Over arbitrary p-adic formal bases, finite locally free p-power torsion commutative group schemes are exactly the perfect F-gauges of Tor amplitude [-1,0] and Hodge-Tate weights 0,1, via an exact, Cartier-duality-compatible equivalence.
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Perfect $F$-gauges and finite flat group schemes
Over arbitrary p-adic formal bases, finite locally free p-power torsion commutative group schemes are exactly the perfect F-gauges of Tor amplitude [-1,0] and Hodge-Tate weights 0,1, via an exact, Cartier-duality-compatible equivalence.