Computing the cohomology groups

Let K be a field, and let T be ACFK , the theory of algebraically closed fields extending K. Then ACFK is strongly minimal, eliminates imaginaries, and has the property that every algebraically closed subset of the monster model is an elementary substructure (i.e., a model). Denote the monster by M. Let G be a connected algebraic group over K. Fix i ≥ 0, and a finite group M with order prime to charK. We are going to give a purely model-theoretic construction which should recover the ith etale cohomology group with coefficients in M of the variety G, base changed to K (or to the monster—these should be the same). If A is a set of parameters, let Gal(A) denote the profinite group Aut(acl(A)/ dcl(A)). Because we eliminate finite imaginaries, the category of sets with continuous action of Gal(A) is equivalent to the category of finite A-definable sets. If L is a field, Gal(L) agrees with the usual Galois group Aut(L/L). If A ⊂ B, there is a natural restriction map Gal(B)→ Gal(A). Let n be sufficiently big. Let g1, . . . , gn, t be a Morley sequence over ∅ (i.e., over K) in the generic type of G. For each non-empty I ⊆ {1, . . . , n}, let