Constructible tori over Dedekind schemes

Adrien Morin; Takashi Suzuki

REVIEW

Constructible tori over Dedekind schemes

Adrien Morin, Takashi Suzuki

Algebraic Geometry · Submitted 6 May 2025 · 8 references

Pith's verdict: Solid extension of the authors' own program that removes tame ramification from a special value formula, with one load-bearing K0 step that needs a real proof.

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arXiv ID2505.03634 (version 1)

Pith IDQ4WF6XV2, Pith's permanent record of this paper and its review

CitationAdrien Morin, Takashi Suzuki. "Constructible tori over Dedekind schemes." arXiv:2505.03634 (2025). https://pith.science/paper/Q4WF6XV2

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Claim paper

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Editor's note

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Colleague,

The one thing to know: this is a serious paper that does what it claims in outline, but the final step from the tame base cases to the general special value formula is compressed into a one-sentence citation, and that is exactly where I would want a referee to dig.

What's new: the exact category of torsion-free constructible tori, described as two-term complexes T -> ⊕ i_*E, and the duality with Z-constructible sheaves (Thms A/B). That part is built carefully on Suzuki's published duality and looks solid. The paper also proves multiplicativity of det Lie for Néron models using Chai-Yu, which is a neat and useful input. The special value formula L*_X(T^•,0)=±χ_X(T^•) is the advertised payoff, extending Morin's earlier tame result.

Where I wince: Prop 11.12 asserts the inclusion of red and blue constructible tori induces an isomorphism on K0, with a one-line 'by the argument of [Mor23a]' proof. This is the load-bearing step that lets χ_X extend to all of Db(CT/X). The stress-test is right: the paper does not show the blue-to-red exact sequences generate all relations in K0, nor that the cited argument goes through without tameness. Theorem 11.13 then reduces by Swan induction to Gm and i_*Z[-1]; if the K0 step fails, the formula for wildly ramified tori doesn't follow. I found no concrete error, but this is a missing verification rather than a trivial omission.

Also, several technical lemmas (4.1, 4.2, 8.1) are sketched. 8.1 has a real proof, 4.2 is a diagram chase, but the paper is dense and at times reads like a sequence of pointers to prior work.

The self-citation pattern is not a problem: the cited theorems are published and are the natural base. The category theory is coherent, and the étale realization functor is handled honestly.

Bottom line: for specialists in arithmetic geometry, this is a valuable contribution and deserves a serious referee. I would send it out, with instructions to verify Prop 11.12 independently, and to expand the derivation of the K0 isomorphism. If that step holds, the paper is a clean advance. I'd probably cite it for the duality theorem regardless.

Referee report

Major objections (3)

[Section 11, Proposition 11.12] The proof of Proposition 11.12 consists of a single sentence asserting that the inclusion of red and blue constructible tori induces an isomorphism on K0 'by the argument of [Mor23a, def. 6.18, prop. 6.19]'. This is load-bearing for Theorem 11.13, because it is what allows χ_X, defined only on red/blue objects, to be extended to all of D^b(CT/X). The paper does not verify that the red/blue classes form an exact subcategory of D^b(CT/X) closed under the kernels, cokernels, and extensions needed for a K0 computation, nor that all relations in K0(D^b(CT/X)) are generated by blue-to-red short exact sequences. Since the definitions of 'red' and 'blue' here are given in terms of finiteness of H^1_B and H^2_B, while the cited proposition concerns étale sheaves in Morin's earlier framework, the transfer is not automatic. A complete proof, or a precise statement of the cited proposition with all hypotheses verified, is required.
[Section 11, Theorem 11.13] The proof of Theorem 11.13 says only: 'By Artin induction and a theorem of Swan [Swa63, Corollary 1], this formally reduces to the case of Gm, or a torsion-free constructible torus of the form T=i_*Z[-1]'. This is the central argument that removes the tame ramification hypothesis, and it is not a formal reduction in any obvious sense. One must at least state and prove the K-theoretic generation statement: every object of D^b(CT/X) lies in the subgroup of K0 generated by objects of the form π_*Gm and i_*Z[-1] (with π finite dominant), with relations coming from exact triangles, and one must show that both the L-function and the Weil-étale Euler characteristic factor through this K0 and behave correctly under finite dominant pushforwards. The references to [Mor23a, cor. 6.8, prop. 6.10] for the function field base cases are not sufficient, since it is not established that the non-tame contributions are killed by the K0 relations. This gap affects the main theorem directly.
[Section 11, Definitions 11.1 and 11.2] The paper defines red and blue constructible tori and asserts in Definition-Proposition 11.2 that RΓ_{W,B} is functorial in blue-to-red morphisms and gives a long exact sequence for blue-to-red short exact sequences. However, it is not shown that the collection of red and blue objects, with the class of blue-to-red short exact sequences, forms an exact category or a Waldhausen category in the sense needed for the K0 and determinant extensions used in Proposition 11.12. Since Proposition 11.12 and hence Theorem 11.13 depend on precisely this structure, a precise categorical framework for red/blue objects should be supplied.

