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The tempered disk and the tempered cohomology

This paper utilizes the ind-Banach framework for derived analytic geometry to define tempered tubular neighborhoods and a new tempered de Rham cohomology for smooth schemes over a residue field, ultimately proving its equivalence to crystalline cohomology for smooth proper schemes while providing a natural continuity interpretation for the transfer theorem of p-adic differential equations.

Original authors: Federico Bambozzi, Bruno Chiarellotto, Pietro Vanni

Published 2026-07-27

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CC BY 4.0

Original authors: Federico Bambozzi, Bruno Chiarellotto, Pietro Vanni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a radio station that is broadcasting from a very strange, foggy world. In this world, the rules of distance and sound are different from our own; it's a place called "p-adic geometry." Here, mathematicians study equations that describe how things change, but these equations behave in ways that seem impossible to us. Sometimes, the solutions to these equations (the "answers" to the math problems) grow so fast that they explode, or they grow so slowly they barely move. To understand these solutions, mathematicians usually use a tool called a "spectrum," which is like a map that shows you where the solutions live and how big their "radius of convergence" is (basically, how far you can trust the answer before it breaks down).

For a long time, this map had some blind spots. It was great at showing you where the solutions were "safe" and well-behaved, but it struggled to describe solutions that grew in a very specific, tricky way: logarithmic growth. Think of logarithmic growth like a plant that grows taller every day, but the amount it grows gets smaller and smaller, slowing down in a way that is hard to pin down on a standard map. The old maps were like rigid grids; they could show you a perfect circle or a square, but they couldn't easily show you these weird, fuzzy shapes where the rules of growth were changing. This made it hard for mathematicians to transfer information from one part of the map to another, like trying to copy a drawing from a piece of paper onto a piece of rubber that keeps stretching.

Now, enter a team of mathematicians who decided to build a brand new, super-fancy map. They didn't just draw a better grid; they changed the very material the map is made of. They used a framework called "derived analytic geometry" and "ind-Banach modules," which you can think of as a magical, stretchy fabric that can hold shapes the old maps couldn't. With this new fabric, they discovered a special kind of open space they call the "tempered disk." This disk is a cozy neighborhood in their new map where functions are allowed to grow logarithmically. It's like finding a special zone in a city where the speed limit isn't a fixed number, but a rule that changes smoothly as you drive.

The big discovery in this paper is that by using this new "tempered" map, a complicated rule about how these logarithmic solutions behave—known as the "Transfer Theorem"—suddenly becomes simple. In the old world, proving this theorem was like trying to explain why a shadow moves by calculating the angle of the sun, the height of the pole, and the texture of the ground all at once. In this new world, it's as simple as saying, "If the shadow is continuous on the map, it moves continuously." The authors show that these logarithmic growth conditions are just a natural, continuous part of the landscape.

Furthermore, they used this new map to build a fresh way of measuring the "shape" of geometric objects, which they call "tempered cohomology." Imagine you have a sculpture made of clay, and you want to know its shape. The old way was to measure it with a ruler that only worked for smooth, perfect surfaces. The new way uses a flexible tape measure that can hug the weird, logarithmic bumps. The authors prove that for smooth, closed shapes (like a perfect sphere or a donut), this new measurement gives the exact same result as the old, trusted methods (called crystalline and rigid cohomology). This is a huge deal because it means their new, fancy map isn't just a fantasy; it's a valid, reliable tool that agrees with the established facts.

However, they also admit that this is just the beginning. While their new map works perfectly for closed shapes, they haven't fully figured out how to use it for open shapes (like a bowl without a bottom) yet. They suggest that to do that, they will need to invent a new kind of "fast-converging" series, which is like a new type of glue to hold the edges of the map together. But for now, they have successfully shown that the "tempered disk" is a real, usable place in the mathematical universe, and that the strange, logarithmic growth of solutions is not a bug, but a feature that fits perfectly into a new, more flexible geometry.

Problem Statement
The paper addresses the geometric interpretation of logarithmic growth conditions in $p$ -adic differential equations and the development of a cohomology theory for schemes over a field $k$ of characteristic $p$ that naturally incorporates these conditions. In classical rigid analytic geometry and Berkovich spaces, information regarding radii of convergence can be transferred from generic points to classical points via continuity arguments, as these radii are represented by analytic domains. However, logarithmic growth conditions (related to the slopes of Frobenius structures) do not correspond to open analytic domains in the same direct way within classical frameworks. While transfer theorems for logarithmic growth exist (e.g., Dwork-Robba, Chiarellotto-Tsuzuki), they lack a unified geometric framework analogous to the study of radii of convergence. Furthermore, there is a need for a cohomology theory that serves as a "tempered" analogue to convergent rigid cohomology, potentially bridging the gap between convergent isocrystals and overconvergent isocrystals, and providing a geometric setting for $p$ -adic representations with specific monodromy properties.

