On the crystalline Artin--Mumford invariant

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $W=W(k)$. We prove that the torsion subgroup of $H_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{s}}^3(X\mathrm{/}W)$ is a birational invariant of smooth projective varieties over $k$. The proof combines the action of correspondences on Hodge–Witt cohomology and on the first Nygaard piece of the de Rham–Witt complex.