Minor objections (6)

[Section 8, proof of Theorem 8.1] The diagram of lattices in the proof of Theorem 8.1 is difficult to read; the top row appears to be missing a tensor factor in the printed version. Please redraw the diagram with clear labels.
[Section 11, footnote 6] The sentence 'we have F^D = T_et[1] where F^D denotes the dual of a Z-constructible sheaf as in loc. cit.' is confusing because the same symbol F^D is used in this paper for the dual of F; please clarify the shift convention in a dedicated paragraph.
[Section 11, Theorem 11.13] The proof refers to 'Swan [Swa63, Corollary 1]' without stating what that corollary asserts; please restate the theorem being used so that the reduction is checkable.
[Appendix A] The appendix contains 'Proposition 0.1', which should be renumbered (for example, Proposition A.1) to match the paper's numbering scheme.
[Section 2, Definition 2.1] The use of 'mapping cone notation [T→⊕...][−1]' while writing the complex as 0→T→...→0 may lead to sign and shift confusion; please state the convention explicitly once in Section 2.
[Section 10, Remark 10.3] The comparison with [GS22] is useful, but the difference between the 1-motive L-function and the Hasse-Weil L-function is not used later in the paper; consider moving this remark to a footnote or expanding it.

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The paper

The paper claims

This paper proves that constructible tori over a Dedekind scheme proper over Spec Z have L-functions whose leading coefficient at s=0 is, up to sign, a Weil-étale Euler characteristic, with no tame ramification hypothesis.

The reading

The paper claims that over any one-dimensional arithmetic scheme with perfect residue fields, the category of constructible tori—objects built from Néron models of tori together with lattices at closed points—is the mirror of the category of $\mathbb{Z}$-constructible sheaves: the two are contravariantly equivalent. On the analytic side, it uses that duality, plus a multiplicativity property of the determinant of the Lie algebra of Néron models, to attach an $L$-function to every constructible torus and prove that at $s=0$ the leading coefficient equals, up to sign, a Weil-étale Euler characteristic. This removes the tame ramification hypothesis that earlier special value formulas for tori required. If the paper is right, special values of these $L$-functions are completely determined by étale cohomology and a Lie-algebra additive factor.

What carries the argument

The load-bearing identity is the special value formula $L^*_X(T^\bullet,0)=\pm\chi_X(T^\bullet)$, with $\chi_X$ defined through the fundamental line $\Delta_X(T)=(\det_{\mathbb Z}\mathrm{R}\Gamma_{W,B}(X,T))^{-1}\otimes \Delta^{\mathrm{add}}_X(T)$. Three mechanisms carry the argument. First, the duality theorems identify constructible tori with the opposite of $\mathbb{Z}$-constructible sheaves, via $T^D=\mathrm{Ext}^0_{X_{\mathrm{sm}}}(T,\mathbb{G}_m)$, so the étale realization is $R\mathrm{Hom}_{X_{\mathrm{et}}}(T^D,\mathbb{G}_m)$. Second, the multiplicativity theorem for $\det\,\mathrm{Lie}$ of Néron models—proved by comparing base change conductors of tori—lets the additive factor $\Delta^{\mathrm{add}}_X$ be defined on the derived category even though the Lie algebra functor itself does not pass to that category. Third, the reduction to the tame base cases $\mathbb{G}_m$ and $i_*\mathbb{Z}[-1]$ is made by group-theoretic induction on finite representations together with the K-theoretic statement that red and blue constructible tori generate the Grothendieck group; if that reduction is valid, the formula for the base cases propagates to all constructible tori.

