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In Lectures on Analytic Geometry, for complex-analytic geometry, seemingly one only considers maps $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ of analytic rings for $0<p\le1$ where $A$ is a "structure ring" like $\mathcal O(\overline D)$ and $M$ is induced by the underlying map of condensed rings. In other words, $\mathcal A$ could be seen as an algebra in $D_{\ge0}(\mathbb C,\mathcal M_{<p})$ (more precisely, a module over the symmetric monad) and $D_{\ge0}(\mathcal A,\mathcal M)$ is simply the $\infty$-category of modules over $\mathcal A$ as an algebra in $D_{\ge0}(\mathbb C,\mathcal M_{<p})$, i.e. no further analytic structure is imposed.

I wonder whether there are "natural" examples of $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ where $\mathcal M$ is not induced by the map $\mathbb C\to\mathcal A$, just as in the rigid analytic geometry, we have finer concepts like Huber pairs?

There is a related question: in Liquid tensor experiment Remark 5.5, we have $p$-liquid analytic rings $(\mathbb Q_\ell,\mathcal M_{<p})$ for all $p\in(0,+\infty]$. When $p=+\infty$, it is the solid theory. I wonder whether this $p$-liquid picture generalizes to Huber pairs? If this is possible, then it might be reasonable to imagine relating the two cases via $\mathbb Z[T]$.

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Great question! So far, we haven't been able to produce analytic ring structures on $\mathbb C$-algebras that are not induced. Similarly, if we equip $\mathbb Q_p$ with a liquid analytic ring structure, we are also only able to use induced analytic ring structures: To "overconvergent rigid spaces"(=Größe-Klönne's dagger spaces), one can associate analytic spaces over a liquid $\mathbb Q_p$, but you need the overconvergence as you can't define open subspaces via conditions like $\{|T|\leq 1\}$ in the liquid theory (contrary to the solid theory). So adic spaces, Huber pairs, etc., work only in the solid theory.

Somehow in the solid case, one can nicely confine to a subset $\{|T|\leq 1\}$, but in the liquid case things will always "spill over" to a small "overconvergent" neighborhood. (This geometric picture is one reason we chose those names...)

Here's another thought. The possibility to pass to $\{|T|\leq 1\}$ in the solid theory comes from the fact that $\mathbb Z((T^{-1}))$ is a compact idempotent $(\mathbb Z[T],\mathbb Z)_{\blacksquare}$-algebra, i.e. the $\mathbb Z$-solidification of $\mathbb Z((T^{-1}))\otimes_{\mathbb Z[T]} \mathbb Z((T^{-1}))$ is still $\mathbb Z((T^{-1}))$. Generally, one could define interesting new analytic ring structures once one has found such compact idempotent algebras (if $\mathcal A$ is an analytic ring and $\mathcal B$ is a compact idempotent $\mathcal A$-algebra, then one can define a new analytic algebra $\mathcal A'$ such that $\mathcal D(\mathcal A')=\mathcal D(\mathcal A)/\mathcal D(\mathcal B)$; starting from $\mathcal A=(\mathbb Z[T],\mathbb Z)_{\blacksquare}$ and $\mathcal B=\mathbb Z((T^{-1}))$, this produces $\mathcal A'=\mathbb Z[T]_{\blacksquare}$). However, in the liquid theory, we have not been able to produce examples of compact idempotent algebras. Trying to adapt this example leads one to questions related to the various types of holomorphic functions on an open or closed unit disc (like Hardy functions etc., maybe the Nevanlinna and Smirnov class), but none of them is compact and idempotent.

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    $\begingroup$ I should however say that one can produce some weird examples if one starts with the liquid analytic ring structure on $\mathbb Z((T))_{>r}$ and then takes the induced analytic ring structure for some subset of the punctured open unit disc over $\mathbb C$: This gives an analytic $\mathbb C$-algebra whose analytic ring structure is not induced from any given $p$-liquid analytic ring structure on $\mathbb C$ (rather, $p$ is varying on this disc). More generally, it seems likely that "hard" methods can produce further analytic ring structures on $\mathbb C$-algebras. $\endgroup$ May 2, 2021 at 19:54
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    $\begingroup$ Just a small note to avoid confusion: the parenthetical comment about producing analytic ring structures from compact idempotent algebras is not exactly accurate, for two reasons. The first reason that one still needs to check that the localization functor preserves connective objects (i.e. internal RHom out of Fib(A->B) needs to preserve connective objects). Of course one could imagine a notion of ``nonconnective analytic ring" where this is not required. $\endgroup$ May 2, 2021 at 20:08
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    $\begingroup$ The second reason is that one needs something which is a priori stronger than compctness of B, namely that internal RHom out of B commutes with colimits. (In the solid case it's the same because the compact objects are closed under tensor products, but we don't know whether this holds in the liquid case, and if it does hold it's at least for a very subtle reason.) $\endgroup$ May 2, 2021 at 20:08
  • $\begingroup$ $\mathcal O(\overline D)\to\mathcal O(\partial D)$ "looks like" $(\mathbb Z[T],\mathbb Z)_\blacksquare\to(\mathbb Z((T^{-1})),\mathbb Z)_\blacksquare$. What is the role of compactness? $\endgroup$
    – Z. M
    May 3, 2021 at 8:47
  • $\begingroup$ The functor $-\otimes_{\mathcal A} \mathcal A'$ is given by $RHom(\mathcal B/\mathcal A,-)$, so compactness is required for this to preserve colimits. (One can actually replace compact idempotents (with the conditions Dustin mentions) by idempotents in pro-compacts, and then some exist; but the ones we've looked at actually happen to give rise to induced analytic ring structures.) In your comment, what is $\mathcal O(\partial D)$? $\endgroup$ May 3, 2021 at 9:54

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