I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times Y_1$ such that $L_1 \cong p_2^*(M_1)$ for some line bundle $M_1$ on $Y_1$, then $p_{2,*}(L_1) \cong M_1$.
Here $X$ is a complete variety over an algebraically closed field $k$, and $Y_1$ is a scheme of finite type over $k$.
The Künneth formula I know relates (co-)homology on product spaces with the (co-)homology on the base spaces, for example as in the stacks-project, tag 0BEC. I don't see how the claim should follow from this.
Does the claim follow from this Künneth formula, or is something else going on here?
It's always hard to reconstruct someone else's train of thought, but here's one attempted explanation.
As aginensky says, the first step seems to be to use the projection formula. This shows that $(p_2)_* L_1 = M_1 \otimes (p_2)_* O_Z$ where I have written $Z$ to denote the product (to save typing).
So now it is enough to show that $(p_2)_* O_Z = O_{Y_1}$. To do this, recall that $(p_2)_* O_Z$ is defined by
\begin{align*} U &\mapsto H^0((p_2)^{-1}(U), O_{(p_2)^{-1}(U)})\\ &= H^0(U \times X, O_{U \times X}) \end{align*} Now Künneth tells us that this is isomorphic to \begin{align*} &H^0(U,O_U) \otimes_k H^0(X,O_X) =H^0(U,O_U)\end{align*} using the fact that $X$ is complete.