# Zariski closed immersions

In p-adic geometry, what should it mean for a morphism to be a Zariski-closed immersion? For locally Noetherian adic spaces, the usual notion of a closed immersion of locally ringed spaces works just fine. For general analytic adic spaces, though, one quickly runs into annoying foundational issues. The issue is roughly as follows. Let $X=\mathrm{Spa}(A,A^+)$ be an (analytic) affinoid adic space. We can certainly define a reasonable notion of Zariski-closed subset, just by following our nose: a subset $Z \subset |X|$ should be Zariski-closed if there is an ideal $I \subset A$ such that $x \in |Z|$ iff $|f|_x = 0\,\forall f \in I$. These are exactly the subsets obtained by pulling back closed subsets of $\mathrm{Spec}(A)$ along the natural map $|\mathrm{Spa}(A,A^+)| \to |\mathrm{Spec}(A)|$. The problem, however, is that such a $Z$ need not come from an actual closed immersion of an affinoid adic space into $X$, because the quotient $A/I$ could just be some junky non-sheafy ring, and maybe there’s no canonical tweak (like replacing $I$ by its closure, or replacing $A/I$ by its uniform completion, or…) which will make it sheafy. And even if we can tweak $A/I$ to make it sheafy, how do we know that $A \to A/I$ is still surjective after going to some rational subset $U \subset X$? You get the picture.

Perhaps surprisingly, the situation for affinoid perfectoid spaces is a lot better. In particular, if $(A,A^+)$ is a perfectoid Tate-Huber pair, there are canonical bijections (satisfying some obvious compatibilities) between
1) closed subsets of $\mathrm{Spec}(A)$,
2) Zariski-closed subsets of $X=\mathrm{Spa}(A,A^+)$,
3) (isomorphism classes of) maps of Tate-Huber pairs $(A,A^+) \to (B,B^+)$ where $B$ is a perfectoid Tate ring, $A \to B$ is surjective, and $B^+$ is the integral closure of the image of $A^+$ in $B$.

We’ve already discussed the bijection 1) <–> 2). For 3) –> 1) or 2), just send $q:A \twoheadrightarrow B$ to the closed subset cut out by the ideal $\ker q$.  The miracle is the association 2) –> 3), which holds by an amazing theorem of Bhatt: if $I$ is a closed ideal in a perfectoid Tate ring $A$, then the uniform completion $B$ of $A/I$ is perfectoid and the natural map $A \to B$ is surjective, cf. Theorem 2.9.12 in Kedlaya’s notes here. Moreover, the map $A \to B$ remains surjective after rational localization on $A$. In particular, if $Z \subset X$ is a Zariski-closed subset, then 2) –> 3) gives an honest closed immersion $\mathrm{Spa}(B,B^+) \to \mathrm{Spa}(A,A^+)$ of locally ringed spaces, and $|\mathrm{Spa}(B,B^+)|$ maps homeomorphically onto $Z$.

The point of all this is that Zariski-closed immersions of affinoid perfectoid spaces behave as well as one could ever dream (with one caveat, which I’ll get to later). The following definition then suggests itself.

Definition. A map of small v-stacks $X \to Y$ is a Zariski-closed immersion if for any affinoid perfectoid space $W$ with a map $W \to Y$, the base change $X \times_{Y} W \to W$ is a Zariski-closed immersion of affinoid perfectoid spaces.

Now of course we’re free to make any definition we want in mathematics, but if it doesn’t capture some essential idea or experimentally observed phenomenon, then who cares? Let me now give some evidence that this definition passes this test.

Example 0. The property of being a Zariski-closed immersion is preserved under composition and base change. If $X \to Y$ is a Zariski-closed immersion and $Y$ is (a small v-sheaf, a diamond, a locally spatial diamond, qc or qs or separated or partially proper over a base $S$), then so is $X$.

Example 1. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces. If $Y$ is affinoid (so $X$ is too), then the map of diamonds $f^{\lozenge} : X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion. This is easy.

Example 2. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces again, but now assume that $f$ is the analytification of a closed immersion of quasiprojective varieties.  Then $f^{\lozenge}: X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion.  For this, we can use the assumption on $f$ to choose a vector bundle $\mathcal{E}$ on $Y$ together with a surjection $\mathcal{E} \twoheadrightarrow \mathcal{I}_{X} \subset \mathcal{O}_Y$. Then for any map $g: W \to Y$ from an affinoid perfectoid, the pullback $g^{\ast}\mathcal{E}$ (in the usual sense of ringed spaces) is a vector bundle on $W$, hence generated by finitely many global sections $e_1,\dots,e_n$ by Kedlaya-Liu. The images of $e_1,\dots,e_n$ along the natural map $(g^{\ast}\mathcal{E})(W) \to \mathcal{O}_{W}(W)$ generate an ideal, and the associated closed immersion of affinoid perfectoids $V \to W$ represents the fiber product $X^{\lozenge} \times_{Y^{\lozenge}} W$. (Hat tip to PS for suggesting this vector bundle trick.)

