Locally finite sets of derivations

Andriy Regeta; Daniel Daigle; Michael Chitayat

arxiv: 2606.08677 · v1 · pith:OENJN2EX · submitted 2026-06-07 · math.AC · math.AG· math.RA

Locally finite sets of derivations

Michael Chitayat , Daniel Daigle , Andriy Regeta This is my paper

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classification math.AC math.AGmath.RA

keywords locally finite derivationsLie subalgebrasquasi-affine varietiesintegrable derivationsDer O(X)solvable Lie algebrascoordinate rings

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The pith

A finitely generated solvable Lie subalgebra consisting of locally finite derivations on the coordinate ring of a quasi-affine variety is itself locally finite.

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

The paper examines conditions that force a Lie subalgebra of derivations on an algebra to be locally finite as a collection. It establishes that when the algebra is the ring of regular functions on a quasi-affine variety over any field, finite generation and solvability of the subalgebra together with local finiteness of each member suffice to make the whole subalgebra locally finite. Under the further restrictions that the field is algebraically closed of characteristic zero and the variety is irreducible and affine, the same hypotheses yield that the subalgebra is integrable. A reader would care because these conclusions supply a criterion that turns local information about individual derivations into global control over the subalgebra they generate.

Core claim

Given an algebra B over a field k, the paper studies conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application, if X is a quasi-affine variety over an arbitrary field k and L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If moreover k is algebraically closed and of characteristic zero and X is irreducible and affine, then L is integrable.

What carries the argument

finitely generated solvable Lie subalgebras of locally finite derivations inside Der O(X) for quasi-affine X

Load-bearing premise

The Lie subalgebra must be finitely generated and solvable, every derivation in it must already be locally finite, and the underlying variety must be quasi-affine.

What would settle it

An explicit example of a finitely generated solvable Lie subalgebra of locally finite derivations on the coordinate ring of some quasi-affine variety whose joint action on the ring fails to be locally finite.

read the original abstract

Given an algebra B over a field k, we study conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application of our results, we show that if X is a quasi-affine variety over an arbitrary field k, and if L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If, moreover, k is algebraically closed and of characteristic zero, and X is irreducible and affine, then L is integrable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean criterion: finitely generated solvable Lie subalgebras of locally finite derivations on quasi-affine varieties are locally finite, and integrable under char-0 algebraically closed affine conditions.

read the letter

The main result organizes conditions under which a Lie subalgebra of derivations is locally finite as a set, then applies them to show that on a quasi-affine variety over any field, a finitely generated solvable L consisting of locally finite derivations must itself be locally finite. With k algebraically closed of characteristic zero and X irreducible affine, L is also integrable.

The work does a reasonable job making the hypotheses explicit and stating the conclusions without extra claims. The application to solvable subalgebras on quasi-affine varieties appears to be the fresh part, building on whatever general conditions they develop earlier.

Soft spots are limited. The general conditions on local finiteness may draw from existing ideas in the area rather than introduce major new machinery, so the advance is mostly in the targeted application. The statement itself shows no circularity or hidden assumptions, and the stress-test confirms the implication is consistent as written. Proof details are not visible here, but the formulation looks direct.

This is for specialists in commutative algebra and algebraic geometry who work with derivation modules and their Lie algebras. Readers already following results on local finiteness or integrability of vector fields would find the precise statement useful.

It deserves peer review to check the proofs and see how much the general conditions add beyond the application.

Referee Report

0 major / 2 minor

Summary. The manuscript studies conditions under which a Lie subalgebra of Der(B) for an algebra B over a field k is locally finite as a set. The central application states that if X is a quasi-affine variety over arbitrary k and L is a finitely generated solvable Lie subalgebra of Der(O(X)) consisting entirely of locally finite derivations, then L is locally finite; moreover, when k is algebraically closed of characteristic zero and X is irreducible and affine, L is integrable.

