On Noether's Degree Bound for Finite Group Schemes

Christiane Ott; Christian Liedtke; Gregor Kemper

REVIEW 1 major objections 4 minor 2 cited by

On Noether's Degree Bound for Finite Group Schemes

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every linearly reductive finite group scheme $G$, the invariant ring of any representation is generated in degrees at most $|G|$.

desk verdict Solid paper: Noether's bound survives for linearly reductive finite group schemes, with a Richman-style unboundedness example for alpha_q and a Molien formula; referee it. read the letter →

arxiv 2505.24752 v1 pith:5R4NYYR3 submitted 2025-05-30 math.AC

Gregor Kemper , Christian Liedtke , Christiane Ott This is my paper

classification math.AC MSC 13A5014L2414L15

keywords invarianttheoryfinitegroupschemeslinearlyreductiveNoetherdegreeboundinfinitesimalMolienformulapositivecharacteristicWittvectors

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

Noether's classical degree bound says that for a finite group acting on a polynomial ring in characteristic zero, the invariant ring is generated by invariants of degree at most the group order. This paper establishes the same bound for every linearly reductive finite group scheme: if $G$ is such a scheme, then $\beta(G) \le |G|$, meaning that for every finite-dimensional representation $V$, the invariant ring $S(V)^G$ is generated by homogeneous invariants of degree at most $|G|$. Because linear reductivity is the scheme-theoretic analogue of the condition that the group order is prime to the characteristic, this is the correct positive-characteristic generalization of Noether's theorem. The paper also shows that the infinitesimal group scheme $\alpha_q$ has $\beta(\alpha_q)=\infty$, so without linear reductivity no bound depending only on the group can exist, and it derives a Molien formula for linearly reductive finite group schemes via a lift to Witt vectors.

What carries the argument

The object that carries the proof is the Noether number $\beta(G)$, defined as the supremum over finite-dimensional representations $\rho$ of the minimal degree $m$ such that $S(V)^G$ is generated by homogeneous invariants of degree at most $m$. The mechanism is the structural decomposition of a linearly reductive finite group scheme $G$ as $G^0 \rtimes G^{\mathrm{\'et}}$, where $G^0$ is a product of $\mu_{p^{e_i}}$ and $G^{\mathrm{\'et}}$ has order prime to $p$; invariant rings are computed in two stages as $S(V)^G=(S(V)^{G^0})^{G^{\mathrm{\'et}}}$, with the first stage controlled by monomial invariants of a diagonalizable action and the second by the classical Noether bound for the finite group $G^{\mathrm{\'et}}(k)$. For the unbounded examples, the device is an explicit family of invariant polynomials $g_l$ in the $\alpha_q$-module $lV_2$ whose degree is forced to be at least $l(q-1)$ by a degree-counting argument.

What would settle it

Find a finite linearly reductive group scheme $G$ and a finite-dimensional representation $V$ for which $S(V)^G$ contains a homogeneous invariant of degree $>|G|$ that is not in the subalgebra generated by invariants of degree $\le |G|$; such a pair would disprove Theorem 3.3. Equivalently, one could look for a linearly reductive $G$ whose abstract decomposition into $G^0 \rtimes G^{\mathrm{\'et}}$ fails over some field, since the proof's reduction depends on that structure.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.3: for a linearly reductive finite group scheme $G$ over any field, $\beta(G) \le |G|$. The proof reduces to algebraically closed fields and uses the classification of such group schemes, which writes $G$ as a semidirect product $G^0 \rtimes G^{\mathrm{\'et}}$, where $G^0$ is a product of multiplicative group schemes $\mu_{p^{e_i}}$ and $G^{\mathrm{\'et}}$ has order prime to $p$. Invariants for $G^0$ are monomial invariants under a diagonal action, and the Noether bound for the finite group $\mathbb{Z}/p^{e_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/p^{e_s}\mathbb{Z}$ bounds the degrees of their generators; invariants for the \'etale part are then bounded by the classical Noether bound for a finite group of order prime to $p$. In the opposite direction, the paper constructs, for each $q=p^e$ and each $l$, a representation of $\alpha_q$ on $2l$ variables whose invariant ring contains a homogeneous invariant of degree $l(q-1)$ that cannot be generated by lower-degree invariants, so $\beta(\alpha_q)=\infty$. Finally, using a flat lift to the ring of Witt vectors, the paper proves a Molien formula expressing the Hilbert series of $S(V)^G$ as the classical Molien series for an associated constant group $G_{\mathrm{abs}}$ acting on $\mathbb{C}^n$.

