The Bogomolov-Prokhorov invariant of surfaces as equivariant cohomology

For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G,H^2(M,\mathbb{Z}))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov “On stable conjugacy of finite subgroups of the plane Cremona group, I”.