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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.

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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.

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  • $\begingroup$ 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 $H^n(X\times_kY, p^*\mathcal{F}\otimes q^*\mathcal{G}) \rightarrow H^n(Z, f^*(p^*\mathcal{F}\otimes q^*\mathcal{G}))$. $\endgroup$
    – john
    Sep 29, 2020 at 9:10
  • $\begingroup$ 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. $\endgroup$
    – Sasha
    Sep 29, 2020 at 10:07

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