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

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 $2$.