arxiv: 2605.03655 · v1 · submitted 2026-05-05 · 🧮 math.AG · math.CT· math.FA· math.NT

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Lectures on Analytic Geometry

Peter Scholze

Pith reviewed 2026-05-07 14:20 UTC · model grok-4.3

classification 🧮 math.AG math.CTmath.FAmath.NT

keywords liquid real vector spacesanalytic geometryadic spacescomplex-analytic spacesanalytic stacksreal vector spaces

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The pith

Liquid real vector spaces provide the foundation for a new approach to complex-analytic geometry that also covers adic spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

These lecture notes develop the basic theory of liquid real vector spaces as part of joint work with Dustin Clausen. The first half lays out this theory and shows how it supports a fresh treatment of complex-analytic geometry. The second half proposes a tentative category of analytic spaces that includes adic spaces and complex-analytic spaces. Although the precise definition was later abandoned as a step toward analytic stacks, the surrounding material continues to offer useful perspectives on unifying different kinds of analytic structures. A sympathetic reader would value the notes for supplying concrete tools to handle analytic spaces in a single framework.

Core claim

The notes develop the basic theory of liquid real vector spaces, which we used in another course to give a new approach to complex-analytic geometry, and give a tentative definition of a category of analytic spaces that contains adic spaces and complex-analytic spaces, while noting that this definition is an abandoned stepping stone on the way to analytic stacks.

What carries the argument

Liquid real vector spaces, which carry the constructions needed for the new approach to analytic geometry, together with the tentative category of analytic spaces that unifies adic spaces and complex-analytic spaces.

If this is right

The theory of liquid real vector spaces directly enables the new constructions in complex-analytic geometry.
The tentative category supplies a single setting in which both adic spaces and complex-analytic spaces can be treated together.
Much of the surrounding discussion continues to guide work toward the eventual definition of analytic stacks.
The notes supply a stable reference point for anyone working with liquid vector spaces in geometric contexts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the liquid vector space formalism scales cleanly, it could be tested on other geometric settings such as rigid analytic spaces over non-archimedean fields.
The emphasis on a unified category may suggest ways to incorporate additional structures like derived or higher-categorical versions without starting from scratch.
Future refinements might replace the abandoned definition with a stack-theoretic version while preserving the liquid vector space foundation.

Load-bearing premise

The surrounding discussion around the abandoned tentative definition of analytic spaces remains relevant and useful for future work on analytic stacks despite the definition itself being set aside.

What would settle it

A concrete check that liquid real vector spaces cannot carry the expected operations for complex-analytic geometry, or that the tentative category fails to contain both adic spaces and complex-analytic spaces as full subcategories, would show the presented framework does not achieve its stated aims.

Figures

Figures reproduced from arXiv: 2605.03655 by Peter Scholze.

**Figure 1.** Figure 1: The Berkovich space M(Z) view at source ↗

read the original abstract

This is a slightly updated version of lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. This is intended as a stable citable version of the material. In the first half of this course, we develop the basic theory of liquid real vector spaces, which we used in another course to give a new approach to complex-analytic geometry. In the second half, we gave a tentative definition of a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces. While the precise definition of analytic spaces represents an abandoned stepping stone on our way to define analytic stacks and hence should be seen as a historical artifact, much of the surrounding discussion stays very relevant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

These are Scholze's 2019/20 Bonn lecture notes documenting the development of liquid real vector spaces and an early tentative definition of analytic spaces that was later dropped in favor of analytic stacks.

read the letter

The main thing to know is that this is expository material from a course, not a research paper pushing a new theorem. The first half works through the basic theory of liquid real vector spaces, which Scholze and Clausen then applied to complex-analytic geometry in other work. That section gives a clear, citable record of the constructions and should be useful for anyone trying to follow their program on analytic geometry and number theory. The notes are careful about citing the joint work with Clausen as the source, and they avoid claiming finality for the second half. The surrounding discussion around the tentative analytic spaces category, which includes adic spaces and complex-analytic spaces, is flagged as historical. The abstract states outright that the precise definition was an abandoned stepping stone on the way to analytic stacks, so readers can treat it as context rather than current machinery. No load-bearing claims are left un-caveated, and there is no sign of circularity or invented entities passed off as settled. The material is aimed at people already engaged with the Scholze-Clausen analytic geometry project who want to see how the ideas developed step by step. It does not stand alone as a self-contained research contribution, so it would not be a candidate for formal peer review in the usual sense. The liquid vector spaces part is worth reading if you are tracking that line of work; the rest is mainly of historical interest.

