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Categorical K\"unneth formulas for analytic stacks

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Pith's one-line read This paper proves that a sheafy 6-functor formalism satisfying the categorical Künneth formula on affine objects extends the formula to every morphism into a stack with a !-cover from an affine, yielding a p-adic Drinfeld lemma.

desk verdict Plausible framework, but the advertised p-adic Drinfeld lemma rests on an unverified assertion that Div1,ct and Div1,la map to AnSpec(Qp) in tilde E. read the letter →

arxiv 2507.08566 v1 pith:FNY6TAWK submitted 2025-07-11 math.AG math.NT

Youshua Kesting This is my paper
classification math.AGmath.NT MSC 14A2014F0814G22
keywords categoricalKünnethformula6-functorformalismanalyticstacksdualisablepresentablecategoriesTannakianreconstructionp-adicDrinfeldlemma!-descentperfect
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

The paper aims to establish, for a broad class of analytic stacks, a categorical Künneth formula: the category of sheaves on a fibre product is the relative tensor product of the categories on the two factors. It introduces a general notion of Künneth morphism inside a 6-functor formalism and shows this property survives composition, base change, open immersions, and !-local descent. The main theorem states that if the formalism satisfies the formula on an affine base site, then every morphism into a sheaf that admits a !-cover from that site is Künneth. Applying this to the analytic-stack formalism yields a Tannakian reconstruction result and a p-adic analogue of Drinfeld's lemma for two stacks related to p-adic Langlands.

What carries the argument

The central object is the Künneth morphism: a map $f: X \to Z$ in a 6-functor formalism $D$ for which $D(X)\otimes_{D(Z)}D(W) \to D(X\times_Z W)$ is an equivalence for every $W\to Z$. The proof mechanism is an inductive extension argument (Theorem 3.24) that starts from the affine base site, where Künneth is assumed, and shows each step in the construction of the extended class of morphisms—adding morphisms that are !-local on source or target, then $*$-local on the target—preserves the property, using closure under composition, base change, and !-descent. A second mechanism is self-duality: when the diagonal Künneth morphism is an equivalence, $D(X)$ is self-dual as a $D(Z)$-module (Proposition 3.15), which gives dualisability and powers the Tannakian reconstruction through 2-descent.

What would settle it

Concretely test the hypotheses of Corollary 4.13.2 by computing the diagonal of $Div^{1,\mathrm{ct}}_{\mathbb{Q}_p}$ or $Div^{1,\mathrm{la}}_{\mathbb{Q}_p}$ in the analytic-stack category: if the diagonal is not representable or no !-cover from an affine analytic stack exists, the p-adic Drinfeld lemma is not established by this theorem. Alternatively, exhibit an analytic stack $W \to \mathrm{AnSpec}(\mathbb{Q}_p)$ for which the equivalence $D_{\mathrm{qc}}(Div^{1,?}_{\mathbb{Q}_p})\otimes_{D_{\mathrm{qc}}(\mathbb{Q}_p)}D_{\mathrm{qc}}(W) \to D_{\mathrm{qc}}(Div^{1,?}_{\mathbb{Q}_p}\times_{\mathrm{AnSpec}(\mathbb{Q}_p)}W)$ fails.

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Extended reading notes

Core claim

For a sheafy 6-functor formalism $D$ on a subcanonical site $\mathcal{C}$, if $D$ satisfies the categorical Künneth formula on $\mathcal{C}$, then its extension $\tilde{D}$ to sheaves on $\mathcal{C}$ satisfies Künneth for every morphism $X\to S$ in the extended class of morphisms, provided $S$ admits a !-cover by an object of $\mathcal{C}$ (Theorem 3.26). In the analytic setting this gives: any morphism into an analytic stack with a !-cover from an affine analytic stack is Künneth and, for analytic $E_\infty$-rings, Tannakian (Corollary 4.13.1). In particular, for the analytic stacks $Div^{1,\mathrm{ct}}_{\mathbb{Q}_p}$ and $Div^{1,\mathrm{la}}_{\mathbb{Q}_p}$, the structural morphism to $\mathrm{AnSpec}(\mathbb{Q}_p)$ is Künneth, giving $D_{\mathrm{qc}}(Div^{1,?}_{\mathbb{Q}_p}) \otimes_{D_{\mathrm{qc}}(\mathbb{Q}_p)} D_{\mathrm{qc}}(W) \cong D_{\mathrm{qc}}(Div^{1,?}_{\mathbb{Q}_p}\times_{\mathrm{AnSpec}(\mathbb{Q}_p)} W)$ for $? \in \{\mathrm{ct}, \mathrm{la}\}$ and every analytic stack $W \to \mathrm{AnSpec}(\mathbb{Q}_p)$ (Corollary 4.13.2).

