9
$\begingroup$

I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any mistakes I write.

Let $\mathrm{Vect}^{an}(\mathbb{C})$ be this category of analytic complex vector spaces of Clausen-Scholze (which I believe is describable as $\aleph_1$-filtered colimits of sequential colimits along compact maps with rapidly decreasing singular values--see section 8 of the complex geometry notes at https://people.mpim-bonn.mpg.de/scholze/Complex.pdf). Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. And let $\mathrm{QC}^{an}(X)$ be the category of analytic quasi-coherent sheaves on the complex manifold $X$, which is tensored over $\mathrm{Vect}^{an}(\mathbb{C})$, and is described by quasicoherent modules over the structure sheaf $\mathcal{O}_X$ inside $\mathrm{Vect}^{an}(\mathbb{C})$.

My first question is the statment of GAGA that Clausen-Scholze prove in this setup. In particular, is the following functor an isomorphism?

$$\mathrm{QC}(X) \otimes_{\mathrm{Vect}(\mathbb{C})} \mathrm{Vect}^{an}(\mathbb{C}) \to \mathrm{QC}^{an}(X)$$

where the $\mathrm{QC}(X)$ is just the usual (big derived) category of quasicoherent sheaves in algebraic geometry.

My second question is about the first part of the theorem where it is claimed that $$K^{an}(X)[\kappa^{-1}]:=K^{alg}(\mathrm{QC}^{an}(X))[\kappa^{-1}] \cong K^{top}(X)$$ Can this isomorphism be upgraded to an equivalence of categories? For example for $X$ being a point, does it come from some colimit of $\mathrm{Vect}^{an}(\mathbb{C})$ with the category of modules over the h-unital ring of compact operators on a Hilbert space $K(H)$? If not how can I see why this isomorphism should hold?

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ I think that the quasicoherent sheaves in Complex.pdf do not assume nuclearity. With that definition, let $X$ be a $\mathbb C$-scheme of finite type, the (symmetric monoidal) category $C^{\operatorname{alg}}(X)$ in Thm 7.3 coincides with the base change of the category $D_{\operatorname{QCoh}}(X)$ of usual quasicoherent sheaves along the functor $D(\mathbb C)\to D^{\operatorname{liq}}(\mathbb C)$, a "non-nuclear" version of the functor that you wrote in the first question. When $X$ is proper, then $C^{\operatorname{an}}(X)=C^{\operatorname{alg}}(X)$, which gives the "non-nuclear" isomorphism.. $\endgroup$
    – Z. M
    Commented Oct 15, 2023 at 17:14
  • 2
    $\begingroup$ (cont'd) However, it is not clear to me what happens if we pass to nuclear objects, i.e. taking $(-)^{\operatorname{rig}}$ to this isomorphism (it is not clear to me whether this operation preserves pushouts). $\endgroup$
    – Z. M
    Commented Oct 15, 2023 at 17:21
  • $\begingroup$ By the way, I do not know why $\mathcal O_X$-modules in sheaves in liquid $\mathbb C$-modules are precisely quasicoherent sheaves (I see that Dustin Clausen claimed this in the video)? Classically, quasicoherence has (or is) an extra condition on $\mathcal O_X$-modules, namely, restrictions along standard opens are given by base changes. $\endgroup$
    – Z. M
    Commented Oct 16, 2023 at 10:44
  • $\begingroup$ Where does Dustin claim that all liquid Ox modules are quasicoherent in the video? I think I'm being imprecise in the question, I rather meant that you can glue the categories R-mod (inside the analytic vector spaces) for (compact?) Steins. $\endgroup$
    – Andy Jiang
    Commented Oct 16, 2023 at 13:22
  • $\begingroup$ youtube.com/watch?v=2xNG4rHUC6U&t=3988s for X being the projective line (maybe something special about it?) $\endgroup$
    – Z. M
    Commented Oct 16, 2023 at 16:50

0

Reset to default

You must log in to answer this question.