Load-bearing premise

The proof of the main bound rests on the imported classification that every linearly reductive finite group scheme splits as a semidirect product $G^0 \rtimes G^{\mathrm{\'et}}$ with $G^0$ a product of $\mu_{p^{e_i}}$ and $G^{\mathrm{\'et}}$ of order prime to $p$; if that classification or the splitting fails, the reduction to the two bounded cases collapses.

Editorial extensions

If this is right

Every linearly reductive finite group scheme $G$ has a finite Noether number, bounded by $|G|$, so its invariant rings admit explicit finite generating sets of bounded degree.
In characteristic zero the theorem recovers Noether's original bound, since every finite group scheme is then linearly reductive and constant.
In positive characteristic the result covers diagonalizable group schemes such as $\mu_{p^e}$ and their semidirect products with prime-to-$p$ \'etale groups, where no such bound was previously known.
For $\alpha_q$, the family of representations $lV_2$ forces $\beta(\alpha_q)=\infty$, showing that the linearly reductive hypothesis cannot be dropped.
The Molien formula computes the Hilbert series of $S(V)^G$ from an abstract group over $\mathbb{C}$, so the series can be obtained without computing invariants even when $G$ has no nontrivial $k$-rational points.

Reading between the lines

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

A natural conjecture suggested by the paper is that every finite group scheme that is not linearly reductive has $\beta(G)=\infty$; the paper proves this for $\alpha_q$ and for schemes whose \'etale quotient is not linearly reductive, but not for mixed cases.
The Witt-vector Molien formula implies that the Hilbert series of invariants is insensitive to whether $\beta$ is finite; Example 5.7 already shows that $\beta$ can jump under lift while the Molien series stays the same, so one could test how far this decoupling goes by searching for linearly reductive schemes where the bound $|G|$ is attained or nearly attained.
The two-stage argument suggests a template for other classes of group schemes: decompose the group into a part whose invariants are monomial and a part governed by classical group invariants; applying the same template to group schemes whose connected part is not multiplicative could yield finite bounds under weaker hypotheses.
Because the unboundedness proof uses only Lucas's theorem and degree counts, the same construction likely gives explicit generators for the invariant rings of $\alpha_q$-modules and could be used to compute the exact Noether number of $lV_2$ rather than only the lower bound $l$.

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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

Colleague, quick take: this is a good paper, better than the arXiv abstract suggests. The central theorem is the natural group-scheme analogue of Noether's bound, and the proof is cleaner than I expected. Lemma 3.1 is the key step: representations of G^0 = product of mu_{p^{e_i}} are diagonalized, invariant monomials are governed by the congruences (1), and those same congruences describe invariants of the corresponding finite abelian group over C, so the characteristic-zero Noether bound transfers. Lemma 3.2 then hands you the etale quotient, and DK15's bound applies because |G^et| is prime to p. The reduction to algebraically closed field via Lemma 2.1 is standard. I don't see a gap in Theorem 3.3. The reliance on the classification of linearly reductive finite group schemes is real, but [AOV08, Ha15, Ch92, Na61] is established literature, not a hidden assumption. The p=0 case is immediate from Lemma 2.1 plus the classical bound, so the theorem covers all characteristics.

Second, the alpha_q example in Proposition 4.1 is the right analogue of Richman's result. The proof is long and binomial-heavy; I did not re-derive every Lucas's theorem step, but the induction has the right shape and the conclusion beta(alpha_q, rho) >= l is consistent with Richman. This is a genuine counterexample showing beta = infinity for an infinitesimal group scheme, which was missing in the literature.

Third, the Molien formula is a nice bonus. It relies on LMM21's canonical lift and specialisation map, which are not yet published; that is a dependency to watch, not a flaw. Example 5.6 makes the point that the Molien series is not a function of k-rational points, and Example 5.7 shows beta can change under lifting while the Molien series does not.

Soft spots: Proposition 4.1 is hard to check line-by-line; there are minor typos (e.g., 'propostion', 'SG0' for 'SG', 'x1,...,n' in 5.1), and the proof of Proposition 5.1 is compressed. None of this touches the central claim. The paper is honest about what is imported and does not oversell the results.

