Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. Let $a_1,a_2,\ldots \in H^\*(E,A)$ be classes that give a basis of $H^\*(F_x,A)$ when restricted to any $F_x$. Assume that the direct image $R^0p_\ast \underline{A}_E$ of the constant sheaf on $E$ is constant. The Leray-Hirsch principle says that $H^\*(E,A)$ is a free $H^\*(B,A)$-module generated by the $a_i$'s.

I would like to ask if anyone knows a reference for a similar result for 'etale cohomology. Ideally I would like to have a statement for $E,B$ varieties over an algebraically closed field $k$ and finite coefficients of order prime to $char (k)$.

Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. Let $a_1,a_2,\ldots \in H^\*(E,A)$ be classes that give a basis of $H^\*(F_x,A)$ when restricted to any $F_x$. Assume that the direct image $R^0p_\ast \underline{A}_E$ of the constant sheaf on $E$ is constant. The Leray-Hirsch principle says that $H^\*(E,A)$ is a free $H^\*(B,A)$-module generated by the $a_i$'s.

I would like to ask if anyone knows a reference for a similar result for étale cohomology. Ideally I would like to have a statement for $E,B$ varieties over an algebraically closed field $k$ and finite coefficients of order prime to $char (k)$.

Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. Let $a_1,a_2,\ldots \in H^\*(E,A)$ be classes that give a basis of $H^\*(F_x,A)$ when restricted to any $F_x$. The Leray-Hirsch principle says that $H^\*(E,A)$ is a free $H^\*(B,A)$-module generated by the $a_i$'s.

I would like to ask if anyone knows a reference for a similar result for 'etale cohomology. Ideally I would like to have a statement for $E,B$ varieties over an algebraically closed field $k$ and finite coefficients of order prime to $char (k)$.

Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. Let $a_1,a_2,\ldots \in H^\*(E,A)$ be classes that give a basis of $H^\*(F_x,A)$ when restricted to any $F_x$. Assume that the direct image $R^0p_\ast \underline{A}_E$ of the constant sheaf on $E$ is constant. The Leray-Hirsch principle says that $H^\*(E,A)$ is a free $H^\*(B,A)$-module generated by the $a_i$'s.

I would like to ask if anyone knows a reference for a similar result for 'etale cohomology. Ideally I would like to have a statement for $E,B$ varieties over an algebraically closed field $k$ and finite coefficients of order prime to $char (k)$.