Tensor Operations on Group Schemes

Mohammad Hadi Hedayatzadeh

REVIEW 3 major objections 5 minor 8 references

Tensor Operations on Group Schemes

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper establishes a multilinear algebra for commutative group schemes and proves that over a perfect field of odd characteristic p, the nth alternating power Λ^n α_{p^n} is isomorphic to α_p for every n ≥ 1.

desk verdict Useful explicit tensor computations for finite group schemes, with the main structural result Λ^n α_{p^n} ≅ α_p likely true but resting on an unproved subgroup-existence assertion in Prop 3.17. read the letter →

arxiv 1908.05452 v1 pith:O46KICQH submitted 2019-08-15 math.NT math.AG

Mohammad Hadi Hedayatzadeh This is my paper · ORCID

classification math.NTmath.AG MSC 14L1514L17

keywords commutativegroupschemesmultilinearmorphismstensorproductsymmetricpowersalternatinglocal-localVerschiebungcharacteristicp

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 develops a multilinear algebra for commutative group schemes, constructing inner Hom, tensor products, and symmetric and alternating powers as pro-finite group schemes over a field. It computes these objects explicitly for the infinitesimal groups $\alpha_{p^n}$: for example, $\mathrm{Hom}(\alpha_{p^n},\alpha_{p^m})$ is a power of $\mathbb{G}_a$ or a mixed product of $\alpha$'s and $\mathbb{G}_a$'s, and $\mathrm{Alt}(\alpha_{p^n}^r,\mathbb{G}_a)\cong \mathbb{G}_a^{\binom{n}{r}}$ when $p>2$. The central result is that over a perfect field of odd characteristic $p$, the $n$-th alternating power satisfies $\Lambda^n\alpha_{p^n}\cong\alpha_p$, and more generally any local-local group scheme of order $p^n$ has $\Lambda^m G=0$ for $m>n$ and $\Lambda^n G$ a quotient of $\alpha_p^{\otimes n}$. A sympathetic reader would care because this provides a concrete, computable tensor calculus for finite group schemes in characteristic $p$, where the exponent $n$ plays the role of the length of a module.

What carries the argument

Three constructions carry the argument. The inner Hom $\underline{\mathrm{Hom}}(G,H)$ represents $T\mapsto\mathrm{Hom}_T(G_T,H_T)$; the tensor product $G_1\otimes\cdots\otimes G_n$ represents multilinear morphisms, with $\mathrm{Hom}(G_1\otimes\cdots\otimes G_n,H)\cong\mathrm{Mult}(G_1\times\cdots\times G_n,H)$; and the symmetric and alternating powers $S^nG$, $\Lambda^nG$ represent $\mathrm{Sym}(G^n,H)$ and $\mathrm{Alt}(G^n,H)$. Their existence over a field is taken from an unpublished manuscript, Theorems 3.10 and 4.3 of [6], which gives pro-finite group schemes and the description $G_1\otimes\cdots\otimes G_n\cong\varprojlim G_\alpha^*$ with $G_\alpha$ running over the finite subgroup schemes of $\mathrm{Mult}(G_1\times\cdots\times G_n,\mathbb{G}_m)$. For the main theorem the decisive mechanism is the Verschiebung, the endomorphism dual to Frobenius, on the pro-finite dual of $\mathbb{G}_a$: the short exact sequence $G_a^{*(p)}\to G_a^*\to\alpha_p\to0$ (Lemma 3.21), together with Lemma 3.25 showing that the Verschiebung annihilates $\Lambda^n\alpha_{p^n}$, forces $\Lambda^n\alpha_{p^n}$ to be a quotient of $\alpha_p$, and since $\alpha_p$ is simple, an isomorphism.

What would settle it

One could compute the order of $\Lambda^n\alpha_{p^n}$ directly, for instance from its Dieudonné module or by classifying alternating multilinear maps $\alpha_{p^n}^n\to H$: if the order is not $p$, or if the Verschiebung on $\Lambda^n\alpha_{p^n}$ is nonzero, Proposition 3.26 is false. Alternatively, finding a local-local group scheme of order $p^n$ with $n>1$ that has no proper subgroup scheme would break the induction behind Proposition 3.17.