Example 3. Let $X^{\ast}$ be a minimally compactified Hodge-type Shimura variety with infinite level at $p$. Then the boundary $Z \to X^{\ast}$ is a Zariski-closed immersion, and so is the diagonal $X^{\ast} \to X^{\ast} \times X^{\ast}$. (These both reduce to the previous example, using a small limit argument in the second case.) In particular, if $U,V \subset X^{\ast}$ are any open affinoid perfectoid subsets, then $U \cap V$ is also affinoid perfectoid. This small observation plays a non-negligible role in my forthcoming paper with Christian Johansson, where (among other things) we prove that any minimally compactified Shimura variety of pre-abelian type with infinite level at $p$ is perfectoid.

Example 4. Fix a perfectoid base field $K$ of characteristic zero. Then the inclusions $\mathrm{Fil}^n \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}} \subset \mathrm{B}_{\mathrm{dR}}$ are Zariski-closed immersions of (ind-)diamonds over $\mathrm{Spd} K$. This can be proved by induction on $n$, and the base case reduces to the fact that the inclusion $\mathrm{Fil}^1 \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}}^{+}$ is the pullback of $\{ 0 \} \to \mathbf{A}^{1}_{K}$ along $\theta$. (To make the induction work, you need to pick an element $\xi \in \mathrm{B_{dR}}^+(K)$ generating $\ker \theta$.)

Example 5. Fix a complete algebraically closed extension $C / \mathbf{Q}_p$. Fix a reductive group $G / \mathbf{Q}_p$ and a geometric conjugacy class of $G$-valued cocharacters $\mu$. Then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C}$ is a Zariski-closed immersion. Also, if $\nu \leq \mu$, then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \nu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C}$ is a Zariski-closed immersion. These claims can be reduced to the case $G = \mathrm{GL}_n$, which in turn reduces to Example 4 by some trickery.

Example 6. Fix a complete algebraically closed nonarchimedean field $C$ of residue characteristic $p$, and let $\mathcal{E} \to \mathcal{F}$ be any injective map of coherent sheaves on the Fargues-Fontaine curve $X_C$. Then the associated map of Banach-Colmez spaces $\mathbb{V}(\mathcal{E}) \to \mathbb{V}(\mathcal{F})$ is a Zariski-closed immersion. This can also be reduced to Example 4.

Let me end with some caveats. First of all, I wasn’t able to prove that if $G \to H$ is a closed immersion of reductive groups, the induced map $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{H, C}$ is a Zariski-closed immersion, although it is surely true. The problem here is (roughly) that an $H$-torsor over some affinoid perfectoid $X$ can only be reduced to a $G$-torsor locally in the analytic topology on $X$, and we then run into the following open question:

Question. Is the property of being Zariski-closed local for the analytic topology? More precisely, if $X$ is affinoid perfectoid with a covering by rational subsets $U_i$, and $Z$ is a closed subset such that $Z \cap U_i$ is Zariski-closed in $U_i$ for all $i$, is $Z$ Zariski-closed?

There are also naturally occurring closed things which probably aren’t Zariski-closed immersions. For instance, I don’t think the map of Banach-Colmez spaces $0 \to H^1(\mathcal{O}(-1)) = \mathbf{A}^{1,\lozenge}_{C} / \underline{\mathbf{Q}_p}$ is a Zariski-closed immersion, because then pulling back would imply that $\underline{\mathbf{Q}_p} \to \mathbf{A}^{1,\lozenge}_{C}$ is a Zariski-closed immersion, which seems extremely unlikely to me. (But I didn’t manage to disprove it! Actually, can one give an explicit example of an affinoid perfectoid $X/C$ and a closed subset $S \subset X$ such that $C$ maps isomorphically to the completed residue field at every point in $S$ and such that $S$ is NOT Zariski-closed? Surely such examples exist.) I also don’t think (closures of) Newton strata in flag varieties are Zariski-closed immersions – they are just too weird.

I also wasn’t able to settle the following compatibility (but admittedly I didn’t try very hard).

Question. Let $f: X \to Y$ be a monomorphism of locally Noetherian adic spaces. If $f^{\lozenge}$ is a Zariski-closed immersion, is $f$ actually a closed immersion?

Happy new year!