Ramification bounds via Wach modules and q-crystalline cohomology

Let $K$ be an absolutely unramified $p$-adic field. We establish a ramification bound, depending only on the given prime $p$ and an integer $i$, for mod $p$ Galois representations associated with Wach modules of height at most $i$. Using an instance of $q$-crystalline cohomology (in its prismatic form), we thus obtain improved bounds on the ramification of ${\mathrm{H}}_{et}^i(X_{{\mathbb{C}}_K},\mathbb{Z}\mathrm{/}p\mathbb{Z})$ for a smooth proper $p$-adic formal scheme $X$ over ${\mathcal{O}}_K$, for arbitrarily large degree $i$.