Functorial splitting of l-adic cohomology of an extension of group varieties

In this document we consider an exact sequence of group varieties e→N→G→Q→ e over an algebraically closed field. We show that for l≠char(k) a prime there exists an isomorphism of graded Ql-algebras Hét∗(G,Ql)≅Hét∗(N,Ql)⊗QlHét∗(Q,Ql) that is compatible with pullback homomorphisms φ∗ of endomorphisms φ:G→G that stabilize N.