Extended reading notes

Core claim

For an irreducible Dedekind scheme $X$ with perfect residue fields, the paper proves two dualities: the functor sending a torsion-free $\mathbb{Z}$-constructible sheaf $F$ to the complex $[T \to \bigoplus_v i_{v,*}E_v]$ (Néron model of the Cartier dual of the generic fibre, with $\mathbb{Z}$-duals of the closed-point stalks) is quasi-inverse to $T\mapsto \mathrm{Ext}^0_{X_{\mathrm{sm}}}(T,\mathbb{G}_m)$, giving a contravariant equivalence between torsion-free constructible tori and torsion-free $\mathbb{Z}$-constructible sheaves; allowing torsion extends this to a contravariant equivalence between all constructible tori and all $\mathbb{Z}$-constructible sheaves. For $X$ proper over $\mathrm{Spec}\,\mathbb{Z}$, it then defines $L_X(T^\bullet,s)$ as an Euler product of geometric-Frobenius determinants on the étale realization, and a Weil-étale fundamental line $\Delta_X(T^\bullet)$ combining the determinant of Weil-étale cohomology with the determinant of the Lie algebra. The central formula is $L^*_X(T^\bullet,0)=\pm\chi_X(T^\bullet)$, where $\chi_X$ is the resulting Weil-étale Euler characteristic; this extends the first-named author's earlier special value theorem by dropping the tame ramification hypothesis.

Load-bearing premise

Everything rests on the assumption that every constructible torus can be assembled from the tame pieces—the multiplicative group and lattices supported at closed points—using exact sequences and direct sums in a way that preserves the Euler characteristic; if that assembly step fails, the general special value formula would not follow from the tame cases.

Editorial extensions

If this is right

Every constructible torus over a number field or function field has its $L$-function's leading coefficient at $s=0$ computed, up to sign, from étale cohomology and a Lie-algebra determinant.
Wildly ramified tori now fall under the same formula that previously needed tame ramification.
The duality theorems translate constructions and questions about Néron models and component groups into constructions on $\mathbb{Z}$-constructible sheaves, and back.
The formula yields a special value statement at $s=1$ for Artin $L$-functions of rational representations, as noted in the paper.

Reading between the lines

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

The same two-step strategy—establish a duality with sheaves, then define a determinant-theoretic Euler characteristic—may extend to higher-dimensional arithmetic schemes if a duality of the same shape exists there; the paper does not claim this.
If the K0-generation step is uniform, one can test the formula numerically on a family of wildly ramified tori over a fixed base, and the equality $L^*_X(T,0)=\pm\chi_X(T)$ would verify the entire chain.
The duality suggests that invariants such as component groups of Néron models have sheaf-theoretic counterparts in $\mathbb{Z}$-constructible sheaves; making that dictionary explicit could give new computational tools for these invariants.

Original abstract

We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively. We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their \'etale realizations and prove a special value formula at $s=0$ using the Weil-\'etale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,535 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively. We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their \'etale realizations and prove a special value formula at $s=0$ using the Weil-\'etale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields.	contribution 1.00	0.94	0.13	asserted 1.00
s1	The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces.	contribution 0.66	0.80	0.06	asserted 1.00
s2	Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively.	result 0.87	0.85	0.23	asserted 1.00
s3	We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their \'etale realizations and prove a special value formula at $s=0$ using the Weil-\'etale formalism developed by the first-named author in arXiv:2210.09102.	contribution 0.73	0.95	0.39	asserted 1.00
s4	This extends the results of the first-named author by removing the tame ramification hypothesis.	interpretation 0.53	0.78	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (8)

[1]

[Bar15] Clark Barwick. “On exact ∞-categories and the Theorem of the Heart”. In: Compositio Mathematica 151.11 (2015), pp. 2160–2186. doi: 10.1112/S0010437X15007447. [BF03] D. Burns and M. Flach. “Tamagawa Numbers for Motives w ith (Non- commutative) Coeﬃcients, II”. In: American Journal of Mathematics 125.3 (2003), pp. 475–512. 9Such a complex would be t...