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Extended reading notes

Core claim

The paper's central claim is that alternating powers of infinitesimal commutative group schemes are controlled by the order exponent. Concretely, Proposition 3.26 asserts that if $k$ is a perfect field of odd characteristic $p$, then for every $n\ge 1$ there is an isomorphism $\Lambda^n\alpha_{p^n}\cong\alpha_p$, where $\Lambda^n$ is defined by the universal property $\mathrm{Alt}(G^n,H)\cong\mathrm{Hom}(\Lambda^n G,H)$. The supporting structural result is Proposition 3.17: for any local-local commutative group scheme $G$ of order $p^n$ with $p$ odd, $\Lambda^m G=0$ for all $m>n$ and $\Lambda^n G$ is a quotient of $\alpha_p^{\otimes n}$. Along the way the paper establishes an adjunction for multilinear morphisms, $\mathrm{Mult}(G_1\times\cdots\times G_r,\mathrm{Mult}(H_1\times\cdots\times H_s,F))\cong\mathrm{Mult}(G_1\times\cdots\times G_r\times H_1\times\cdots\times H_s,F)$, and explicit formulas such as $\mathrm{Sym}(\alpha_{p^n}^r,\mathbb{G}_a)\cong\mathbb{G}_a^{\binom{n+r-1}{r-1}}$ and $\mathrm{Alt}(\alpha_{p^n}^r,\mathbb{G}_a)\cong\mathbb{G}_a^{\binom{n}{r}}$ for $p>2$.

Load-bearing premise

The proof assumes without justification that every local-local commutative group scheme of order $p^n$ with $n>1$ contains a proper subgroup scheme, and it relies on an unpublished manuscript [6] for the existence of inner Hom and tensor products; if either premise fails, Proposition 3.17 and the main calculation do not follow.

Editorial extensions

If this is right

For every local-local group scheme $G$ of order $p^n$ with $p$ odd, $\Lambda^mG=0$ for $m>n$: alternating powers vanish above the order exponent.
The top alternating power $\Lambda^nG$ is always a quotient of $\alpha_p^{\otimes n}$, so it is built from the simplest infinitesimal group scheme.
For direct sums of local-local group schemes of orders $p^n$ and $p^m$, $\Lambda^{n+m}(G\oplus H)\cong\Lambda^nG\otimes\Lambda^mH$.
The specific calculation $\Lambda^n\alpha_{p^n}\cong\alpha_p$ gives a family of nonzero alternating powers of unbounded degree, all isomorphic to the same small group scheme $\alpha_p$.
Multilinear morphisms to $\mathbb{G}_a$ have explicit dimensions: $\mathrm{Sym}(\alpha_{p^n}^r,\mathbb{G}_a)\cong\mathbb{G}_a^{\binom{n+r-1}{r-1}}$ and $\mathrm{Alt}(\alpha_{p^n}^r,\mathbb{G}_a)\cong\mathbb{G}_a^{\binom{n}{r}}$ for $p>2$.

Reading between the lines

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

If the main theorem is right, the alternating power functor gives each local-local group scheme of order $p^n$ a canonical 'top exterior power' map to $\alpha_p$, which could serve as a determinant-like invariant; one could test whether it is compatible with Cartier duality or with composition of group schemes.
Because the tensor product is shown not to commute with arbitrary base change, the isomorphism $\Lambda^n\alpha_{p^n}\cong\alpha_p$ should be viewed as field-specific; a natural extension is to compute $\Lambda^n\alpha_{p^n}$ over non-perfect fields or over rings, where the conclusion may fail.
The vanishing pattern $\Lambda^mG=0$ for $m>n$ mirrors exterior powers of a length-$n$ module, so a Dieudonné-module interpretation is plausible: under the usual correspondence, $\Lambda^nG$ should correspond to the $n$-th exterior power of the Dieudonné module; checking this would give a computational route and explain the quotient-by-$\alpha_p$ structure.
The unproved proper-subgroup premise is likely derivable from the Dieudonné module classification of local-local group schemes; supplying that proof would remove the main gap in the induction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the explicit computations and the structural result Λ^n α_{p^n} ≅ α_p are the real content here, and they look right; the main weakness is a load-bearing unproved assertion about subgroup existence in Prop 3.17. The framework is Pink's, and the paper acknowledges that.

