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Let $f:X\to Y$ be a morphism of ringed spaces. We define a functor $f_!:\mathcal{O}_X\mathsf{-Mod}\to\mathcal{O}_Y\mathsf{-Mod}$ as $$\Gamma(U,f_!\mathscr{F}):=\{s\in \Gamma(f^{-1}(U),\mathscr{F})\:|\: f|_{\operatorname{supp}s}:\operatorname{supp}s\to U\text{ is proper}\},$$ where we say that a continuous morphism (between general topological spaces) is proper if it is universally closed.

Defined in this way, the proper direct image only seems to be used for locally compact spaces, where we have all the formalism of Verdier duality. In SGA4, P. Deligne says that, for schemes, the derived functors of $f_!$ are pathological. (Section XVII.3.1.) This is possibly why in étale cohomology and for $\mathcal{D}$-modules (possibly in other scenarios) we use other definitions for $f_!$. Moreover, this should be the reason why when studying Grothendieck duality we search for an adjoint of $\mathsf{R}f_*$, instead of $\mathsf{R}f_!$.

What exactly goes wrong with $f_!$ and its derived functors when the ringed spaces in question are not locally compact?

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  • $\begingroup$ You may find some ideas about this by looking in Chapter 2.5 in the book "Sheaves on Manifolds" by Kashiwara and Schapira. $\endgroup$
    – Vivek Shende
    Commented Aug 16, 2021 at 9:16
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    $\begingroup$ At the level of derived categories, we know what we want $f_!$ to be when $f$ is either proper or an open immersion, which forces the definition for compactifiable morphisms. Now in the proper case we have a right derived functor, and in the open immersion case we have a left derived functor, so we shouldn't expect the composite to be a derived functor in general. For nice enough topological spaces it turns out to still be the right derived functor of underived $f_!$, but this uses partitions of unity and does not work for schemes: see SGA4 XVII Contre-exemple 6.1.6. $\endgroup$
    – Marc Hoyois
    Commented Aug 16, 2021 at 16:32
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    $\begingroup$ In short, the issue for schemes is that $R(p_*\circ j_!)\neq R(p_*)\circ j_!$, and it's the right-hand side we're interested in. $\endgroup$
    – Marc Hoyois
    Commented Aug 16, 2021 at 16:34
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    $\begingroup$ It is a property of the map $f$ which is needed, not a property of the spaces. Reasonably $f$ must be both separated and locally proper. Any mapping between locally compact Hausdorff spaces has these properties. $\endgroup$
    – Tyrone
    Commented Aug 16, 2021 at 22:54
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    $\begingroup$ In the setting of étale constructible sheaves, Laszlo-Olsson define functors $Rf_!$ and $f^!$ for any finite type morphism between algebraic stacks locally of finite type. No separation hypotheses are needed. $\endgroup$
    – Dan Petersen
    Commented Aug 17, 2021 at 10:38

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