For the question, everything is over an algebraically closed field and by a scheme we mean a scheme of finite type. The theorem of the cube is the following: Let $X$ be a complete variety, $Y$ a scheme, and $L$ a line bundle on $X\times Y$. Then there is a unique closed subscheme $Y_1$ of $Y$ satisfying the following properties: (a) If $L_1$ is the restriction of $L$ to $X\times Y_1$, there is a line bundle $M_1$ on $Y_1$ and an isomorphism $p_2^*M_1\rightarrow L_1$ on $X\times Y_1$. (b), since (b) has little to do with the question, let us focus on (a). I am puzzled by the following thing: by the Künneth formula, if $p_2^*M_1\rightarrow L_1$ is an isomorphism, then $M_1 \rightarrow {p_2}_*(L_1)$ is an isomorphism. Does anyone knows what is the content of the Künneth formula mentioned here and more importantly, why the statement is true? Thanks!

For the question, everything is over an algebraically closed field and by a scheme we mean a scheme of finite type. The theorem of the cube is the following: Let $X$ be a complete variety, $Y$ a scheme, and $L$ a line bundle on $X\times Y$. Then there is a unique closed subscheme $Y_1$ of $Y$ satisfying the following properties: (a) If $L_1$ is the restriction of $L$ to $X\times Y_1$, there is a line bundle $M_1$ on $Y_1$ and an isomorphism $p_2^*M_1\rightarrow L_1$ on $X\times Y_1$. (b), since (b) has little to do with the question, let us focus on (a). I am puzzled by the following thing: by the Künneth formula, if $p_2^*M_1\rightarrow L_1$ is an isomorphism (and $M_1$ is a line bundle on $Y_1$), then $M_1 \rightarrow {p_2}_*(L_1)$ is an isomorphism. Does anyone knows what is the content of the Künneth formula mentioned here and more importantly, why the statement is true? Thanks!