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This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves.

Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ of finite type over $k$. I consider $\phi: G\rightarrow G$ a group homomorphism, and I would like to know under which conditions (on $G$, $k$) such a morphism is always proper (resp. smooth).

My goal is to understand the behaviour of convolution under group homomorphisms and to show some properties, for instance that for any group homomorphism $\phi: G\rightarrow G$ and all objects $K$, $L$ of $D^b_c(G,\overline{\mathbb{Q}_l})$, one has $$R(\phi,\phi)_*(K\times L)=(R\phi_*K\times R\phi_*L)$$

(which seems to be true, at least in the case of $\mathcal{D}$-modules).

Or that, given $\phi: G\rightarrow H$ a group homomorphism between separated smooth groupschemes of finite type over $k$, then $$\phi^*((R\phi_!K)\star_!L)=K\star_! (\phi^*L)$$ for any $K$ on $G$ and $L$ on $H$.

This would be alright if I could apply smooth or proper base changes to diagrams involving $\phi$, which I'm not sure I can (at least for proper base change).

I thank you for your help !

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    $\begingroup$ Your first equation looks like a special case of the general Kunneth formula, which is in SGA somewhere. For your second equation it is OK to use proper base change because proper base change holds for an arbitrary morphism if you use cohomology with compact supports. (This follows from the ordinary proper base change and the definition of cohomology with compact supports). $\endgroup$
    – Will Sawin
    Commented Nov 6, 2017 at 17:58
  • $\begingroup$ Thanks for your answer ! Even if this is less conceptual, can't we also say that one has $R(\phi,\phi)_*pr_1^*=pr_1^*R\phi_*$ (by smooth base change of $pr_1$), and the same for $pr_2$, therefore $$R(\phi,\phi)_*(K\times L)=R(\phi,\phi)_*(pr_1^*K\otimes pr_2^*L)=pr_1^*R\phi_*K\otimes pr_2^*R\phi_*L=(R\phi_*K\times R\phi_*L) ?$$ $\endgroup$
    – Yoël
    Commented Nov 6, 2017 at 18:22
  • $\begingroup$ The middle equality is the Kunneth formula. Also, I think you mean to use $\boxtimes$ instead of $\times$ in several places, unless there is some standard notation I am very unfamiliar with. $\endgroup$
    – Will Sawin
    Commented Nov 6, 2017 at 19:20
  • $\begingroup$ Does Künneth formula says that the equality in the middle holds even if we replace $(\phi,\phi)$ by some $(f,g)$, for a pair of $k$-morphisms $f$, $g: G\rightarrow G$ (not necessarily equal or homomorphisms) ? If not, do you know which conditions on $f$, $g$ we need ? Thanks again ;-) $\endgroup$
    – Yoël
    Commented Nov 6, 2017 at 19:48
  • $\begingroup$ The notation I use, $K\times L:=pr_1^*K\otimes pr_2^*L$ is the one Katz has in "Rigid local system" (chapter 2, page 4) $\endgroup$
    – Yoël
    Commented Nov 6, 2017 at 20:15

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