# Stalks of derived tensor product (in the Kunneth formula)

So, essentially here's what I'm curious about. Suppose that $k$ is a (separably closed/algebraic closed) field $X_i,Y_i/k$ are finite type and $f_i:X_i\to Y_i$ are $k$-maps (all of this for $i=1,2$). Then, the Kunneth formula implies that

$$R(f_1\times f_2)_\ast \mathbb{Q}_\ell =\left(q_1^\ast Rf_{1\ast}\mathbb{Q}_\ell\right)\otimes_{\mathbb{Q}_\ell}^L \left(q_2^\ast Rf_{2\ast}\mathbb{Q}_\ell\right)$$

where this is all happening in, say, $D^b_c(-,\mathbb{Q}_\ell)$ (for the appropriate scheme) and $q_i:Y_1\times_k Y_2\to Y_i$ are the projection maps.

Now, I'd like to use this to compute $(R^m(f_1\times f_2)_\ast\mathbb{Q}_\ell)_{\overline{(y_1,y_2)}}$ for some geometric point $\overline{(y_1,y_2)}$ of $Y_1\times Y_2$ (really a $k$-point!) and, really, I only really know what $(R^m f_{i\ast}\mathbb{Q}_\ell)_{\overline{y_i}}$ is. Is this doable? Namely, is there some condition I could know about the $f_i$, $X_i$, or $Y_i$ that would guarantee that

$$(R^m(f_1\times f_2)_\ast\mathbb{Q}_\ell)_{\overline{(y_1,y_2)}}=H_m(\mathcal{F})_{\overline{(y_1,y_2)}}=\bigoplus_{a+b=m}(R^af_{1\ast}\mathbb{Q}_\ell)_{\overline{y_1}}\otimes (R^bf_{2\ast}\mathbb{Q}_\ell)_{\overline{y_2}}$$

where

$$\mathcal{F}=\left(q_1^\ast Rf_{1\ast}\mathbb{Q}_\ell\right)\otimes_{\mathbb{Q}_\ell}^L \left(q_2^\ast Rf_{2\ast}\mathbb{Q}_\ell\right)$$

And, again, if not, how would one go about computing these objects in practice?

NB: Even though I am curious about this in the case of general $f_i$ note that, in my case, neither $f_i$, $X_i$, nor $Y_i$ are proper--in fact, the $f_i$ are open embeddings.

Thanks!

EDIT: A local elder has suggested that there should be a spectral sequence relating $$(R^m(f_1\times f_2)_\ast\mathbb{Q}_\ell)_{\overline{(y_1,y_2)}}=H_m(\mathcal{F})_{\overline{(y_1,y_2)}}$$ to $$\bigoplus_{a+b=m}(R^af_{1\ast}\mathbb{Q}_\ell)_{\overline{y_1}}\otimes (R^bf_{2\ast}\mathbb{Q}_\ell)_{\overline{y_2}}$$ and that this should just totally collapse since the off-terms are Tor-functors which should vanish here since we're working with $\mathbb{Q}_\ell$-spaces.

Does this sound reasonable? Can anyone confirm?

Thanks!