# Mixing solids and liquids

Is there a nontrivial way to consider products of archimedean and non-archimedean spaces in the context of Clausen–Scholze's analytic geometry?

Context: Last week during a conference in Essen (School on Arithmetic Geometry) Peter Scholze has given a talk about his (joint with Dustin Clausen) work on analytic geometry. One of the main conceptual achievements of this theory is to provide a single framework encompassing both nonarchimedean geometry and complex-analytic geometry, using the theory of solid and liquid analytic ring structures, respectively.

After the talk one of the participants has asked whether there is a way in this theory to take a product of spaces living in those two worlds, one nonarchimedean (over some $$\mathbb Z_p$$) and one archimedean (over $$\mathbb R$$). The obvious answer, as explained by prof. Scholze, is that yes, you can take such a product, but it will be empty. This makes some sense — it is justified for instance by the fact that the solid tensor product $$\mathbb Z_p\otimes^\blacksquare\mathbb R$$ is zero (see Example 6.4 Scholze (joint with Clausen) - Lectures on condensed mathematics) (one may object using solid tensor product when involving $$\mathbb R$$, but let me just mention that a similar obstruction occurs when trying to consider product of objects over $$\mathbb Z_p,\mathbb Z_q$$ for $$p\neq q$$).

However, spaces of this flavor do appear, even within arithmetic geometry — the most standard examples are various adelic spaces appearing e.g. when defining Shimura varieties, which are (restricted) products of spaces over local fields. (Note: a natural suggestion would be to just put some analytic ring structure on the ring of adeles, but I don't believe it would give us the "right" geometric spaces we care about, see comments.)

Another example (which also appeared in that conference, and I'm tempted to believe is the one that prompted the question) is the product of a (Drinfeld) $$p$$-adic upper half plane with complex upper half plane, as it admits a discrete action of a group $$\mathrm{SL}_2(\mathbb Z[\frac{1}{p}])$$.

Given the "unifying" nature of condensed mathematics and analytic geometry, I would expect this framework to be able to accommodate such spaces. The regular ("non-completed") tensor product $$\mathbb Z_p\otimes\mathbb R$$ is still nontrivial over condensed group, so one could still hope that there is some way to produce analytic ring structures to accommodate those. Any ideas in that direction would be appreciated.

• I am not sure whether your argument is valid on the nose, but it could be fixed as follows: consider the map $\mathbb R\to\mathbb R_{\operatorname{Liq}}$ of analytic rings where the first is the condensed ring equipped with the trivial analytic structure. Taking the coproduct with $\mathbb Z_{p,\blacksquare}$ in the category of analytic rings, you get a map of analytic rings, whose source is the zero ring, and therefore the target is also zero.
– Z. M
Sep 10, 2022 at 15:05
• Let me mention that there is a liquid structure on $\mathbb Q_\ell$ described in Remark 5.5 in liquid tensor experiments.
– Z. M
Sep 10, 2022 at 16:00
• @Z.M Thank you for both of these comments! RE first, yeah, obviously the tensor product in the category of analytic rings is more relevant here than just that of (solid) abelian groups, and your argument shows its triviality. RE second, this is a good point. I imagine tensoring those with each other or with $\mathbb R_{Liq}$ also gives zero by essentially the same argument? Sep 10, 2022 at 17:15
• The ring of adeles is built on something more like the direct product rather than the tensor product. Sep 10, 2022 at 17:23
• Perhaps someone should come up with a "restricted (direct) product" for analytic rings which would yield adeles as such an object. Sep 10, 2022 at 17:28

I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. These sometimes involve something like a product of a $$p$$-adic upper half plane and a complex upper half plane.
It is kind of tricky to give a good answer. The short answer is that I hope that in the future, some form of the formalism will be able to accomodate the spaces that Darmon cares about. But there are many slightly different categories in place, and these spaces will not be analytic spaces; they will not admit any kind of structure sheaf. A good analogy, for those familiar with $$p$$-adic geometry, is that analytic spaces are some generalization of adic spaces (which in particular only admit one map to $$\mathrm{Spec}(\mathbb Z)$$), while Darmon's spaces ought to be something like diamonds (which can admit several maps to (the diamond version of) $$\mathrm{Spec}(\mathbb Z)$$, but do not really have a structure sheaf). Diamonds are defined on the test category of perfectoid spaces $$S$$ in characteristic $$p$$, and for any such $$S$$ there is an adic space ''$$S\times \mathrm{Spa}(\mathbb Z_p)$$''; similarly, there ought to be an elusive test category $$\mathcal{P}$$ on which Darmon's spaces are defined, and for any $$S$$ in that test category $$\mathcal{P}$$, one can define an analytic space ''$$S\times \mathrm{Spec}(\mathbb Z)$$''. Objects in $$\mathcal{P}$$ should admit a notion of ''map to $$\mathrm{Spec}(\mathbb Z)$$'', which ought to give rise to a graph, which is a closed subspace ''$$S^\sharp$$'' of ''$$S\times \mathrm{Spec}(\mathbb Z)$$'', where now $$S^\sharp$$ is actually an analytic space, and then more generally for any analytic space $$X$$ one can define a functor (analogous to $$X^\diamond$$) on $$\mathcal{P}$$, which classifies such $$S^\sharp$$ plus a map $$S^\sharp\to X$$. In this context, such nontrivial products as desired by Darmon could be defined.
• diamonds (which can admit several maps to (the diamond version of) Spec(Z), but do not really have a structure sheaf) — sorry, I am a bit confused. Is the category defined by Lucas Mann thought as a quasicoherent stack over diamonds (or v-stacks)?
• Well, diamonds have a characteristic $p$ sheaf -- they are defined on perfectoid spaces in characteristic $p$, after all -- but no structure sheaf that, under the present analogy, should live over $\mathbb Z_p\otimes\mathbb R$. (I.e., for a diamond with two maps to $\mathrm{Spd}(\mathbb Z_p)$, there's no structure sheaf corresponding to that.) Sep 11, 2022 at 19:30