What is actually new: the explicit Hopf algebra calculations in Section 2 — Hom(α_{p^n}, α_{p^m}), Mult for products of α_{p^{n_i}} into G_a, and the Sym/Alt formulas for G_a — are clean and self-contained, and I checked a couple of them line by line. They give concrete instances of Pink's inner Hom formalism for non-reduced finite group schemes, which is useful. Proposition 3.26 (Λ^n α_{p^n} ≅ α_p over perfect odd-characteristic fields) is a nice structural payoff; the proof strategy via Verschiebung and Lemma 3.21 is sensible.

Where it gets soft. Proposition 3.17 is the key step for the main theorem. The induction needs to split a local-local group scheme of order p^n (n>1) by a proper subgroup scheme, and also uses that every subgroup of a local-local is local-local. Both facts are asserted without proof or citation. The subgroup-existence fact is probably true (it follows from Dieudonné theory or from existence of order-p subgroup schemes in finite connected group schemes), but the paper does not show it, and the assertion is not in Pink's notes as far as the text says. As written, the induction cannot start for n>1. This gap is internal to the paper, not just a missing reference, and it propagates to Prop 3.26 because that proof uses the epimorphism α_p^{⊗n} → Λ^n α_{p^n} from Prop 3.17. A referee should insist on a lemma here — or at least a precise citation.

Second, the foundational objects (inner Hom, tensor product, Sym/Alt) come from Pink's unpublished manuscript [6]. That is not a flaw by itself, but it does mean the paper's foundations are not publicly checkable. Combined with the duplicated Prop 2.14 (same statement as Prop 2.12) and the self-citation in Lemma 3.14's proof (should be Lemma 3.13), the manuscript reads as a polished but not final draft.

The direct tensor calculations for α_p (Examples 3.9 and 3.10) are good, and the fact that tensor product does not commute with base change is honestly flagged.

Bottom line: this is the kind of paper I would send to a referee with a request to fix the Prop 3.17 gap and check the subgroup assertions. The computations alone justify the time.

Referee Report

3 major / 5 minor

Summary. The paper develops a multilinear algebra formalism for commutative group schemes: inner Hom, multilinear morphism schemes, tensor products, symmetric powers, and alternating powers. The first half establishes adjunction-type isomorphisms for multilinear morphisms and computes explicit examples over fields of characteristic p, including Hom(α_{p^n}, α_{p^m}) and Mult(α_{p^{n_1}}×⋯×α_{p^{n_r}}, G_a). The second half proves structural results about alternating powers, including a short exact sequence analogue of tensor products (Theorem 3.16), and culminates in Proposition 3.26: over a perfect field k of odd characteristic p, Λ^n α_{p^n} ≅ α_p for every n ≥ 1.

Significance. If the gaps are repaired, this would be a useful contribution: it gives a systematic treatment of multilinear constructions for commutative group schemes, with explicit, checkable Hopf-algebra computations, and it establishes a striking analogue of exterior powers over finite local-local group schemes. The explicit formulas for Hom(α_{p^n}, α_{p^m}) and the alternating-power calculation Λ^n α_{p^n} ≅ α_p are concrete and likely to be of independent interest. However, the main theorem currently depends on an unproved structural assertion about local-local group schemes, and the foundational objects are quoted from an unpublished manuscript, so the paper is not yet self-contained enough for the claims as they stand.

major comments (3)

[Proposition 3.17, proof (p. 40)] The induction step asserts without proof that every local-local commutative group scheme of order p^n with n > 1 contains a proper subgroup scheme G′, and that every subgroup (and quotient) of a local-local group scheme is local-local. This is load-bearing: part (b) of Proposition 3.17, and hence the epimorphism α_p^{⊗n} → Λ^n α_{p^n} used in Proposition 3.26, rely on this induction. Please either prove the existence of such G′ (e.g., via the kernel of Frobenius or Verschiebung) or supply a published reference; as written, the induction cannot start for n > 1.
[Lemma 3.14, proof (p. 36)] The proof invokes the lemma being proved: 'We can therefore apply Lemma 3.14, so there is a multilinear morphism…'. This is circular. The intended reference appears to be Lemma 3.13, which gives the factorization through π when the restriction to G′ is zero. Also, the sentence 'by Lemma 3.13 φ′ is also alternating' should likely cite Lemma 3.11 instead. Because Lemma 3.14 is used in Theorem 3.16(a) and hence in Propositions 3.17 and 3.26, these citations must be corrected before the proof becomes valid.
[Section 3, foundational results quoted from [6]] The existence of inner Hom, tensor products, and symmetric and alternating powers is quoted from Pink's unpublished notes [6, Theorems 3.10 and 4.3]. Since [6] is listed as 'in preparation' and is not publicly accessible, the paper's foundational objects are not independently verifiable. The author should either include complete statements (or proofs) of the quoted results or arrange for the cited manuscript to be made available; otherwise the main results are conditional on an inaccessible source.

