  In the present paper, we provide a lower bound of the essential dimension
over a field of positive characteristic via Kato's cohomology group, defined by
cokernel of a general Artin-Schreier operator. Combining this with Tignol's
result 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 of
characteristic $2$.
