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Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$-schemes such that $\lvert X\rvert=\bigcup_i f_i(\lvert Z_i\rvert)$ (notice that $\lvert Y\rvert$ denotes the underlying topological space of a scheme $Y$).

I am interested in the following inter-related questions:

Has sheaf theory on this topology been studied? If so, what do we know about it? In particular, can we compare it somehow to the sheaves on the Zariski site?

If this facilitates things, assume that we work on affine schemes, and our coverings are also given by affine morphisms in the same way, so that everything is reduced to rings and quotient of rings.

Maybe this is a very basic question and maybe it is fully settled in the literature, but I am not aware of a good reference. Any references/comments are highly appreciated!

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    $\begingroup$ One problem with this topology is that it is not subcanonical, i.e., the representable functors are not sheaves. $\endgroup$
    – Martin Brandenburg
    Commented Jan 22, 2023 at 20:52
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    $\begingroup$ Any motivation for this topology? $\endgroup$
    – Z. M
    Commented Jan 22, 2023 at 21:07
  • $\begingroup$ @MartinBrandenburg, thank you so much for indicating that. Is this easy to see? if not, is it possible to give some reference about this point? $\endgroup$
    – FNH
    Commented Jan 22, 2023 at 21:36
  • $\begingroup$ @Z.M, well, in my work I aim to construct a variance of schemes for some purposes. However, in that setting, defining a topology using open sets of non-vanishing functions is not well-behaved when you go to sheaves (i.e. you can not associate structure sheaf on those open sets), however, using vanishing loci is okay: the type of rings one should associate to such a vanishing set is clear. To make this sensible, I thought one should try to rebuild schemes using sheaves on closed immersions instead of open immersions as is standard, and see if the resulting theory is equivalent to schemes. $\endgroup$
    – FNH
    Commented Jan 22, 2023 at 21:40
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    $\begingroup$ There is something similar in [Clausen–Scholze, Condensed Mathematics and Complex Geometry, Lecture VII], where quasicompact open subsets are closed in that topology, or equivalently, the Hochster dual of the Zariski topology, but open subsets are not given by closed immersions, but roughly speaking, by the formal completion along some closed subset (whose complement is quasicompact). $\endgroup$
    – Z. M
    Commented Jan 22, 2023 at 22:18

1 Answer 1

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The topology you mention is called the "closed topology" $J_\mathrm{cl}$ in "Points in algebraic geometry" by Gabber–Kelly. See also the comparison diagram by Pieter Belmans. Note that Gabber and Kelly additionally ask that the covering families are finite. But since they ask for the base scheme to be Noetherian (and separable), I don't think there is a difference, because any covering family automatically has a finite refinement.

You can apply the Comparison Lemma to simplify the site $(\mathrm{Sch}/S,J_\mathrm{cl})$ without changing the category of sheaves on it. To make the notation easier, I will consider $J_\mathrm{cl}$ on $\mathrm{Sch}$, but the argument can be modified to work for $\mathrm{Sch}/S$ as well. The claim is that the category of sheaves $\mathbf{Sh}(\mathrm{Sch},J_\mathrm{cl})$ is equivalent to the category of presheaves $\mathbf{PSh}(\mathrm{IntSch})$ on the category $\mathrm{IntSch}$ of integral schemes.

Any scheme $X$ can be written as a union $$X = \bigcup_{x \in |X|} \overline{\{x\}}$$ and each of the closed subsets $\overline{\{x\}}$ can be given the reduced induced subscheme structure. In this way, any scheme can be covered by integral (i.e. irreducible and reduced) schemes. The Comparison Lemma then says that the category of sheaves for your topology can equivalently be described as a category of sheaves on the full subcategory of integral schemes. The Grothendieck topology that we have to take on this category of integral schemes is the restriction of the one on schemes: we again look at covers by closed subschemes. But for an integral scheme $X$, if you have such a covering by closed subschemes, one of the closed subschemes will contain the generic point, and then this closed subscheme must be equal to $X$. In this way, you can see that the restricted Grothendieck topology is precisely the presheaf topology.

In the same way, you can also find $\mathbf{Sh}(\mathrm{Sch}/S,J_\mathrm{cl})\simeq \mathbf{PSh}(\mathrm{IntSch}/S)$.

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  • $\begingroup$ I am confused by the comment on the indifference of the finiteness of the topology. I don't see why any cover is refined by a finite one — although the base is Noetherian, the covered scheme is not necessarily so. $\endgroup$
    – Z. M
    Commented Jan 23, 2023 at 22:38
  • $\begingroup$ Yes, you are right. I didn't mention that in the article by Gabber and Kelly, $\mathrm{Sch}/S$ is defined to be the category of finite type separated $S$-schemes. With this notation, each scheme in $\mathrm{Sch}/S$ is again Noetherian. $\endgroup$
    – Jens Hemelaer
    Commented Jan 24, 2023 at 8:52
  • $\begingroup$ In general, there are size issues that I didn't take into account. My answer assumes that $\mathrm{Sch}/S$ is a small category, either by restricting to schemes of limited cardinality (universes), or by restricting to schemes that are of finite type over $S$ (as in Gabber–Kelly). $\endgroup$
    – Jens Hemelaer
    Commented Jan 24, 2023 at 9:01

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