minor comments (5)

[Proposition 2.14 (p. 13)] Proposition 2.14 duplicates Proposition 2.12 verbatim; if a different statement was intended (e.g., the underlined version), please provide it, otherwise remove the duplicate.
[Proof of Proposition 2.27 (p. 22)] In the displayed formula near the end of the proof, the term s^{p^{n_1}} should be s^{p^{i_1}}; the exponent must match the summation index i_1.
[Proof of Lemma 3.14 (p. 36)] In addition to the incorrect internal references, the line 'by Lemma 3.13 φ′ is also alternating' should probably refer to Lemma 3.11, which establishes that precomposing with an epimorphism preserves the alternating property. Please check all lemma citations in this proof.
[Proof of Proposition 3.26 (p. 49)] The line 'Hom(Λ^n α_{p^n}, G_a) ≅ G_a(k) = k' conflates the group scheme Hom with its k-points; the notation should distinguish the scheme from its value at k, e.g., by writing Hom(Λ^n α_{p^n}, G_a)(k) ≅ k.
[Throughout] There are numerous typographical issues, including 'Qr' instead of 'Q' in the proof of Proposition 2.3, inconsistent spacing in 'G1...,G r,H 1...,H s', and broken hyphenation. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are direct Hopf-algebra and universal-property arguments, with external foundational citations that are not self-citations.

full rationale

The derivation chain is self-contained once the external foundational theorems of Pink's manuscript [6] are accepted as input. Inner Hom, Mult, Sym, Alt, tensor products, symmetric powers, and alternating powers are all introduced by universal properties, and the subsequent computations (e.g., Propositions 2.24, 2.25, 2.27, 2.29, and Examples 3.9, 3.10) are explicit Hopf-algebra calculations with no fitted parameters and no appeal to the target results. The central Proposition 3.26 is proved from Proposition 3.17, Lemma 3.21, and Lemma 3.25; none of these presupposes the isomorphism Lambda^n alpha_{p^n} = alpha_p. In particular, Proposition 3.17 is an induction using Theorem 3.16 and right exactness of tensor products, and Lemma 3.25 independently proves vanishing of the Verschiebung by reducing from general H to finite-type H to finite H. No 'prediction' is renamed input, and no load-bearing step reduces by construction to its own conclusion. There is one genuinely unproved structural assertion in the proof of Proposition 3.17, namely that a nontrivial local-local group scheme of order p^n, n>1, possesses a proper subgroup scheme, and that every subgroup of a local-local group scheme is again local-local; this is a possible mathematical gap rather than circularity, since the assertion is independent of the statements being proved and is not justified by citing the paper's own results. The heavy reliance on Pink's unpublished manuscript is a verification-risk concern, not a self-citation chain: Pink is not an author of the present paper, and the cited theorems are used as external existence results, not as a way of importing the paper's own conclusions.

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

The paper's new results rest on foundational existence theorems quoted from Pink's unpublished notes [6]; without these, the objects Hom, Mult, tensor product, and powers are not defined. Additional unproved standard facts, such as the existence of proper subgroups in local-local group schemes, are also used without citation.

assumptions (4)

domain assumption Theorems 3.10 and 4.3 of Pink [6]: inner Hom of finite flat group schemes exists with the stated properties, and tensor products of finite group schemes over a field exist as pro-finite group schemes.
Used throughout the paper (e.g., Remark 2.2, Remark 3.2) without proof; the new results presuppose these existence theorems from an unpublished manuscript.
domain assumption Every local-local finite commutative group scheme of order p^n with n > 1 contains a proper subgroup scheme.
Assumed in the proof of Proposition 3.17 to start the induction; not proved or cited in the paper.
standard math For a perfect field k, the Frobenius pullback functor (−)^{(p)} is an equivalence of categories on the category of affine commutative group schemes over k.
Used in Lemma 3.24 to identify Λ^n(G^{(p)}) with (Λ^nG)^{(p)}; this is standard for perfect fields.
domain assumption The group scheme α_p is simple, i.e., its only subgroup schemes are 0 and itself.
Used in Proposition 3.26 to conclude that a nonzero epimorphism from α_p is an isomorphism; this is standard but not proved in the paper.