Referee Report

0 major / 2 minor

Summary. The manuscript consists of lecture notes from a 2019/20 course at the University of Bonn on analytic geometry, part of joint work with Dustin Clausen. The first half develops the basic theory of liquid real vector spaces, which the authors used elsewhere to approach complex-analytic geometry. The second half presents a tentative definition of a category of analytic spaces containing adic spaces and complex-analytic spaces; this definition is explicitly described as an abandoned intermediate step toward analytic stacks, with the surrounding discussion retained for its potential relevance.

Significance. These notes provide a stable, citable record of foundational material on liquid real vector spaces that has already been applied in subsequent work on complex-analytic geometry. The explicit framing of the second-half construction as historical context rather than a current claim reduces the risk of overinterpretation and may aid researchers tracing the development of analytic stacks.

minor comments (2)

The abstract states that the tentative definition 'should be seen as a historical artifact' while claiming surrounding discussion 'stays very relevant'; a brief paragraph in the introduction (or a dedicated subsection) explicitly mapping which parts of the second half remain applicable to current analytic-stack constructions would improve clarity for readers.
Notation for liquid real vector spaces is introduced without a consolidated reference table or index; adding one (even a short one) would help readers cross-reference definitions across the two halves of the notes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The referee's summary correctly captures both the scope of the lecture notes and their status as a historical record rather than a current claim.

Circularity Check

0 steps flagged

Expository lecture notes with self-contained foundational development

full rationale

The manuscript is framed explicitly as lecture notes documenting an intermediate, historical construction (the tentative analytic spaces category, now abandoned in favor of analytic stacks). It develops the theory of liquid real vector spaces as foundational material used elsewhere, with transparent caveats that the second-half definition is a stepping stone rather than a current claim. No derivation chain reduces by construction to fitted inputs, self-definitions, or load-bearing self-citations; the content is presented as independent exposition of joint work, with no equations or theorems asserted as predictions that collapse to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The notes introduce liquid real vector spaces as a new foundational object and a tentative category of analytic spaces; these appear to be the main invented entities. No explicit free parameters or standard axioms are mentioned in the abstract.

invented entities (2)

liquid real vector spaces no independent evidence
purpose: To provide a basic theory usable for a new approach to complex-analytic geometry
Described in the abstract as the core of the first half of the course; treated as a novel construction in the joint work.
category of analytic spaces no independent evidence
purpose: Tentative definition intended to contain adic spaces and complex-analytic spaces
Presented in the abstract as the second half of the material, later abandoned in favor of analytic stacks.

pith-pipeline@v0.9.0 · 5425 in / 1354 out tokens · 49536 ms · 2026-05-07T14:20:51.468959+00:00 · methodology

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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math.NT 2026-05 unverdicted novelty 8.0

The Weil-Moore anima refines the Weil group into a space with higher homotopy groups to improve its cohomological behavior for number fields.

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Works this paper leans on

38 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

Andreychev, Pseudocoherent and P erfect C omplexes and V ector B undles on A nalytic A dic S paces , arXiv:2105.12591, 2021

G. Andreychev, Pseudocoherent and P erfect C omplexes and V ector B undles on A nalytic A dic S paces , 2021, https://arxiv.org/abs/2105.12591

work page arXiv 2021
[2]

Aoki, (semi)topological K -theory via solidification , 2024, https://arxiv.org/abs/2409.01462

K. Aoki, (semi)topological K -theory via solidification , 2024, https://arxiv.org/abs/2409.01462

work page arXiv 2024
[3]

Avil\' e s, F

A. Avil\' e s, F. C. S\' a nchez, J. M. F. Castillo, M. Gonz\' a lez, and Y. Moreno, Separably injective B anach spaces , Lecture Notes in Mathematics, vol. 2132, Springer, [Cham], 2016

2016
[4]

Arhangel'skii and M

A. Arhangel'skii and M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008

2008
[5]

Ben-Bassat and K

O. Ben-Bassat and K. Kremnizer, Non- A rchimedean analytic geometry as relative algebraic geometry , Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 1, 49--126

2017
[6]

Beilinson, Topological -factors , Pure Appl

A. Beilinson, Topological -factors , Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson. Part 3, 357--391

2007
[7]

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972

1972
[8]

Bhatt and P

B. Bhatt and P. Scholze, The pro-\'etale topology for schemes, Ast\'erisque (2015), no. 369, 99--201

2015
[9]

Commun., vol

Alain Connes and Caterina Consani, Universal thickening of the field of real numbers, Advances in the theory of numbers, Fields Inst. Commun., vol. 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, pp. 11--74

2015
[10]

R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, A theory of complex interpolation for families of B anach spaces , Adv. in Math. 43 (1982), no. 3, 203--229

1982
[11]

Cembranos, C(K,\,E) contains a complemented copy of c_ 0 , Proc

P. Cembranos, C(K,\,E) contains a complemented copy of c_ 0 , Proc. Amer. Math. Soc. 91 (1984), no. 4, 556--558

1984
[12]

