Artin Approximation and Proper Base Change

and an abelian sheaf F of torsion abelian groups on X, the natural base change map gRf∗F ∼ −→ Rf ′ ∗(g F) is an isomorphism. By limit arguments and considerations with geometric stalks as discussed last time, we can arrange that F is a Z/nZ-sheaf for some n > 0 and it suffices to treat the case where S = Spec (A) for a strictly henselian local ring and S′ = Spec (k′) for a separably closed field k′ over the separably closed residue field k of A. Next, we reduce to the “core case” k′ = k (i.e., S′ is the closed point s of S): Lemma 1.2. If the core case is proved in general then for any proper scheme X over a separably closed field k and any torsion abelian sheaf F on X, the map H(X,F)→ H(XK ,FK) is an isomorphism for any separably closed extension K/k. ∗Notes taken by Tony Feng