REVIEW 5 minor 1 cited by
Cohen's theorem in tensor triangular geometry
T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper proves that in an essentially small tensor triangulated category with weakly Noetherian Balmer spectrum, finite generation of every radical ideal (equivalently every prime ideal) forces the spectrum to be finite.
desk verdict A short, correct, and useful tt-geometric analogue of Cohen's theorem, cleanly proven from standard spectral-space facts; the only unverifiable piece is an illustrative example. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Three mechanisms carry the argument. First, Balmer's classification theorem identifies radical ideals with Thomason subsets of the Balmer spectrum, and Lemma 5 shows that a radical ideal is finitely generated exactly when the complement of its support is constructible. Second, Hochster duality rewrites Thomason subsets as inverse-closed subsets of the inverse spectral space, so finite generation of all radical ideals becomes the statement that every inverse-closed subset is constructible; Proposition 6 identifies this with inverse-Noetherianity and with finite generation of all prime ideals. Third, Lemma 3 is the point-set hinge: a spectral space is finite if and only if it is weakly Noetherian and inverse-Noetherian, because weak Noetherianity makes every point locally closed in the inverse topology and inverse-Noetherianity then forces the patch space to be discrete and compact, hence finite.
What would settle it
Find an essentially small tt-category $\mathscr{K}$ whose Balmer spectrum $\mathrm{Spc}(\mathscr{K})$ is weakly Noetherian and inverse-Noetherian but has infinitely many points. Equivalently, exhibit a weakly Noetherian, infinite Balmer spectrum in which every radical ideal is generated by a single object; the main theorem predicts no such category exists.
Extended reading notes
Core claim
The central discovery is Theorem 7: for an essentially small tt-category $\mathscr{K}$ with weakly Noetherian Balmer spectrum, the following are equivalent: (a) every radical ideal in $\mathscr{K}$ is finitely generated; (b) every prime ideal in $\mathscr{K}$ is finitely generated; (c) $\mathrm{Spc}(\mathscr{K})$ is finite. The equivalence of (a) and (b) is unconditional and forms the tt-geometric version of Cohen's theorem, while the bridge to finiteness is a new point-set characterization of finite spectral spaces. As a direct consequence, any tt-stratified rigidly-compactly generated tt-category with infinite Balmer spectrum must contain a prime ideal of compact objects that is not finitely generated.
Load-bearing premise
The load-bearing premise is a point-set topological fact: any spectral space that is both weakly Noetherian and inverse-Noetherian must actually be finite. If this fact fails, the chain from finitely generated ideals to finiteness of the prime spectrum collapses.
Editorial extensions
If this is right
- If the main theorem is correct, then in every tt-stratified rigidly-compactly generated tt-category whose spectrum is infinite, there exists a prime ideal of compact objects that cannot be generated by finitely many objects.
- For the derived category of a Noetherian commutative ring $R$, every ideal in $D(R)^\omega$ is finitely generated exactly when $\mathrm{Spec}(R)$ is finite; the tt-theoretic analogue therefore does not detect Noetherianity of $R$ itself.
- For finite $p$-local spectra, the thick subcategory theorem gives an infinite spectrum in which every thick tensor ideal is generated by a single object, showing that the weakly Noetherian hypothesis is essential in Theorem 7.
- For equivariant spectra of compact Lie groups, all ideals are finitely generated when the group is finite and not when the group is infinite.
- For the rational global spectra of finite elementary abelian $p$-groups, the spectrum is infinite and inverse-Noetherian, so every prime ideal is principal even though the spectrum is not weakly Noetherian.
Reading between the lines
- The theorem suggests that finite generation of all ideals is not a useful analogue of Noetherianity in tt-geometry; any attempt to formulate a 'Noetherian-like' finiteness condition should look at weaker or differently shaped generation properties to avoid collapsing to finiteness.
- Because Proposition 6 is unconditional, one can decide whether a tt-category has a non-finitely-generated ideal purely from the inverse topology of its Balmer spectrum; this gives a topological search strategy for finding such ideals in new examples.
- A testable extension would be to ask whether a bounded-cardinality version of generation, rather than finite generation, still forces finiteness of weakly Noetherian spectra, or whether intermediate cardinalities allow infinite but controlled spectra.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This note proves Cohen-type equivalences for tensor triangulated categories. The main result (Theorem 7) shows that for an essentially small tt-category K, Spc(K) is weakly Noetherian and all radical ideals are finitely generated (equivalently, all prime ideals are finitely generated) exactly when Spc(K) is finite. The proof combines Proposition 6, which uses Balmer's classification and Hochster duality to identify finite generation of all radical ideals with inverse-Noetherianness of the spectrum, with Lemma 3, a point-set statement saying that a spectral space is finite iff it is weakly Noetherian and inverse-Noetherian. The paper also derives a stratification consequence (Corollary 10) and gives several examples, including p-local finite spectra, equivariant spectra, and a conditional example from global homotopy theory.
