Crystallinity for syntomic cohomology, \'etale cohomology, and algebraic $K$-theory

Abstract

We prove for $n\geq c-1$ that the functor taking an animated ring $R$ to its mod $(p^c,v_1^{p^n})$ syntomic cohomology factors through the functor $R \mapsto R/p^{c(n+2)}$, a phenomenon we term crystallinity for mod $(p^c,v_1^{p^n})$ syntomic cohomology. As an application, we completely and explicitly compute the mod $(p,v_1 ^{p^{n}-1})$ algebraic $K$-theory of $\mathbb Z/p^{k}$ whenever $k \geq n+2$ and $p>2$. As a second application, we deduce crystallinity for the mod $p^c$ syntomic complexes associated to smooth $p$-adic formal schemes, and in particular for the Galois equivariant mod $p^c$ \'etale cohomologies of their adic generic fibers. Finally, we strengthen known $p$-adic convergence theorems for the topological Hochschild homology of ring spectra, and as a result relate crystallinity for algebraic $K$-theory to Lichtenbaum--Quillen theorems.