View on Pith doi:10.1112/s0010437x15007447 2015
[123]

Math. Surveys Monogr. Amer. Math. Soc., Providen ce, RI, 2005, pp. 235–321. REFERENCES 29 [Knu02] Finn F. Knudsen. “Determinant functors on exact cat egories and their extensions to categories of bounded complexes”. In: Michigan Mathemat- ical Journal 50.2 (2002), pp. 407–445. doi: 10.1307/mmj/1028575741. [KS06] Masaki Kashiwara and Pierre Schapira. Categ...

View on Pith arXiv 2002
[151]

Lecture Notes in Math. Berlin: Springer, 1970, pp. xv+564. [DGB70] M. Demazure, P. Gabriel, and J. E. Bertin. Schemas en Groupes. Sem- inaire de Geometrie Algebrique du Bois Marie 1962/64 (SGA 3). I: Proprietes Generales des Schemas en Groupes . 1st ed. lecture Notes in Mathematics. Springer Berlin, Heidelberg,

View on Pith 1970
[332]

Grundlehren der mathematischen Wissenschaften [Fundamen tal Princi- ples of Mathematical Sciences]. Springer-Verlag, Berlin, 2006, pp. x+497. isbn: 978-3-540-27949-5. doi: 10.1007/3-540-27950-4 . [Mil06] James S. Milne. Arithmetic duality theorems . Second. BookSurge, LLC, Charleston, SC, 2006, pp. viii+339. isbn: 1-4196-4274-X. [Mor23a] Adrien Morin. “Sp...

View on Pith doi:10.1007/3-540-27950-4 2023
[1970]

doi: 10.1007/BFb0058993. [DM97] W.G. Dwyer and S.A. Mitchell. “On the K-theory spectr um of a smooth curve over a ﬁnite ﬁeld”. In: Topology 36.4 (1997), pp. 899–929. issn: 0040-9383. doi: https://doi.org/10.1016/S0040-9383(96)00030-4. url: https://www.sciencedirect.com/science/article/pii/S0040938396000304. [Fon92] Jean-Marc Fontaine. “Valeurs spéciales d...

View on Pith doi:10.1007/bfb0058993 1997
[1990]

isbn: 978-3-540-50587-7. doi: 10.1007/978-3-642-51438-8 . [Bre11] Manuel Breuning. “Determinant functors on triangu lated categories”. In: Journal of K-Theory 8.2 (2011), pp. 251–291. doi: 10.1017/is010006009jkt120. [BS15] Bhargav Bhatt and Peter Scholze. “The Pro-étale topo logy for schemes”. In: Asterisque 369 (Sept. 2015), pp. 99–201. doi: 10.24033/ast...

View on Pith doi:10.1007/978-3-642-51438-8 2011
[1992]

url: http://www.numdam.org/item/SB_1991-1992__34__205_0 [GS22] Thomas H. Geisser and Takashi Suzuki. “Special value s of L-functions of one-motives over function ﬁelds”. In: Journal für die reine und ange- wandte Mathematik (Crelles Journal) 2022.793 (2022), pp. 281–304. doi: doi:10.1515/crelle-2022-0081. [Kle05] Steven L. Kleiman. “The Picard scheme”. In...

View on Pith doi:10.1515/crelle-2022-0081 2022
[2022]

[Suz19] Takashi Suzuki. “Néron models of 1-motives and dual ity”. In: Kodai Math. J. 42.3 (2019), pp. 431–475. issn: 0386-5991. doi: 10.2996/kmj/1572487228. [Swa63] Richard G. Swan. “The Grothendieck ring of a ﬁnite gr oup”. In: Topology 2.1 (1963), pp. 85–110. issn: 0040-9383. doi: https://doi.org/10.1016/0040-9383(63)90025-9. [Ver96] Jean-Louis Verdier....

