Perfect $F$-gauges and finite flat group schemes

We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\Int/p^n\Int$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By relating fppf cohomology of group schemes and syntomic cohomology of $F$-gauges, we deduce some consequences: These include the representability of relative fppf cohomology of finite flat group schemes under proper smooth maps of $p$-adic formal schemes, as well as a reproof of a purity result of \v{C}esnavi\v{c}ius-Scholze. We also give a general criterion for a classification in terms of objects closely related to Zink's windows over frames and Lau's divided Dieudonn\'e crystals, and we use this to recover several known classifications, and also give some new ones.