Load-bearing premise

The result depends on the claim that the two specific stacks $Div^{1,\mathrm{ct}}_{\mathbb{Q}_p}$ and $Div^{1,\mathrm{la}}_{\mathbb{Q}_p}$ have representable diagonals and admit a !-cover from an affine analytic stack, which Corollary 4.13.2 asserts without a detailed verification in the paper.

Editorial extensions

If this is right

  • If the main theorem is correct, the categorical Künneth formula holds for every morphism of analytic stacks whose target admits a !-cover from an affine analytic stack, with no compact-generation or perfectness assumption.
  • The p-adic Drinfeld lemma follows for $Div^{1,\mathrm{ct}}_{\mathbb{Q}_p}$ and $Div^{1,\mathrm{la}}_{\mathbb{Q}_p}$: for any analytic stack $W$ over $\mathrm{AnSpec}(\mathbb{Q}_p)$, tensoring sheaf categories over $D_{\mathrm{qc}}(\mathbb{Q}_p)$ computes the sheaf category of the fibre product.
  • A Tannakian reconstruction result holds: for stacks covered by affines in this way, morphisms of stacks are recovered from symmetric monoidal colimit-preserving functors between their categories of sheaves.
  • The Künneth property propagates along covers: it is closed under composition, arbitrary base change, open immersions, and !-locality on source and target, so a cover reduces the general case to affines.
  • The elliptic-curve quotient $* \to */E^{\triangleright}$ shows the obstruction: without the representable-diagonal or !-cover hypothesis, the formula can fail.

Reading between the lines

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

  • The criterion gives a practical recipe for new stacks: to obtain a p-adic Drinfeld lemma for another coefficient stack, it suffices to exhibit a !-cover by an affine and a representable diagonal, sidestepping direct dualisability checks.
  • If a p-adic spectral action is eventually constructed, the theorem supplies the tensor-product compatibility it would require; the paper states this hope rather than deriving the action.
  • One could test whether the argument extends to stacks with only quasi-affine diagonals by using the fact that open immersions are Künneth, a generalisation the paper flags but does not prove.
  • The non-Künneth elliptic-curve example suggests the representability of the diagonal, not the choice of sheaf formalism, is the essential obstruction; examining other quotients with non-representable diagonals would sharpen this boundary.
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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

4 major / 6 minor

Summary. The paper proposes an abstract criterion for categorical Künneth formulas in 6-functor formalisms and applies it to the Clausen–Scholze theory of analytic stacks. It defines Künneth morphisms, proves permanence properties (composition, base change, and !-locality), states an extension theorem (Theorem 3.24) ensuring that a sheafy Künneth formalism on a subcanonical site extends to sheaves while preserving Künneth for morphisms into objects admitting an affine !-cover, and derives a Tannakian reconstruction theorem (Theorem 3.33). In the analytic setting the paper claims a p-adic Drinfeld lemma, namely the equivalence Dqc(Div1,?_{Qp}) ⊗_{Dqc(Qp)} Dqc(W) ≅ Dqc(Div1,?_{Qp} ×_{AnSpec(Qp)} W) for ? ∈ {ct, la}. The general framework is potentially useful, but the advertised analytic application is not supplied with the necessary verification.

Significance. The abstract part of the paper, if made rigorous, would provide a clean conceptual explanation of Künneth formulas for quasi-coherent sheaves on analytic stacks, extending [DN10] to a setting where compact generation is replaced by dualisability. The permanence properties (Lemmas 3.4, 3.5, Propositions 3.21, 3.22) and the use of Ramzi's dualisability criterion are sensible and likely correct. The paper also gives credit to the relevant prior work ([HM24], [Ram24], [Cam24], [ABM24]) and does not overclaim the novelty of the 6-functor machinery. However, the proofs of the main theorems are sketches, and the p-adic Drinfeld lemma — the headline application — is asserted without checking the analytic stack hypotheses. The paper is therefore a promising framework needing substantial completion rather than a finished proof.

