# Relative Künneth formula

Let $$X \to X^\prime \to S$$ and $$Y \to Y^\prime \to S$$ be morphisms of schemes. Then we obtain the following diagram.

Is it true that under suitable conditions, the equation $$\mathbf R \varphi_\ast(\mathbf L p^\ast(\mathcal F) \otimes^{\mathbf L} \mathbf L q^\ast(\mathcal G)) \cong \mathbf L p^{\prime, \ast}(\mathbf R f_\ast(\mathcal F)) \otimes^{\mathbf L} \mathbf L q^{\prime, \ast}(\mathbf R g_\ast(\mathcal G))$$ holds for any $$\mathcal F \in \mathbf D_{\mathsf{Coh}}^b(\mathcal O_X)$$ and $$\mathcal G \in \mathbf D_{\mathsf{Coh}}^b(\mathcal O_Y)$$. In other words, if we write $$\mathcal F \boxtimes \mathcal G = \mathbf L p^\ast(\mathcal F) \otimes^{\mathbf L} \mathbf L q^\ast(\mathcal G)$$, then is it true that $$\mathbf R \varphi_\ast(\mathcal F \boxtimes \mathcal G) \cong \mathbf R f_\ast(\mathcal F) \boxtimes \mathbf R g_\ast(\mathcal G)$$? If $$X^\prime = S = Y^\prime$$, then this equation reduces to the Künneth formula in 0FLN.

• A tangential comment for the bounded derived category of contsructible sheaves: If $S$ is the spectrum of a field $k$ and your schemes are of finite type over $k$, then this is true for $D_{ctf}^b(-, A)$, where $A$ is some torsion noetherian ring (see [Lei Fu, Étale cohomology theory, 9.3.5]. The proof in the reference uses the universal strong local acyclicity of finite type $k$-schemes relative to $D_{ctf}^b(-, A)$.
– A.B.
Jun 24 at 18:47
• Maybe I should add, continuing my previous tangential comment, that with proper morphisms, you can remove the assumption on S and allow yourself more general complexes, see 7.4.9 of the refered book. Sorry for the tangents.
– A.B.
Jun 25 at 7:40

Consider the diagram with cartesian squares $$\require{AMScd} \begin{CD} X\times_S Y @> \Delta >> X \times Y\\ @V \varphi V V @VV f \times g V\\ X'\times_S Y' @> \Delta' >> X' \times Y'\\ @V V V @VV V\\ S @>>\Delta_S > S \times S \end{CD}$$ In its terms the isomorphism you want to obtain can be rewritten as $$\mathbf{R}\varphi_* \circ \mathbf{L}\Delta^* \cong \mathbf{L}{\Delta'}^* \circ \mathbf{R}(f \times g)_*$$ which holds true as soon as the top square is Tor-independent.