# Künneth formula for scheme over product of varieties

Suppose we are given two varieties $$X, Y$$ over a field $$k$$ (probably assumed to be projective or proper over $$k$$). Denote the projections by $$p\colon X\times_k Y \rightarrow X$$ and $$q\colon X\times_k Y \rightarrow Y$$ and say we are given a quasi-coherent $$\mathcal{O}_X$$-module $$\mathcal{F}$$ and a quasi-coherent $$\mathcal{O}_Y$$-module $$\mathcal{G}$$ (probably assumed to be finite locally free or something weaker).

Then we have the Künneth formula, giving us a canonical isomorphism $$H^n(X \times_k Y, p^*\mathcal{F}\otimes_{\mathcal{O}_{X\times_k Y}} q^*\mathcal{G})=\bigoplus_{i+j=n}H^i(X, \mathcal{F})\otimes_kH^j(Y, \mathcal{G})$$.

Suppose further we are given a (probably proper or some other assumtion) morphism $$f\colon Z \rightarrow X\times_k Y$$ of varieties over $$k$$, then we can consider the pullback $$f^*(p^*\mathcal{F}\otimes_{\mathcal{O}_{X\times_k Y}} q^*\mathcal{G})=f^*p^*\mathcal{F}\otimes_{\mathcal{O}_{Z}} f^*q^*\mathcal{G}$$. I would like to compute $$H^n(Z,f^*p^*\mathcal{F}\otimes_{\mathcal{O}_{Z}} f^*q^*\mathcal{G})$$; so my question is, if there exists something similar to the Künneth formula in this situation?

My attemt was to write $$H^n(Z,f^*p^*\mathcal{F}\otimes_{\mathcal{O}_{Z}} f^*q^*\mathcal{G})=H^n(X \times_k Y, Rf_*(f^*p^*\mathcal{F}\otimes_{\mathcal{O}_{Z}} f^*q^*\mathcal{G}))$$ and then use the projection formula, aiming to apply the Künneth formula over $$X \times_k Y$$. But I was not able to reach a suitable expression in order to apply the Künneth formula.

I am willing to assume some extra assumtions, if needed.

There is no Kunneth formula for $$Z$$, but using the projection formula you can rewrite $$Rf_*(f^*p^*\mathcal{F} \otimes f^*q^*\mathcal{G}) \cong p^*\mathcal{F} \otimes q^*\mathcal{G} \otimes Rf_*\mathcal{O}_Z.$$ Now if you have a resolution for $$Rf_*\mathcal{O}_Z$$ formed by sheaves that look like $$p^*\mathcal{F_i} \otimes q^*\mathcal{G_i}$$ you can use the Kunneth formula termwise and then use the hypercohomology spectral sequence. This holds true, for instance, if $$Z \subset X \times Y$$ is the zero locus of a regular section of a vector bundle of the form $$p^*\mathcal{F_1^\vee} \otimes q^*\mathcal{G_1^\vee}$$ (then the Koszul resolution takes the required form) and in many other cases.
• Thank's for the explanation. Is there a common way to characterize the existence of such resolution in terms of properties of $f$ or $X$ and $Y$? In my particular setting, (in the most general situation) I would like to take an arbitrary morphism $f$ (with probably some assumptions) and would like to decompose $H^n(X\times_kY, p^*\mathcal{F}\otimes q^*\mathcal{G}) \rightarrow H^n(Z, f^*(p^*\mathcal{F}\otimes q^*\mathcal{G}))$.
• I don't think there is a general way to characterize when such a resolution exists. A simple example when it does not exist is the case when $X = Y$ is a curve of positive genus and $Z$ is the diagonal. Commented Sep 29, 2020 at 10:07