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Duality in tensor-triangular geometry via proxy-smallness

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Pith's one-line read A proxy-small geometric functor satisfies Grothendieck duality on all f_*-rigid objects, even when its right adjoint does not preserve compact objects.

desk verdict A substantial, well-written framework for duality in tt-geometry via proxy-smallness, with one genuine but repairable proof gap in Proposition 3.23; the main theorems hold up and the paper deserves serious refereeing. read the letter →

arxiv 2510.24415 v2 pith:VLZIKBLV submitted 2025-10-28 math.CT math.ACmath.ATmath.RT

Thomas Peirce , Jordan Williamson This is my paper · ORCID
classification math.CTmath.ACmath.ATmath.RT MSC 18G8018M0555P42
keywords tensor-triangulargeometryproxy-smallnessGrothendieckdualityGorensteinMatlisdualisingobjectstorsionandcompletionfunctorsPicardgrouprigid
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper replaces a restrictive hypothesis in tensor-triangular duality—that a geometric functor's right adjoint preserves compact objects—with the weaker condition that the functor be proxy-small. Under this condition it proves Grothendieck duality on the canonical subcategory of f_*-rigid objects, for which the image under f_* is rigid, and it classifies the rigid and invertible objects in the associated torsion category. This yields a uniform perspective on local algebra, chromatic and equivariant stable homotopy theory, and the Gorenstein property, capturing Matlis duality, Gorenstein duality, and Watanabe's theorem on invariant rings. The paper also introduces Matlis dualising objects and Gorenstein geometric functors, and relates orientability of Matlis lifts to the difference between the Gorenstein property and Gorenstein duality.

What carries the argument

The load-bearing device is the proxy-small geometric functor: a geometric functor $f_* : C \to D$ such that the unit $1_D$ is proxy-small relative to $f_*$, meaning $f_* 1_D$ is proxy-small in $C$ and the localising tensor ideal generated by the image of $f_*1_D$ under $f^*$ is all of $D$. Proxy-smallness is a weak finiteness condition—an object is proxy-small when its tensor-triangular information is captured by compact objects, as with the residue field of a local ring witnessed by a Koszul complex. This hypothesis lets the paper construct the torsion category $\Gamma_f C = \mathrm{Loc}^{\otimes}(f_* 1_D)$ and the completion category $\Lambda_f C$, with the MGM equivalence between them, and to factor $f_*$ through $\Gamma_f C$. The proof of Grothendieck duality then runs by comparing rigidity in $\Gamma_f C$, $\Lambda_f C$, and $D$, using the fact that strong monoidal functors preserve rigid objects and that $f_*$-isomorphisms can be detected by tensoring with a proxy-smallness witness.

What would settle it

Produce an enhanced geometric functor $f_* : C \to D$ for which $f_* 1_D$ is proxy-small in $C$ but $\mathrm{Loc}^{\otimes}(f^* f_* 1_D) \neq D$; if such a functor exists, Theorem 3.24 is false and the paper's main theorems, while conditionally true, would no longer apply to any example whose proxy-smallness was verified only through Theorem 3.24. A natural place to look is a map of commutative ring spectra where the ring is proxy-small but the induced module category is not generated by the image of the unit.

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Extended reading notes

Core claim

The central claim is Theorem 4.23: if $f_* : C \to D$ is a proxy-small geometric functor and $M \in C$ is such that $f_* M$ is rigid in $D$, then there is a natural isomorphism $f_! X \otimes f_* M \simeq f_!(X \otimes M)$ for all $X$, and in particular $\omega_f \otimes f_* M \simeq f_! M$ with $\omega_f = f_! 1_C$. This says Grothendieck duality holds on the full subcategory of $f_*$-rigid objects even when the right adjoint does not preserve compact objects. The paper further claims that an object is $f_*$-rigid exactly when its torsion, completion, and $f_*$-image are rigid (Theorem 4.14), and that, for Gorenstein proxy-small functors with pure-semisimple target, Matlis dualising objects are exactly those whose $f_*$-image is invertible (Theorem 6.18).

Load-bearing premise

The main theorems rest on the assumption that proxy-smallness of $f_* 1_D$ automatically implies a compatibility condition (that the left-adjoint image of $f_* 1_D$ generates the target category), and the argument for this automaticity is terse; if it fails, the conditional theorems survive but the examples must be checked by hand.

