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?
Best Answer
Isn't the dualizing complex defined in general in the proper case by taking applying the right adjoint of $\pi_\ast$ to $k$? That's what I'll take as the definition anyways. The Gorenstein property just means that the dualizing complex is invertible.
It is true that this can be computed by the formula you write down, and one way to see this is as in our lecture notes. (We do define compactly supported cohomology without the smoothness assumption.) So yes, this can be interpreted in our setting.
I know only few references on those compactly supported cohomology groups, and I'm not aware of any that work $\infty$-categorically.
And as you guessed, you actually don't have to sheafify this construction, it's already a sheaf.