Who it's for: people working in invariant theory, quotient singularities, and finite group schemes. It deserves a serious referee. My recommendation: send it to review; the referee should verify the binomial manipulations carefully and check LMM21's status, but the main theorem is solid.

Referee Report

1 major / 4 minor

Summary. The paper studies the Noether degree bound and Molien series for finite group schemes. Its main theorem (Theorem 3.3) states that for every finite linearly reductive group scheme G over a field, the invariant ring of any finite-dimensional representation is generated in degrees at most |G|, generalizing Noether's classical bound. The proof reduces to the algebraically closed case, splits the group into its connected multiplicative part and its étale quotient, and applies the classical Noether bound for finite groups of order prime to p. The paper also constructs, for the infinitesimal group scheme α_q, representations for which the generating degree grows linearly with the number of copies, proving β(α_q)=∞ (Proposition 4.1), and it gives a Molien formula for finite linearly reductive group schemes via lifting to Witt vectors (Theorem 5.4).

Significance. Theorem 3.3 is a clean and natural generalization of classical results, and the proof is sound: it uses the standard classification of linearly reductive finite group schemes and reduces the key Lemma 3.1 to a monomial argument over the corresponding product of cyclic groups. Proposition 4.1 provides the first infinitesimal analogue of Richman's unboundedness result and is a valuable contribution. The Molien formula, once corrected as noted below, is also useful. The paper's reliance on external classification theorems and on [LMM21] for lifting facts is appropriate and clearly stated. The proofs are detailed and, apart from the issues in Section 5 discussed below, appear correct.

major comments (1)

[Section 5] Theorem 5.4(1) as stated is false for non-algebraically closed fields. The formula 1/|G| ∑_{g∈G(k)} 1/det(1−g^{−1}t, V_k) uses only k-rational points, but for example over k=Q and n odd, μ_n has μ_n(Q)={1} while the Molien series is 1/(1−t^n); this contradicts the displayed formula and is in fact acknowledged in Example 5.6. The proof itself reduces to the algebraically closed case, so the statement should either assume k is algebraically closed or replace G(k) by G(\bar k). The same issue appears in the definition of the abstract group G_abs: for p=0 the paper defines G_abs≅G(k), but the abstract Molien series must use geometric points, not rational points (again see Example 5.6). These two points need to be corrected for the Molien formula to be stated and applied correctly.

minor comments (4)

[Lemma 3.1 (page 3)] The condition e_i > 1 should be e_i ≥ 1: μ_p is also a finite linearly reductive group scheme of multiplicative type, and the proof works for e_i=1 as well.
[Throughout the paper] There are several typos: 'propostion' for 'proposition', 'diagramme' for 'diagram', 'linarly' for 'linearly', 'simplicitiy' for 'simplicity', and in Proposition 5.1 the polynomial ring is written as k[x_1,...,n] instead of k[x_1,...,x_n].
[Proof of Proposition 4.1 (final display, page 9)] The line 'q−1 (3) = deg_{x_1,...,x_l} h_r' is confusing because (3) refers to an earlier binomial identity; it should refer to assumption (3) in the list of reductions. Please renumber or clarify.
[Theorem 3.3 and Section 3] The proof of Theorem 3.3 is written only for characteristic p>0; the p=0 case is immediate from Lemma 2.1 and the classical Noether bound, but this is not stated explicitly. A sentence in Section 3 would remove ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main bound reduces to the classical char-0 Noether bound, the Fleischmann-Fogarty theorem, and an external classification, none of which assume the paper's conclusion.