Editorial extensions

If this is right

  • Grothendieck duality $\omega_f \otimes f_* M \simeq f_! M$ holds for every $M \in C$ whose image $f_* M$ is rigid, so the duality formula is available without any compact-preservation assumption.
  • Rigidity transfers across the torsion/completion boundary: $M$ is $f_*$-rigid if and only if $\Gamma_f M$ is rigid in $\Gamma_f C$, if and only if $\Lambda_f M$ is rigid in $\Lambda_f C$, if and only if $f_* M$ is rigid in $D$.
  • Invertibility transfers similarly (Theorem 5.8), giving abstract criteria for when torsion or completion objects are invertible; this recovers the $K(n)$-local invertibility criterion and the local algebra dualisability results.
  • For a Gorenstein proxy-small functor with pure-semisimple target, the Matlis dualising objects are classified: $f_! M$ is invertible if and only if $f_* M$ is invertible, if and only if $\Gamma_f M$ and $\Lambda_f M$ are invertible (Theorem 6.18).
  • Gorenstein duality holds exactly when a specific Matlis lift of $\omega_f$ is orientable (Proposition 8.9), and automatic orientability makes the Gorenstein condition equivalent to Gorenstein duality; this is what powers the formal-DGA and invariant-theory applications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The transfer principle '$\Gamma_f X$ has property P iff $f_* X$ has property P' is shown for rigidity and invertibility but not reflexivity (Remark 5.12); a natural next step is to determine exactly which objects properties satisfy such a transfer, starting with compactness or proxy-smallness themselves.
  • Theorem 4.23 gives a partial answer to the question of forcing Grothendieck duality by localising the source; one could now ask whether the subcategory of $f_*$-rigid objects is the largest full subcategory on which $\omega_f \otimes f_* M \simeq f_! M$ holds, and whether it has an intrinsic description in the Balmer spectrum.
  • The fixed-point argument in Proposition 10.20 is a template for producing new Gorenstein duality statements from equivariant ones; applying it to compact Lie groups with non-trivial adjoint action should yield non-equivariant duality statements that are not visible from the non-equivariant Gorenstein condition alone.
  • The paper leaves open whether relative proxy-smallness is independent of the chosen witness in general (Remark 3.10); a positive answer would simplify Definition 3.15 and would make the compatibility condition purely a statement about $f_* 1_D$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper develops a tensor-triangular framework for duality phenomena, centred on the new notion of a proxy-small geometric functor. The main theorems give an automatic compatibility statement for enhanced functors (Theorem 3.24), a classification of rigid objects in the associated torsion category (Theorem 4.14), a Grothendieck duality statement on the subcategory of f_*-rigid objects without assuming that f_* preserves compact objects (Theorem 4.23), a Picard-group classification of invertible torsion objects (Theorem 5.8), a classification of Matlis dualising objects (Theorem 6.18), Morita-theoretic classification of Matlis lifts (Theorem 7.19), and a theory of Gorenstein duality for geometric functors with sections (Theorem 8.9). The paper also contains ascent and descent results and applications to commutative algebra, formal DGAs, chromatic and equivariant stable homotopy theory, and polynomial invariant theory including a new perspective on Watanabe's theorem.

Significance. If the results hold, this is a significant contribution: it systematically generalises and complements the duality frameworks of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar, and it replaces the restrictive compact-preservation hypothesis with the more flexible proxy-smallness condition. The paper is a long, carefully structured proof document with many detailed coherence arguments and concrete applications, and the claimed theorems are substantial and useful. However, a load-bearing proof step in Proposition 3.23 is not justified as written, although the statement itself is true and admits a short repair; this does not undermine the overall architecture of the paper but requires correction before publication.

major comments (1)
  1. [§3.B, Proposition 3.23] The proof of Proposition 3.23 invokes Lemma 3.22 to transfer the conclusion 1_D ∈ Loc⊗(f_*f^*1_D) from the category of A-modules back to D, where A = f_*f^*1_D. This does not follow from Lemma 3.22. For the functor F = A⊗− : D → Mod(A), whose right adjoint is restriction of scalars and is conservative, Lemma 3.22 gives an inclusion for the left adjoint F, namely F(Loc(X)) ⊆ Loc(F X); it does not give res(Loc(A)) ⊆ Loc(res A). Thus the cited transfer in the direction used is not justified. The proposition itself is nevertheless correct: the composite ε∘α is the identity on 1_D, so 1_D is a retract of A in D; hence 1_D ∈ thick(A) ⊆ Loc⊗(A), and therefore Loc⊗(A) = D. This two-line repair does not need enhancements. Because Proposition 3.23 is used in Theorem 3.24 to make condition (2) of Definition 3.15 automatic for enhanced geometric functors, this is a genuine gap in a load-bearing passage, and the proof in the manuscript should be replaced rather than merely annotated.
minor comments (4)
  1. [Example 5.5(iii)] The sentence 'The Picard group of spectra is trivial, and is isomorphic to Z' is internally confusing: the second clause says the group is Z, while Definition 5.4 defines 'trivial' as surjectivity of Z → Pic(C). Please rephrase, for example 'The Picard group of spectra is Z, and hence is trivial in the sense of Definition 5.4.'
  2. [Example 8.12] There is a typo in 'We that the Wirthmüller isomorphism means f_* is Gorenstein'; this should read 'We note that' or 'We have'.
  3. [Proposition 3.23] The definition of the map α as f_*(η) is compressed; explicitly writing η: 1_C → f^*f_*1_C and the identification f_*1_C ≃ 1_D before defining α would make the argument easier to follow.
  4. [Corollary 10.16] The phrase 'Since res^G_1 k[V] is a polynomial ring' relies on an implicit identification of modules over the restriction of k[V]; adding a brief clarifying sentence would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