major comments (4)
  1. [§3.4, Theorem 3.24] The proof of Theorem 3.24 is incomplete. The step (∗) is only sketched: it asserts that any f ∈ E''_S lies in Ku(E''_S) by Proposition 3.21, but the cover (Xi → X) produced by the definition of E'' must be shown to be a universal !-cover for the formalism D′ in the sense required by Proposition 3.21, and this is not done. The step (∗′) is dismissed with 'applying (∗) repeatedly', and (∗∗) is derived from the equality E′_{∗S} = E′_S, which is asserted without proof. In addition, the construction of ˜E as a union over n ≥ 1 of finite iterations does not address possible limit steps in the Heyer–Mann construction, so the conclusion Ku(˜E_S) = ˜E_S is not justified for the actual class ˜E.
  2. [§3.4, Theorem 3.26] The proof of Theorem 3.26 is not detailed enough for the main abstract result. To apply Theorem 3.24 to the Cech objects S_{[n],i•}, one must know that D_{S_{[n],i•}} is symmetric monoidal for each ([n],i•); the manuscript does not prove this, nor does it explain how it follows from the sheafy 6-functor formalism. The step 'the result follows now by Theorem 3.24 and Proposition 3.22' also needs to justify that the base-changed morphisms f_{[n],i•} lie in ˜E and that the !-cover data are compatible with the inductive definition of ˜E. Since this theorem is the only bridge from the affine Künneth property to arbitrary analytic stacks, the missing details are load-bearing.
  3. [§3.5, Theorem 3.33] The proof of Theorem 3.33 is a sketch rather than a proof. The reduction 'by iterating the following argument we can assume S = ∗' is neither explained nor justified, the identification of the pullback P with the subcategory Mod^aff_{D(X)} PrL is asserted without a definition of that subcategory or a proof of the identification, and the final descent step ('this gives an arrow Z → X by !-descent') omits the actual descent data. These are central to the Tannakian reconstruction claim.
  4. [§4.3, Corollary 4.13.2] The p-adic Drinfeld lemma is not established. The corollary asserts that the structural morphisms Div1,?_{Qp} → AnSpec(Qp,□) lie in ˜E, but no proof is given that the objects Div1,ct_{Qp} and Div1,la_{Qp} are analytic stacks or that these morphisms lie in ˜E. To verify membership in ˜E for a morphism with affine target, one needs by Proposition 3.23(iv) and Proposition 3.18 a universal !-cover U → Div1,? with U affine analytic and U → AnSpec(Qp) in E, together with a check that the quotient maps by the group actions remain universal !-covers; none of this is supplied. Moreover, the definition of Div1,la_{Qp} contains an undefined expression: the limit 'lim_{T ↦ (T+1)^p−1} D□\{0}' is not over a specified diagram, and the notation D□\{0} is not introduced. The advertised Künneth equivalence therefore does not follow from the results of Section 3.
minor comments (6)
  1. [§4.3, before Corollary 4.13.2] The citation 'By Theorem 3.24' is inaccurate: for S ∈ AnSt one needs Theorem 3.26 or Corollary 4.13.1, not Theorem 3.24, which is stated for S ∈ C.
  2. [§4.1, Lemma 4.4] Lemma 4.4(2) contains the unresolved placeholder '!!!REF!!!' for the description of pushouts in AnRing; a precise reference should be supplied.
  3. [§3.4, Remark 3.25] The notation 'coMod_{h^*(1)} D(∗)' in Remark 3.25 is not defined; the comodule category over an algebra object should be explained or referenced.
  4. [§4.3, Example 4.5] Example 4.5 asserts that f_*(1) ≅ 1 for the elliptic curve stack and that D(∗) → Dqc(E▷) is not an equivalence, but neither statement is proved; the example should be completed or explicitly labeled as heuristic.
  5. [Throughout] There are numerous typographical issues: 'thourough' in Section 2, 'K¨ uneth' in Proposition 3.22, 'arbitry' in Remark 3.25, and 'descents' in Theorem 3.33; the manuscript should be proofread.
  6. [§4.2, Examples 4.3 and 4.4] Example 4.3 states that (−)▷ sends fpqc covers to !-covers without giving a reference, and Example 4.4 relies on [And21, Proposition 3.34] but the target of the functor from AnH to AnSt should be stated more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the only flagged issue is an unproved membership assertion in Corollary 4.13.2, which is a missing proof rather than a circular reduction.