View on Pith arXiv 2019

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

[Bar15] Clark Barwick. “On exact ∞-categories and the Theorem of the Heart”. In: Compositio Mathematica 151.11 (2015), pp. 2160–2186. doi: 10.1112/S0010437X15007447. [BF03] D. Burns and M. Flach. “Tamagawa Numbers for Motives w ith (Non- commutative) Coeﬃcients, II”. In: American Journal of Mathematics 125.3 (2003), pp. 475–512. 9Such a complex would be t...

View on Pith doi:10.1112/s0010437x15007447 2015

[2] [123]

Math. Surveys Monogr. Amer. Math. Soc., Providen ce, RI, 2005, pp. 235–321. REFERENCES 29 [Knu02] Finn F. Knudsen. “Determinant functors on exact cat egories and their extensions to categories of bounded complexes”. In: Michigan Mathemat- ical Journal 50.2 (2002), pp. 407–445. doi: 10.1307/mmj/1028575741. [KS06] Masaki Kashiwara and Pierre Schapira. Categ...

View on Pith arXiv 2002

[3] [151]

Lecture Notes in Math. Berlin: Springer, 1970, pp. xv+564. [DGB70] M. Demazure, P. Gabriel, and J. E. Bertin. Schemas en Groupes. Sem- inaire de Geometrie Algebrique du Bois Marie 1962/64 (SGA 3). I: Proprietes Generales des Schemas en Groupes . 1st ed. lecture Notes in Mathematics. Springer Berlin, Heidelberg,

View on Pith 1970

[4] [332]

Grundlehren der mathematischen Wissenschaften [Fundamen tal Princi- ples of Mathematical Sciences]. Springer-Verlag, Berlin, 2006, pp. x+497. isbn: 978-3-540-27949-5. doi: 10.1007/3-540-27950-4 . [Mil06] James S. Milne. Arithmetic duality theorems . Second. BookSurge, LLC, Charleston, SC, 2006, pp. viii+339. isbn: 1-4196-4274-X. [Mor23a] Adrien Morin. “Sp...

View on Pith doi:10.1007/3-540-27950-4 2023

[5] [1970]

doi: 10.1007/BFb0058993. [DM97] W.G. Dwyer and S.A. Mitchell. “On the K-theory spectr um of a smooth curve over a ﬁnite ﬁeld”. In: Topology 36.4 (1997), pp. 899–929. issn: 0040-9383. doi: https://doi.org/10.1016/S0040-9383(96)00030-4. url: https://www.sciencedirect.com/science/article/pii/S0040938396000304. [Fon92] Jean-Marc Fontaine. “Valeurs spéciales d...

View on Pith doi:10.1007/bfb0058993 1997

[6] [1990]

isbn: 978-3-540-50587-7. doi: 10.1007/978-3-642-51438-8 . [Bre11] Manuel Breuning. “Determinant functors on triangu lated categories”. In: Journal of K-Theory 8.2 (2011), pp. 251–291. doi: 10.1017/is010006009jkt120. [BS15] Bhargav Bhatt and Peter Scholze. “The Pro-étale topo logy for schemes”. In: Asterisque 369 (Sept. 2015), pp. 99–201. doi: 10.24033/ast...

View on Pith doi:10.1007/978-3-642-51438-8 2011

[7] [1992]

url: http://www.numdam.org/item/SB_1991-1992__34__205_0 [GS22] Thomas H. Geisser and Takashi Suzuki. “Special value s of L-functions of one-motives over function ﬁelds”. In: Journal für die reine und ange- wandte Mathematik (Crelles Journal) 2022.793 (2022), pp. 281–304. doi: doi:10.1515/crelle-2022-0081. [Kle05] Steven L. Kleiman. “The Picard scheme”. In...

View on Pith doi:10.1515/crelle-2022-0081 2022

[8] [2022]

[Suz19] Takashi Suzuki. “Néron models of 1-motives and dual ity”. In: Kodai Math. J. 42.3 (2019), pp. 431–475. issn: 0386-5991. doi: 10.2996/kmj/1572487228. [Swa63] Richard G. Swan. “The Grothendieck ring of a ﬁnite gr oup”. In: Topology 2.1 (1963), pp. 85–110. issn: 0040-9383. doi: https://doi.org/10.1016/0040-9383(63)90025-9. [Ver96] Jean-Louis Verdier....

View on Pith arXiv 2019