The Stacks project

Remark 75.43.4. We can ask if in Grothendieck's algebraization theorem (in the form of Lemma 75.43.3), we can get by with weaker separation axioms on the target. Let us be more precise. Let $A$, $I$, $S$, $S_ n$, $X$, $Y$, $X_ n$, $Y_ n$, and $g_ n$ be as in the statement of Lemma 75.43.3 and assume that

  1. $X \to S$ is proper, and

  2. $Y \to S$ is locally of finite type.

Does there exist a morphism of algebraic spaces $g : X \to Y$ over $S$ such that $g_ n$ is the base change of $g$ to $S_ n$? We don't know the answer in general; if you do please email stacks.project@gmail.com. If $Y \to S$ is separated, then the result holds by the lemma (there is an immediate reduction to the case where $X$ is finite type over $S$, by choosing a quasi-compact open containing the image of $g_1$). If we only assume $Y \to S$ is quasi-separated, then the result is true as well. First, as before we may assume $Y$ is quasi-compact as well as quasi-separated. Then we can use either [Bhatt-Algebraize] or from [Hall-Rydh-coherent] to algebraize $(g_ n)$. Namely, to apply the first reference, we use

\[ D_{perf}(X) \to \mathop{\mathrm{lim}}\nolimits D_{perf}(X_ n) \xrightarrow {\mathop{\mathrm{lim}}\nolimits Lg_ n^*} \mathop{\mathrm{lim}}\nolimits D_{perf}(Y_ n) = D_{perf}(Y) \]

where the last step uses a Grothendieck existence result for the derived category of the proper algebraic space $Y$ over $R$ (compare with Flatness on Spaces, Remark 76.13.7). The paper cited shows that this arrow determines a morphism $Y \to X$ as desired. To apply the second reference we use the same argument with coherent modules:

\[ \textit{Coh}(\mathcal{O}_ X) \to \mathop{\mathrm{lim}}\nolimits \textit{Coh}(\mathcal{O}_{X_ n}) \xrightarrow {\mathop{\mathrm{lim}}\nolimits g_ n^*} \mathop{\mathrm{lim}}\nolimits \textit{Coh}(\mathcal{O}_{Y_ n}) = \textit{Coh}(\mathcal{O}_ Y) \]

where the final equality is a consequence of Grothendieck's existence theorem (Theorem 75.42.11). The second reference tells us that this functor corresponds to a morphism $Y \to X$ over $R$. If we ever need this generalization we will precisely state and carefully prove the result here.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GHK. Beware of the difference between the letter 'O' and the digit '0'.