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In the present paper, we provide a lower bound of the essential dimensionover a field of positive characteristic via Kato's cohomology group, defined bycokernel of a general Artin-Schreier operator. Combining this with Tignol'sresult on the second trace form of simple algebras of degree $4$, we show that$\operatorname{ed}(\operatorname{\mathbf{PGL}}_{4})\geq 4$ over a field ofcharacteristic $2$.

A lower bound on the essential dimension of $\operatorname{\mathbf{PGL}}_{4}$ in characteristic $2$