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Cite this review

Pith. "Pith review of Tensor Operations on Group Schemes." pith.science (2026). https://pith.science/paper/O46KICQH

@misc{pith2026190805452,
  author       = {Pith},
  title        = {Pith review of: Tensor Operations on Group Schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O46KICQH}},
  note         = {Machine review of arXiv:1908.05452}
}

read the original abstract

In this paper we study multilinear morphisms between commutative group schemes and the associated tensor constructions. We will also do some explicit calculations and give examples that show that this theory behaves in a way that one would naturally expect.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[6]

Pink, Multilinear theory of commutative group schemes, in preparation

R. Pink, Multilinear theory of commutative group schemes, in preparation

work page
[1]

Demazure, Lectures onp-divisable groups, Springer-Verlag, 1972

M. Demazure, Lectures onp-divisable groups, Springer-Verlag, 1972

work page 1972
[2]

Demazure and P

M. Demazure and P. Gabriel, Groupes Alg `ebriques, Tome I , North- Holland, 1970

work page 1970
[3]

Demazure and A

M. Demazure and A. Grothendieck et al., S´eminaire de G ´eom´etrie Alg`ebrique: Sch ´emas en Groupes , Lecture Notes in Mathematics 151, 152, 153, Springer-Verlag, 1970

work page 1970
[4]

Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer-Verlag, 1966 49

F. Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer-Verlag, 1966 49

work page 1966
[5]

Oort and J

F. Oort and J. Tate, Group schemes of prime order, Annales scientiﬁques de l’´E.N.S. 4e s´erie, tome 3,n◦1, pp. 1-21, 1970

work page 1970
[7]

Pink, Finite group schems and p-divisable groups , Lecture notes in WS 2004/05 at ETH Z ¨urich, available on the website http://www.math.ethz.ch/∼pink/FiniteGroupSchemes.html

R. Pink, Finite group schems and p-divisable groups , Lecture notes in WS 2004/05 at ETH Z ¨urich, available on the website http://www.math.ethz.ch/∼pink/FiniteGroupSchemes.html

work page 2004
[8]

W. C. Waterhouse, Introduction to Afﬁne Group schemes , Springer- Verlag, 1979 50

work page 1979

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Pink, Multilinear theory of commutative group schemes, in preparation

R. Pink, Multilinear theory of commutative group schemes, in preparation

work page

[2] [1]

Demazure, Lectures onp-divisable groups, Springer-Verlag, 1972

M. Demazure, Lectures onp-divisable groups, Springer-Verlag, 1972

work page 1972

[3] [2]

Demazure and P

M. Demazure and P. Gabriel, Groupes Alg `ebriques, Tome I , North- Holland, 1970

work page 1970

[4] [3]

Demazure and A

M. Demazure and A. Grothendieck et al., S´eminaire de G ´eom´etrie Alg`ebrique: Sch ´emas en Groupes , Lecture Notes in Mathematics 151, 152, 153, Springer-Verlag, 1970

work page 1970

[5] [4]

Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer-Verlag, 1966 49

F. Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer-Verlag, 1966 49

work page 1966

[6] [5]

Oort and J

F. Oort and J. Tate, Group schemes of prime order, Annales scientiﬁques de l’´E.N.S. 4e s´erie, tome 3,n◦1, pp. 1-21, 1970

work page 1970

[7] [7]

Pink, Finite group schems and p-divisable groups , Lecture notes in WS 2004/05 at ETH Z ¨urich, available on the website http://www.math.ethz.ch/∼pink/FiniteGroupSchemes.html

R. Pink, Finite group schems and p-divisable groups , Lecture notes in WS 2004/05 at ETH Z ¨urich, available on the website http://www.math.ethz.ch/∼pink/FiniteGroupSchemes.html

work page 2004

[8] [8]

W. C. Waterhouse, Introduction to Afﬁne Group schemes , Springer- Verlag, 1979 50

work page 1979