Significance. If correct, the paper provides a clean and citable characterization: under a mild topological hypothesis, categorical finite generation of all radical ideals collapses to finiteness of the Balmer spectrum. The proofs are short and transparent, with careful citations to Balmer's classification and to Dickmann-Schwartz-Tressl; Proposition 6 and Lemma 3 are well matched, and Lemma 3 is a neat point-set fact of independent interest. The stratification consequence is a natural and attractive payoff. The paper introduces no ad hoc parameters and its central derivation rests on standard material, which makes the result credible and convenient for future work.
minor comments (5)
- [Topological preliminaries, Lemma 3] The proof invokes [DST19, Corollary 8.1.19(i)] without stating the content of that result. Since the step from local closedness of points to discreteness of the patch space is load-bearing, please quote the corollary or state explicitly the property being used.
- [Proposition 6] The cited equivalence of items (i), (iii), and (v) of [DST19, Proposition 8.1.11] is not reproduced. The reader cannot check that (iii) means 'every inverse-closed subset is constructible' and (v) means 'gen(P) is constructible for every P' without consulting the book; please restate those equivalences or give a precise reference to the relevant statements.
- [Example 11] The sentence 'D(R) is rigidly-compactly generated so that every ideal in D(R)^ω is radical' needs a reference or a brief justification; rigidity alone is not an automatic guarantee of radicality for all thick tensor ideals in an arbitrary tt-category, and the reader may not know the specific classification fact for D(R).
- [Example 13] The key homeomorphism in Equation (14) is attributed to 'forthcoming joint work' without a preprint identifier or a statement of the precise conditions under which it is established. Since this example is presented as the initial inspiration, please make its conditional status explicit; this does not affect the main theorem.
- [Throughout] There are several presentation issues: the notation alternates between 'supp' and 'Supp', the display in Example 12 appears garbled, and the term 'standard' in Example 13 is used without definition.
Circularity Check
No significant circularity: the main equivalence is assembled from independent point-set and Balmer-spectrum theorems, with self-citations serving only as definitions or context, not as load-bearing derivation steps.
full rationale
The derivation chain runs from Definition 1 (weakly Noetherian, quoted from the author's earlier [BHS23]) as a hypothesis, not as a conclusion. Lemma 2 and Lemma 3 supply the point-set bridge: a spectral space is finite iff it is weakly Noetherian and inverse-Noetherian, with the forward direction using the external [DST19, Corollary 8.1.19(i)] and compactness of the patch space. Lemma 5 translates finite generation of a radical ideal into constructibility of the support complement via Balmer's classification [Bal05, Theorem 4.10] and [DST19, Theorem 1.5.4(iii)]. Proposition 6 then invokes [DST19, Proposition 8.1.11] to equate inverse-Noetherianness with constructibility of all inverse-closed subsets and of the generalization closures of points; the paper's contribution is the translation into ideal generation, not a circular restatement. Theorem 7 composes Proposition 6 with Lemma 3, with no step assuming the target result or defining a key term in terms of the conclusion. The self-citations ([BHS23] for weakly Noetherian; [BBG23], [BCHS24], [BGH20] in examples) are used for definitions or external theorems and do not close a derivation loop. The only deferred support is Example 13's homeomorphism (14), attributed to forthcoming joint work; that example is explicitly illustrative and is not used in the proof of Theorem 7, Proposition 6, or Corollary 10, so it does not affect the central derivation. No fitted parameters, renamed predictions, or imported uniqueness claims occur. The central claim is therefore self-contained against external point-set and tt-geometry results.
Assumptions & free parameters
assumptions (4)
- domain assumption Balmer's classification theorem ([Bal05, Thm 4.10]): radical thick tensor ideals of an essentially small tt-category K are in bijection with Thomason subsets of Spc(K) via support.
- standard math Hochster duality and the spectral space toolkit from DST19, in particular Prop 8.1.11 and Cor 8.1.19(i), characterizing inverse-Noetherian spaces and locally closed points.
- domain assumption The notion of weakly Noetherian spectral space and its characterization (weakly visible points) from BHS23, used as a hypothesis.
- domain assumption Zou's theorem (Zou23, Thm 8.13): any tt-stratified rigidly-compactly generated tt-category has weakly Noetherian spectrum.
Cite this review
Pith. "Pith review of Cohen's theorem in tensor triangular geometry." pith.science (2026). https://pith.science/paper/5V26C5SV
@misc{pith2026250515786,
author = {Pith},
title = {Pith review of: Cohen's theorem in tensor triangular geometry},
year = {2026},
howpublished = {\url{https://pith.science/paper/5V26C5SV}},
note = {Machine review of arXiv:2505.15786}
}
abstract
A theorem of Cohen from 1950 states that a commutative ring is Noetherian if and only if every prime ideal is finitely generated. In this note, we establish analogues of this result in tensor triangular geometry. In particular, for an essentially small tensor triangulated category $\mathscr{K}$ with weakly Noetherian spectrum, we show that every prime ideal in $\mathscr{K}$ can be generated by finitely many objects if and only if the set of prime ideals of $\mathscr{K}$ is finite.
Forward citations
Cited by 1 Pith paper
-
The spectrum of global representations for families of bounded rank and VI-modules
Compact derived VI-modules are classified by their support types, and the Balmer spectrum for families of bounded-rank abelian p-groups is computed explicitly.
