Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to *$ be the map to the point (though one can consider more general maps as well). There is a classically known pro-algebraic "compactly supported global sections" functor defined by Deligne in the appendix to Hartshorne's "Residues and Duality". Namely, one has a functor $$\pi_!: \mathrm{Coh}_U\to D^b \mathrm{ProFinVect},$$ given informally by taking a sheaf $F$ to the fiber of the map $$\Gamma(U, F)\to \Gamma(\mathring{\delta}_X, F),$$ where $\mathring{\delta}_X: = \widehat{X\setminus U}\cap U$ is the punctured formal boundary. (More explicitly, one defines a functor on coherent sheaves on $X$ and applies it to any continuation of $F$, or equivalently, to $j_*F$ as an ind-object. It's easy to see that this can be done fully inside the $\infty$-category of complexes of pro-coherent sheaves on $X$.) This functor is described in the condensed language in Lecture 11 of Scholze and Clausen's Lectures, though here my understanding is limited (and the lecture works under a smoothness condition, which is surely unnecessary in defining $\pi_!$).
Since $\pi_!$ is valued in complexes of pro-vector spaces, one can define the dual contravariant functor $(\pi_!)^*:D^b Coh(U)\to D^b Vect$ (as a functor of $\infty$-categories). As $U$ varies, this forms a presheaf of "distributions on $F$" (which I think is a sheaf in general) which, for $X$ smooth, agrees with the Serre dualizing sheaf. Write $S_X$ for the sheaf of distributions on the constant sheaf, i.e. (the sheafification of) $$S_X:U\mapsto (\pi_!^U)^*(\mathcal{O}_X).$$
I have three questions about this construction.
- What is the name of the resulting complex for a general proper (and arbitrarily singular) $X$? I want to call it the dualizing complex, but have only seen that defined under some homological singularity restrictions on $X$ like the Gorenstein property.
- Is there a way to interpret this construction in the condensed language (e.g., is this what would be called $\pi^!\mathcal{O}$ in the Clausen-Scholze lectures? Is it clear that their construction works in the non-smooth context?)
- The construction in the Deligne appendix only defines $\pi_! F$ as a complex of pro-vector spaces, not pro-finite vector spaces. It is very easy to make it pro-finite-dimensional (and thus with an ind-finite dimensional dual) by rewriting his construction $\infty$-categorically, but I have never seen this done. Is there a reference to this?
