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

Abstract

In this document we consider an exact sequence of group varieties $e\to N\to G\to Q\to$ $e$ over an algebraically closed field. We show that for $l\ne \mathrm{char}\left(k\right)$ a prime there exists an isomorphism of graded $Q_l$-algebras $H_{\acute{e}t}^{\ast }(G,Q_l)\cong H_{\acute{e}t}^{\ast }(N,Q_l){\otimes }_{Q_l}{H}_{\acute{e}t}^{\ast }(Q,Q_l)$ that is compatible with pullback homomorphisms ${\phi }^{\ast }$ of endomorphisms $\phi :G\to G$ that stabilize $N$.