full rationale

The central theorem (Theorem 3.3, beta(G) <= |G| for linearly reductive finite group schemes) is obtained from three external inputs, not from its own conclusion. First, the connected-etale decomposition G = G^0 ⋊ G^et with G^0 multiplicative and |G^et| prime to p is imported from the classification literature [AOV08, Ha15, Ch92, Na61], none of which is authored by Kemper, Liedtke, or Ott. Second, Lemma 3.1 transfers the characteristic-zero Noether bound for the finite abelian group G~ = ∏ Z/p^{e_i} acting on C^n to the group scheme G^0 = ∏ μ_{p^{e_i}} acting on k[x_1,...,x_n]: the invariant monomials of the two actions are governed by exactly the same congruences (equation (1)), so the classical bound (cited independently to [Sch91, Lemma 2.1]) yields generation in degrees <= |G~| = |G^0|. This is a genuine reduction, not a restatement, because the target result for infinitesimal μ_{p^e}-schemes is not assumed anywhere in the classical statement. Third, the step from S^{G^0} to (S^{G^0})^{Ĝ} uses [DK15, Theorem 3.22], the classical Fleischmann-Fogarty bound for finite groups of order prime to p, originally [Fl00, Fo01], both cited in the introduction; the fact that Gregor Kemper co-authored the textbook presentation does not make the theorem a product of this paper. Lemma 3.2 is proven in-paper, and the p = 0 case is the classical Noether bound. Section 4 (β(α_q) = ∞) is a self-contained computation with a direct invariant construction, independent of everything else. Section 5 cites [LMM21] repeatedly; this is a self-citation (Liedtke is a co-author), and those lifting and specialisation results are structurally load-bearing for the proof of the Molien formula as written. However, they are independent support: they predate this paper, have stated assumptions that include neither Noether's bound nor Molien's formula for group schemes, and are used as tools to transport the classical Molien formula (cited to [Mo97]/[DK15]) to the Witt-vector setting. No fitted parameter is renamed as a prediction, no known result is repackaged under new coordinates, and no uniqueness theorem from the authors' own prior work is invoked to force a choice. The only flagged dependency, the classification of linearly reductive finite group schemes, is a standard external literature input; if it failed the proof would fail, but that is a correctness risk, not circularity.

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

The paper introduces no free numerical parameters and no new postulated entities. The only inputs are the group scheme G, a representation V, and the established structural theorems listed above. The canonical lift and the abstract group G_abs are definitions imported from prior work, not new entities.

assumptions (6)

standard math Classification of finite linearly reductive group schemes (Chin, Hashimoto, Nagata, Abramovich-Olsson-Vistoli): G is linearly reductive iff G^ét has order prime to p and G^0 is of multiplicative type.
Section 2, paragraph after Lemma 2.1; used to write G^0 as product of μ_{p^{e_i}} and to treat G^ét as a finite group of order prime to p in Section 3.
standard math Connected-étale sequence splits over perfect fields: G ≅ G^0 ⋊ G^ét.
Invoked in Section 3 (By [Ha15, Section (2.10)] we have G = G^0 ⋊ G^ét) to realize invariants as iterated invariants (Lemma 3.2(b)).
standard math Noether bound for finite groups in characteristic 0 and for finite groups of order prime to p (Fleischmann, Fogarty).
Used in Lemma 3.1 for the model abelian group \tilde G and after Lemma 3.2 for the étale quotient; also used in characteristic 0 in the proof of Theorem 5.4.
standard math Lucas's theorem on binomial coefficients modulo p.
Used repeatedly in Proposition 4.1 to evaluate sums of binomial coefficients modulo p and to conclude that certain sums vanish.
standard math Molien's theorem for finite groups in characteristic zero.
Invoked in Section 5.3 to compute the Molien series of the lifted constant group scheme G_K over a characteristic zero field.
standard math Existence of canonical flat lifts of linearly reductive finite group schemes over the Witt ring W(k) and properties of the specialization map (LMM21, Propositions 2.4 and 2.9).
The Molien formula proof in Section 5 relies on lifting G to characteristic zero and on sp preserving degrees, direct sums, tensor products and duals. This is a theorem by one of the authors with others, not proved here.

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Pith. "Pith review of On Noether's Degree Bound for Finite Group Schemes." pith.science (2026). https://pith.science/paper/5R4NYYR3

@misc{pith2026250524752,
  author       = {Pith},
  title        = {Pith review of: On Noether's Degree Bound for Finite Group Schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5R4NYYR3}},
  note         = {Machine review of arXiv:2505.24752}
}

abstract

This paper establishes Noether's classical degree bound $\beta(G) \le |G|$ for finite and linearly reductive group schemes. On the other hand, we provide examples of infinitesimal group schemes where $\beta(G)$ is unbounded. We also generalize Molien's formula to finite and linearly reductive group schemes.

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Forward citations

Cited by 2 Pith papers

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Works this paper leans on

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Reviewed August 7, 2026 · model on record in the stance chip above.

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work page 2015

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J. Fogarty, On Noether's bound for polynomial invariants of a finite group, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5--7

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