Microlocalization and singular supports of constructible étale sheaves

Let $X$ be a smooth scheme over a perfect field of characteristic $p>0$, and let $F$ be a constructible complex of finite Tor-dimension with finite coefficients of characteristic prime to $p$. We prove $${\mathrm{S}\mathrm{S}}_{\mu }(F)=\mathrm{S}\mathrm{S}(F),$$ where ${\mathrm{S}\mathrm{S}}_{\mu }(F)$ is Saito’s microlocal singular support and $\mathrm{S}\mathrm{S}(F)$ is Beilinson’s singular support. This answers a question of Saito. As applications, we resolve another question of Saito regarding support estimates for microlocalization along a smooth closed subscheme, and we prove Saito’s conjecture on characteristic classes (\emph{Invent. Math. 207: 597-695, 2017}), showing that the cohomological characteristic classes of Abbes and Saito are the cycle classes associated with the corresponding characteristic cycles.