Math., vol

Alain Connes, The W itt construction in characteristic one and quantization , Noncommutative geometry and global analysis, Contemp. Math., vol. 546, Amer. Math. Soc., Providence, RI, 2011, pp. 83--113. 2815131

2011
[13]

C esnavi c ius and P

K. C esnavi c ius and P. Scholze, Purity for flat cohomology, arXiv:1912.10932, 2019

work page arXiv 1912
[14]

Commelin and P

J. Commelin and P. Scholze, Liquid tensor experiment: B lueprint for the reduction to T heorem 9.4 , https://math.commelin.net/files/LTE.pdf, 2021

2021
[15]

Clausen and P

D. Clausen and P. Scholze, C ondensed M athematics and C omplex G eometry , https://people.mpim-bonn.mpg.de/scholze/Complex.pdf, 2022

2022
[16]

, Analytic stacks, 2026, book in preparation

2026
[17]

Cabello S\' a nchez, J

F. Cabello S\' a nchez, J. M. F. Castillo, and N. J. Kalton, Complex interpolation and twisted twisted H ilbert spaces , Pacific J. Math. 276 (2015), no. 2, 287--307

2015
[18]

Diestel, J

J. Diestel, J. H. Fourie, and J. Swart, The metric theory of tensor products, American Mathematical Society, Providence, RI, 2008, Grothendieck's r\' e sum\' e revisited

2008
[19]

Dold and D

A. Dold and D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen. , Ann. Inst. Fourier 11 (1961), 201--312 (German)

1961
[20]

A. I. Efimov, K-theory and localizing invariants of large categories, https://arxiv.org/abs/2405.12169, 2025

work page arXiv 2025
[21]

Enflo, A counterexample to the approximation problem in B anach spaces , Acta Math

P. Enflo, A counterexample to the approximation problem in B anach spaces , Acta Math. 130 (1973), 309--317

1973
[22]

Grothendieck, Produits tensoriels topologiques et espaces nucl\' e aires , Mem

A. Grothendieck, Produits tensoriels topologiques et espaces nucl\' e aires , Mem. Amer. Math. Soc. No. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140

1955
[23]

Harbater, Convergent arithmetic power series, Amer

D. Harbater, Convergent arithmetic power series, Amer. J. Math. 106 (1984), no. 4, 801--846

1984
[24]

Huber, Continuous valuations, Math

R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455--477

1993
[25]

N. J. Kalton, Convexity, type and the three space problem, Studia Math. 69 (1980/81), no. 3, 247--287

1980
[26]

K. S. Kedlaya and R. Liu, Relative p -adic H odge theory, II : Imperfect period rings , arXiv:1602.06899v3, 2019

work page arXiv 2019
[27]

N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1--30

1979
[28]

Lurie, Higher topos theory, Annals of Mathematics Studies, vol

J. Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009

2009
[29]

, Higher A lgebra , https://www.math.ias.edu/ lurie/papers/HA.pdf, 2017

2017
[30]

, Spectral algebraic geometry, www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf, 2018

2018
[31]

Mac Lane, Homologie des anneaux et des modules, Colloque de topologie alg\'ebrique, L ouvain, 1956, Georges Thone, Li\`ege, 1957, pp

S. Mac Lane, Homologie des anneaux et des modules, Colloque de topologie alg\'ebrique, L ouvain, 1956, Georges Thone, Li\`ege, 1957, pp. 55--80

1956
[32]

Nikolaus and P

T. Nikolaus and P. Scholze, On topological cyclic homology, Acta Math. 221 (2018), no. 2, 203--409

2018
[33]

Ribe, Examples for the nonlocally convex three space problem, Proc

M. Ribe, Examples for the nonlocally convex three space problem, Proc. Amer. Math. Soc. 73 (1979), no. 3, 351--355

1979
[34]

Rosick\' y , On homotopy varieties, Adv

J. Rosick\' y , On homotopy varieties, Adv. Math. 214 (2007), no. 2, 525--550

2007
[35]

Scholze, Lectures on C ondensed M athematics , https://people.mpim-bonn.mpg.de/scholze/Condensed.pdf, 2019

P. Scholze, Lectures on C ondensed M athematics , https://people.mpim-bonn.mpg.de/scholze/Condensed.pdf, 2019

2019
[36]

, Geometrization of the real l ocal L anglands C orrespondence , https://people.mpim-bonn.mpg.de/scholze/RealLocalLanglands.pdf, 2024

2024
[37]

M. F. Smith, The P ontrjagin duality theorem in linear spaces , Ann. of Math. (2) 56 (1952), 248--253

1952
[38]

To\" e n and G

B. To\" e n and G. Vezzosi, Homotopical algebraic geometry. II . G eometric stacks and applications , Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. 2394633

2008