Sato. On the one hand, this generalizes comparisons between syntomic complexes and p-adic vanishing cycles; on the other hand, it is closely related to Beilinson--Lichtenbaum statements on the generic fiber. (Joint with Bhargav Bhatt.)