Methodology
The authors employ the framework of derived analytic geometry based on ind-Banach algebras (specifically the category $\text{Ind}(\text{Ban}_R)$ ).

Ind-Banach Framework: They utilize the theory of ind-Banach modules over a non-Archimedean Banach ring $R$ , viewing them as a quasi-abelian category. They establish that the derived $\infty$ -category of these modules, $D_\infty(\text{Ind}(\text{Ban}_R))$ , forms a closed symmetric monoidal stable $\infty$ -category. They also relate this to the category of complete bornological vector spaces ( $\text{CBorn}_K$ ) via a dissection functor, allowing for the application of functional analytic methods.
Derived Analytic Spectrum: They define a spectrum functor $S(E)$ for a closed symmetric monoidal stable $\infty$ -category $E$ . This spectrum is constructed from the lattice of homotopy epimorphisms from the unit object. Unlike the Clausen-Scholze spectrum where these correspond to closed subsets, the authors adopt a "Hochster-dual" convention where homotopy epimorphisms correspond to compact open subsets. This allows for the construction of derived analytic spaces where structure sheaves can contain functions with specific growth conditions.
Tempered Power Series: A central technical tool is the introduction of the algebra of tempered power series, $R[[t]]_{\text{temp}}$ , defined as the ind-Banach colimit of series with logarithmic growth of order $n$ . This algebra defines an open subset of the derived analytic affine line, termed the tempered disk.
Tempered Tubes: For a smooth $k$ -scheme $X$ embedded in the special fiber of a smooth formal $V$ -scheme $\mathcal{P}$ , the authors define the tempered tube $]X[\mathcal{P}_{\text{temp}}$ as a derived analytic space. This is constructed by gluing affine tempered tubes, which are defined via quotients of tempered power series algebras by ideals generated by liftings of the defining equations of $X$ .
Cohomology Definition: The tempered convergent rigid cohomology is defined as the derived de Rham cohomology (specifically the Hodge-completed version) of the tempered tube. They utilize the analytic cotangent complex and the derived de Rham complex within the ind-Banach setting.

Key Contributions and Results

Geometric Interpretation of Transfer Theorems: The authors demonstrate that the classical transfer theorem for logarithmic growth of solutions to $p$ -adic differential equations can be reinterpreted as a simple continuity statement in the derived analytic setting. Specifically, the existence of solutions with log-growth at a generic point implies their existence at a classical point because the map between the corresponding spectra is continuous and the tempered disk is an open subset.
Construction of Tempered Disks and Decompositions: They explicitly construct the tempered open disk as an open subset of the derived analytic affine line $A^1_R$ . They also analyze the complement of the tempered disk at infinity, identifying it with the algebra of fast converging series ( $K[[t]]_{\text{fast}}$ ). They prove that the affine line can be covered by the punctured tempered disk at infinity and the fast converging disk at zero, with a non-empty intersection.
Independence of Embedding: A crucial result is the proof that the tempered tube $]X[\mathcal{P}_{\text{temp}}$ is independent of the choice of the regular embedding of $X$ into the formal scheme $\mathcal{P}$ and the choice of liftings of the defining equations. This relies on a tempered version of the "weak fibration theorem" (Theorem 8.19), showing that the tempered tube behaves well under smooth morphisms of formal schemes.
Comparison Theorem: For smooth proper $k$ -schemes, the authors establish a comparison theorem (Proposition 9.29) showing that the tempered convergent rigid cohomology is naturally isomorphic to the crystalline cohomology (and thus the rigid cohomology) tensored with the fraction field $K$ .
Poincaré Lemma: They prove a "Tempered Poincaré Lemma" (Lemma 9.28), establishing the strict exactness of the de Rham complex for tempered power series, which is essential for proving the independence of the cohomology from the embedding.

Significance
The paper claims to provide a natural analytic setting for studying the continuity of logarithmic growth conditions, analogous to how Berkovich spaces handle radii of convergence. By introducing tempered tubes and tempered cohomology, the authors offer a refined geometric framework that sits between classical rigid geometry and more general derived settings.

Refinement of Spectrum: The approach yields a spectrum richer than classical rigid geometry, accommodating open subsets whose structure sheaves contain functions of logarithmic growth.
Unification: It unifies the study of $p$ -adic differential equations with geometric cohomology theories, interpreting transfer theorems as continuity properties.
Future Directions: The authors suggest that this framework could provide a geometric foundation for $p$ -adic representations with finite local monodromy (which correspond to overconvergent isocrystals) and other intermediate types of representations associated with logarithmic decay series. They note that extending this theory to non-proper varieties would require a tempered analogue of Monsky-Washnitzer cohomology, involving logarithmic decay at infinity, which is planned for future work.

The paper does not claim to solve the general Riemann-Hilbert correspondence for all $p$ -adic representations but positions its construction as a necessary geometric step toward such a goal, particularly for representations that do not fit strictly into the convergent or overconvergent categories.

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