full rationale

The paper's central derivation is structural: Definition 3.3 defines Künneth morphisms, and Theorem 3.24 (with the extension theorem from Heyer–Mann [HM24]) proves that a Künneth 6-functor formalism extends to sheaves while preserving Künneth for morphisms into objects admitting a !-cover from the base site. The proof in Theorem 3.24 is an induction over the extension steps of [HM24] using Proposition 3.21, not a restatement of the conclusion. The base case for analytic rings, Proposition 4.11 (D on Aff is Künneth), is cited from Clausen–Scholze [AS] and Camargo [Cam24], which are external sources rather than self-citations. The advertised p-adic Drinfeld lemma (Corollary 1.5.2) is then a direct application of Corollary 4.13.1/Theorem 3.26, conditional on the structural morphism Div1,?Qp → AnSpec(Qp,□) lying in the extended class ˜E. That membership is asserted without proof in Section 4.3; no detailed verification is given that the quotients are in AnSt, that the cover maps are in E0, or that the structural morphisms lie in ˜E. This is an omitted proof and a real correctness risk, but it is not a circular step: the theorem does not define tilde-E membership into existence, and the assertion is not derived from the conclusion it purports to prove. There is no fitted parameter renamed as a prediction, no self-citation chain carrying the argument, and no uniqueness theorem imported from the authors' own prior work. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the 6-functor formalism of Clausen-Scholze and the extension machinery of Heyer-Mann; the paper's contribution is the definition of Künneth morphisms and the observation that they are preserved under the extension. No free parameters or invented entities are introduced.

assumptions (4)
  • standard math The framework of ∞-categories, presentable categories, and the Lurie tensor product.
    Used throughout Sections 2-4 without proof; standard background in derived algebraic geometry.
  • domain assumption The 6-functor formalism D on (Aff, E) is Künneth and Tannakian, as stated in Proposition 4.11.
    Attributed to [AS, Lecture 17] and [Cam24, Lemma 2.1.3]; this is the input to the extension theorem and is not proven from scratch in the paper.
  • standard math The extension construction [HM24, Theorem 3.4.11] produces a sheafy 6-functor formalism on AnSt.
    The main theorem 3.24 is proven by reviewing the inductive steps of this construction; the paper relies on its validity.
  • domain assumption The analytic stacks Div1,ct and Div1,la admit a !-cover from an affine analytic stack and have representable diagonal.
    These hypotheses are needed for Corollary 4.13.2 (p-adic Drinfeld lemma) but are asserted without detailed verification in Section 4.3.

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Pith. "Pith review of Categorical K\"unneth formulas for analytic stacks." pith.science (2026). https://pith.science/paper/FNY6TAWK

@misc{pith2026250708566,
  author       = {Pith},
  title        = {Pith review of: Categorical K\"unneth formulas for analytic stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNY6TAWK}},
  note         = {Machine review of arXiv:2507.08566}
}
abstract

In arXiv:0805.0157v5, the authors define a class of derived stacks, called "perfect stacks" and show that for this class the categories of quasi-coherent sheaves satisfy a categorical K\"unneth formula. Motivated to extend their results to the theory of analytic stacks as developed by Clausen-Scholze, we investigate categorical K\"unneth formulas for general $6$-functor formalisms. As applications we show a general Tannakian reconstruction result for analytic stacks and, following recent work of Ansch\"utz, Le Bras and Mann arXiv:2412.20968v1, show a $p$-adic version of Drinfeld's lemma for certain stacks that appear conjecturally in a categorical $p$-adic Langlands program.

Figures

Figures reproduced from arXiv: 2507.08566 by the authors.

Figure 1
Figure 1. Note that we suppressed the equivalence mX,Z : D(X) ⊗D(Z) D(Z) ∼= D(X) in the lower left corner and the equivalence mZ,X : D(Z) ⊗D(Z) D(X) ∼= D(X) in the lower right corner of [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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Works this paper leans on

4 extracted references · 2 canonical work pages · cited by 2 Pith papers

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    A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

    [ABM24] Johannes Ansch¨ utz, Arthur-C´ esar Le Bras, and Lucas Mann. “A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve”. In: arXiv preprint arXiv:2412.20968 (2024). [And21] Grigory Andreychev. “Pseudocoherent and perfect complexes and vector bundles on analytic adic spaces”. In:arXiv preprint arXiv:2105.12591 (2021). 33...

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