The Stacks project

Proof. Axiom (B)(a) is immediate from axiom (A5). Let $X$ and $Y$ be nonempty smooth projective schemes over $k$ equidimensional of dimensions $d$ and $e$. To see that axiom (B)(b) holds, observe that the diagonal $\Delta _{X \times Y}$ of $X \times Y$ is the intersection product of the pullbacks of the diagonals $\Delta _ X$ of $X$ and $\Delta _ Y$ of $Y$ by the projections $p : X \times Y \times X \times Y \to X \times X$ and $q : X \times Y \times X \times Y \to Y \times Y$. Compatibility of $\gamma $ with intersection products then gives that

is the cup product of the pullbacks of $\gamma ([\Delta _ X])$ and $\gamma ([\Delta _ Y])$ by $p$ and $q$. Write

and similarly $\gamma ([\Delta _ X]) = \sum \eta _{X, i}$ and $\gamma ([\Delta _ Y]) = \sum \eta _{Y, i}$. The observation above implies we have

(If our cohomology theory vanishes in negative degrees, which will be true in almost all cases, then only the term for $i = 0$ contributes and $\eta _{X \times Y, 0}$ lies in $H^0(X) \otimes H^0(Y) \otimes H^{2d}(X)(d) \otimes H^{2e}(Y)(e)$ as expected, but we don't need this.) Since $\lambda _ X : H^{2d}(X)(d) \to F$ and $\lambda _ Y : H^{2e}(Y)(e) \to F$ send $\eta _{X, 0}$ and $\eta _{Y, 0}$ to $1$ in $H^*(X)$ and $H^*(Y)$, we see that $\lambda _ X \otimes \lambda _ Y$ sends $\eta _{X \times Y, 0}$ to $1$ in $H^*(X) \otimes H^*(Y) = H^*(X \times Y)$ and the proof is complete. $\square$