Reference: [Lectures on Analytic Geometry](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf)


Let $f\colon(\mathcal A,\mathcal M)\to(\mathcal B,\mathcal N)$ be a map of analytic ring. There are several possible ways to pose the flatness:


1. Flatness as the base change functor $D\_{\ge0}(\mathcal A,\mathcal M)\to D\_{\ge0}(\mathcal B,\mathcal N)$ maps static objects (i.e. those in the heart) to static objects.


I hope that I am not mistaken. The base change functor preserves nuclearity: since the forgetful functor preserves small colimits, the base change functor preserves compact objects. Furthermore, the base change functor preserves trace-class maps. It follows that the base change functor maps basic nuclear objects to basic nuclear, and therefore preserves nuclearity as well.


2. Flatness as the base change functor maps nuclear static objects to nuclear static objects.


In commutative algebra, the flatness guarantees the Tor-independence of pushout diagrams of rings. The analytic analogue does not seem to be true in general:


3. Flatness: for all maps $(\mathcal A,\mathcal M)\to(\mathcal C,\mathcal L)$ where $\mathcal C$ is static, the pushout $(\mathcal B,\mathcal N)\otimes\_{(\mathcal A,\mathcal M)}^L(\mathcal C,\mathcal L)$ is static.


The first implies the second. When the map $f$ is steady, the underlying condensed ring of the pushout coincides with $\mathcal C\otimes\_{(\mathcal A,\mathcal M)}^L(\mathcal B,\mathcal N)$, and in this case, the first implies the third. Is it reasonable to only consider steady maps?


I wonder which one should be the "correct" one to be considered in general, and more relations (like inverse implications under reasonable assumptions) between these three.




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**Update.** I thought that there is "the correct flatness" which works in nearly all situations, which does not seem to be the case, following Clausen's comment. In fact, I asked this question as a first step to understand whether classical algebro-geometric concepts like smooth maps, étale maps and syntomic maps generalize to all maps of analytic rings. This might be too optimistic, so maybe I could start with understanding the following special case:


Suppose that $A\to B$ is a flat map of discrete rings. Then when the map $A\_\blacksquare\to B\_\blacksquare$ satisfies the three conditions above respectively?



Flatness of maps of analytic rings
Flatness in analytic geometry is an interesting question! As Dustin says, it comes with several important caveats.


First, open immersions may not be flat even in the weakest sense of the word. Here is an instructive example. Let $K$ be your favourite analytic field ($\mathbb C$ or $\mathbb Q\_p$ will do) equipped with a reasonable analytic ring structure. Let $D$ be an open unit disc over $K$, which contains two disjoint open discs $D\_1,D\_2\subset D$. Then $M=\mathcal O(D\_1)/\mathcal O(D)$ is a nuclear $\mathcal O(D)$-module; note that by analytic continuation, the map $\mathcal O(D)\to \mathcal O(D\_1)$ is injective. After base change to $D\_2$, this becomes the quotient of $\mathcal O(D\_1\cap D\_2)$ by $\mathcal O(D\_2)$. As $D\_1\cap D\_2=\emptyset$, this is $0/^{\mathbb L} \mathcal O(D\_2) = \mathcal O(D\_2)[1]$. Thus, flatness fails, and there's no way to rescue this if you want a very general notion of "quasicoherent sheaves".


The good news is that at least the Tor-dimension for open immersions is bounded in these "usual" examples, by the dimension of the space.


Another interesting fact is due to my PhD student Lucas Mann. If $A\to B$ is a map of discrete rings that is syntomic (=flat + local complete intersection) and $A$ is a valuation ring, then $A\_{\blacksquare}\to B\_{\blacksquare}$ is of Tor-dimension $\leq 1$; if $A$ is a field, it is even flat. Here, I use "flat" (and Tor-dimension) in the sense of 1) in your question.


Regarding your last question: Note that if $A$ is discrete, then nuclear modules are just usual discrete $A$-modules, so usual flat maps of discrete rings are always flat in the sense of 2). For flatness in the sense of 3): Consideration of induced analytic ring structures shows that it implies flatness in the sense of 1). If one restricts to steady maps, it reduces to flatness in the sense of 1). For more general maps, I think this will get insanely hard to control.


