# What is the Künneth formula for complete varieties?

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?

• Certainly the projection formula is going on here. Perhaps to apply it properly, one needs to use Kunneth. – aginensky Mar 21 at 14:38
• @aginensky Yeah by the projection formula we get: $p_{2,*} p_2^* M_1 = p_{2,*}(\mathcal{O} \otimes p_2^*M_1) = p_{2,*}(\mathcal{O}_{X\times Y_1}) \otimes M_1$, so we reduce this to show that $p_{2,*}\mathcal{O}_{X \times Y_1} \cong \mathcal{O}_{Y_1}$. I think I can show this using "Cohomology and Base Change" (Hartshorne, p.290+291), but I still don't see any connection to the Künneth formula. – red_trumpet Mar 21 at 14:58
• Perhaps in saying that the structure sheaf of the product is the tensor product of the pullbacks of the two structure sheaves. – aginensky Mar 21 at 14:59

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.
• Seems correct. Two notes: In the last equation, it should be $\otimes$ instead of $\times$, and afaik $U \mapsto H^0(U\times X, \mathcal{O})$ already is a sheaf, so no need to sheafifiy here. – red_trumpet Mar 21 at 20:54