Reference graph
Works this paper leans on
-
[1]
The spectrum of excisive functors
Gregory Arone , Tobias Barthel , Drew Heard , and Beren Sanders . The spectrum of excisive functors . arXiv e-prints, accepted for publication in Invent. Math. , page arXiv:2402.04244, February 2024
-
[2]
The spectrum of prime ideals in tensor triangulated categories
Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. , 588:149--168, 2005
2005
-
[3]
Profinite equivariant spectra and their tensor-triangular geometry
Scott Balchin , David Barnes , and Tobias Barthel . Profinite equivariant spectra and their tensor-triangular geometry . arXiv e-prints , page arXiv:2401.01878, January 2024
arXiv 2024
-
[4]
Scott Balchin , Tobias Barthel , and J. P. C. Greenlees . Prismatic decompositions and rational G -spectra . arXiv e-prints , page arXiv:2311.18808, November 2023
arXiv 2023
-
[5]
Iyengar , Henning Krause , and Julia Pevtsova
Tobias Barthel , Dave Benson , Srikanth B. Iyengar , Henning Krause , and Julia Pevtsova . Lattices over finite group schemes and stratification . arXiv e-prints , page arXiv:2307.16271, July 2023
arXiv 2023
-
[6]
Quillen stratification in equivariant homotopy theory
Tobias Barthel, Nat\`alia Castellana, Drew Heard, Niko Naumann, and Luca Pol. Quillen stratification in equivariant homotopy theory. Invent. Math. , 239(1):219--285, 2025
2025
-
[7]
Cosupport in tensor triangular geometry
Tobias Barthel , Natalia Castellana , Drew Heard , and Beren Sanders . Cosupport in tensor triangular geometry . arXiv e-prints, accepted for publication in Ast\'erisque , page arXiv:2303.13480, March 2023
arXiv 2023
-
[8]
On surjectivity in tensor triangular geometry
Tobias Barthel, Nat\`alia Castellana, Drew Heard, and Beren Sanders. On surjectivity in tensor triangular geometry. Math. Z. , 308(4):Paper No. 65, 7, 2024
2024
Show all 23 references
-
[9]
Generalized tensor idempotents and the telescope conjecture
Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3) , 102(6):1161--1185, 2011
2011
-
[10]
The tt-geometry of permutation modules
Paul Balmer and Martin Gallauer . The tt-geometry of permutation modules. Part I: Stratification . arXiv e-prints , page arXiv:2210.08311, October 2022
2022 arXiv
-
[11]
The spectrum of A rtin motives
Paul Balmer and Martin Gallauer. The spectrum of A rtin motives. Trans. Amer. Math. Soc. , 378(3):1733--1754, 2025
2025
-
[12]
Tobias Barthel, J. P. C. Greenlees, and Markus Hausmann. On the B almer spectrum for compact L ie groups. Compos. Math. , 156(1):39--76, 2020
2020
-
[13]
Stratification in tensor triangular geometry with applications to spectral M ackey functors
Tobias Barthel, Drew Heard, and Beren Sanders. Stratification in tensor triangular geometry with applications to spectral M ackey functors. Camb. J. Math. , 11(4):829--915, 2023
2023
-
[14]
Benson, Srikanth B
David J. Benson, Srikanth B. Iyengar, and Henning Krause. Stratifying modular representations of finite groups. Ann. of Math. (2) , 174(3):1643--1684, 2011
2011
-
[15]
I. S. Cohen. Commutative rings with restricted minimum condition. Duke Math. J. , 17:27--42, 1950
1950
-
[16]
Spectral spaces , volume 35 of New Mathematical Monographs
Max Dickmann, Niels Schwartz, and Marcus Tressl. Spectral spaces , volume 35 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2019
2019
-
[17]
Hochster
M. Hochster. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. , 142:43--60, 1969
1969
-
[18]
Hopkins and Jeffrey H
Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II . Ann. of Math. (2) , 148(1):1--49, 1998
1998
-
[19]
The connection between the K -theory localization theorem of T homason, T robaugh and Y ao and the smashing subcategories of B ousfield and R avenel
Amnon Neeman. The connection between the K -theory localization theorem of T homason, T robaugh and Y ao and the smashing subcategories of B ousfield and R avenel. Ann. Sci. \'Ecole Norm. Sup. (4) , 25(5):547--566, 1992
1992
-
[20]
William T. Sanders . Support and vanishing for non-Noetherian rings and tensor triangulated categories . arXiv e-prints , page arXiv:1710.10199, October 2017
2017 arXiv
-
[21]
Global homotopy theory , volume 34 of New Mathematical Monographs
Stefan Schwede. Global homotopy theory , volume 34 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2018
2018
-
[22]
R. W. Thomason. The classification of triangulated subcategories. Compositio Math. , 105(1):1--27, 1997
1997
-
[23]
Support theories for non-Noetherian tensor triangulated categories
Changhan Zou . Support theories for non-Noetherian tensor triangulated categories . arXiv e-prints , page arXiv:2312.08596, December 2023
2023 arXiv
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.