The paper's central theorems are derived from its stated definitions and external results, with no step reducing to its own inputs by construction.

full rationale

I find no circularity in this paper's derivation chain. The main results (Theorem 4.14, 4.23, 5.8, 6.18) are proven from the definition of proxy-small geometric functor (Definition 3.15) plus standard facts about torsion/completion (Hovey–Palmieri–Strickland) and adjoint functors. The characterization of f_*-rigid objects via f_*M rigid is a substantive theorem, not a definitional equivalence: the proof transfers rigidity through the closed monoidal completion functor and the proxy-small witness set. The Grothendieck duality formula (4.24) is then deduced from the coherence lemmas and the rigidity classification, not assumed. The paper's reliance on external results (Balmer–Dell'Ambrogio–Sanders, Dwyer–Greenlees–Iyengar) is standard mathematical dependency, and the self-citations (Peirce 2025, Pol–Williamson 2023) appear only as motivational outlook or suggested techniques, never as load-bearing inputs. The one genuine issue is a proof gap in Proposition 3.23: the written proof appears to apply Lemma 3.22 in the wrong direction (from the right adjoint) to conclude 1_D ∈ Loc⊗(f_*f^*1_D). This is a correctness problem in the exposition, but it is not circularity: the claim is true by an independent two-line retract argument (ε∘α = id), and the subsequent theorems remain valid once that repair is inserted. No fitted parameters, self-citations, or uniqueness theorems imported from the authors' prior work are used to force the conclusions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claims rest on standard categorical background (Brown representability, MGM equivalence) plus explicit hypotheses of proxy-smallness, pure semisimplicity, and conservativity. No free parameters are fitted. No invented entities are introduced; the new notions are definitions, not postulates.

assumptions (6)
  • domain assumption Rigidly-compactly generated tt-categories as the ambient setting
    Section 2 establishes the categorical framework, quoting compact generation and rigidity as standing assumptions, following [25] and [4].
  • standard math Brown representability yields the adjoint triple f_* ⊣ f_* ⊣ f_! for geometric functors
    Invoked in Section 2.5 and throughout; standard in compactly generated triangulated categories.
  • domain assumption The MGM equivalence for torsion and completion (Theorem 3.6)
    Quoted from [25, Theorem 3.3.5], adjusted to proxy-small generators; this is the backbone of the torsion/completion framework.
  • domain assumption Pure semisimplicity of D in Theorems 6.18 and 7.23, and conservativity of f_* in several corollaries
    These are explicit hypotheses in the statements, used to make Krull-Schmidt arguments and to identify weak and strong Matlis lifts.
  • domain assumption Specific proxy-smallness of fields, K(n)_*-modules, and totalisation functors in examples
    Examples 4.16, 4.17, 10.3 rely on known proxy-smallness results (Koszul complexes, [15, Proposition 2.12], etc.).
  • domain assumption The forgetful functor from modules over a commutative algebra in an enhanced tt-category is conservative
    Used in the proof of Proposition 3.23; the paper states this without proof.

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Pith. "Pith review of Duality in tensor-triangular geometry via proxy-smallness." pith.science (2026). https://pith.science/paper/VLZIKBLV

@misc{pith2026251024415,
  author       = {Pith},
  title        = {Pith review of: Duality in tensor-triangular geometry via proxy-smallness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLZIKBLV}},
  note         = {Machine review of arXiv:2510.24415}
}
read the original abstract

We make a systematic study of duality phenomena in tensor-triangular geometry, generalising and complementing previous results of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar. A key feature of our approach is the use of proxy-smallness to remove assumptions on functors preserving compact objects, and to this end we introduce proxy-small geometric functors and establish their key properties. Given such a functor, we classify the rigid objects in its associated torsion category, giving a new perspective on results of Benson--Iyengar--Krause--Pevtsova. As a consequence, we show that any proxy-small geometric functor satisfies Grothendieck duality on a canonical subcategory of objects, irrespective of whether its right adjoint preserves compact objects. We use this as a tool to classify Matlis dualising objects and to provide a suitable generalisation of the Gorenstein ring spectra of Dwyer--Greenlees--Iyengar in tensor-triangular geometry. We illustrate the framework developed with various examples and applications, showing that it captures Matlis duality and Gorenstein duality in commutative algebra, duality phenomena in chromatic and equivariant stable homotopy theory, and Watanabe's theorem in polynomial invariant theory.

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