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

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


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

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.


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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.