Significance. If the proofs hold, the result supplies an explicit criterion linking finite generation, solvability, and local finiteness of derivation Lie algebras on quasi-affine varieties, with a stronger integrability conclusion under standard hypotheses on k and X. This may be useful in contexts where local finiteness controls the structure of automorphism groups or algebraic actions.

minor comments (2)

The abstract and introduction repeat the main statement verbatim; a single concise formulation would improve readability.
Notation for the structure sheaf O(X) and the derivation module Der(O(X)) should be introduced explicitly in the first section if the paper targets a broad audience in commutative algebra.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of the results, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states and proves a conditional theorem: under explicit hypotheses on the quasi-affine variety X, the finitely generated solvable Lie subalgebra L of Der(O(X)), and the field k, L is locally finite (and integrable under extra conditions). No equations, parameters, or claims reduce by construction to fitted inputs or self-referential definitions. The result is a standard algebraic proof resting on general conditions rather than any of the enumerated circularity patterns. No load-bearing self-citations or ansatzes are invoked in a way that collapses the derivation to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the result appears to rely on standard notions of derivations, Lie algebras, and varieties without introducing new entities.

pith-pipeline@v0.9.1-grok · 5612 in / 1098 out tokens · 17350 ms · 2026-06-27T17:22:53.830214+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 3 canonical work pages · 3 internal anchors

[1]

Borel subalgebras of Lie algebras of vector fields

I. Arzhantsev and M. Zaidenberg, Borel subalgebras of L ie algebras of vector fields , arXiv:2510.17223v3, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Daigle, Locally nilpotent sets of derivations, Polynomial rings and affine algebraic geometry, Springer Proc

D. Daigle, Locally nilpotent sets of derivations, Polynomial rings and affine algebraic geometry, Springer Proc. Math. Stat., vol. 319, Springer, Cham, [2020] 2020, pp. 41--71. 4113941

2020
[3]

J. P. Furter and H. Kraft, On the geometry of the automorphism groups of affine varieties, arXiv:1809.04175, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018
[4]

E. S. Golod, Some problems of B urnside type , Proc. I nternat. C ongr. M ath. ( M oscow, 1966), Izdat. ``Mir'', Moscow, 1968, (Russian), pp. 284--289

1966
[5]

Hartshorne, Algebraic G eometry , GTM , vol

R. Hartshorne, Algebraic G eometry , GTM , vol. 52, Springer- V erlag, 1977

1977
[6]

H.\ Kraft and M.\ Zaidenberg, Algebraically generated groups and their L ie algebras , J. Lond. Math. Soc. (2) 109 (2024), no. 2, Paper No. e12866, 39. 4704157

2024
[7]

L. H. Rowen, Ring theory. V ol. II , Pure and Applied Mathematics, vol. 128, Academic Press, Inc., Boston, MA, 1988

1988
[8]

Skutin, Maximal L ie algebras among locally nilpotent derivations , Mat

A. Skutin, Maximal L ie algebras among locally nilpotent derivations , Mat. Sb. 212 (2021), no. 2, 138--146. 4223965

2021
[9]

Locally finite solvable Lie algebras of derivations

M. Zaidenberg, Locally finite solvable L ie algebras of derivations , arXiv:2604.02864v3, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

Borel subalgebras of Lie algebras of vector fields

I. Arzhantsev and M. Zaidenberg, Borel subalgebras of L ie algebras of vector fields , arXiv:2510.17223v3, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Daigle, Locally nilpotent sets of derivations, Polynomial rings and affine algebraic geometry, Springer Proc

D. Daigle, Locally nilpotent sets of derivations, Polynomial rings and affine algebraic geometry, Springer Proc. Math. Stat., vol. 319, Springer, Cham, [2020] 2020, pp. 41--71. 4113941

2020

[3] [3]

J. P. Furter and H. Kraft, On the geometry of the automorphism groups of affine varieties, arXiv:1809.04175, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[4] [4]

E. S. Golod, Some problems of B urnside type , Proc. I nternat. C ongr. M ath. ( M oscow, 1966), Izdat. ``Mir'', Moscow, 1968, (Russian), pp. 284--289

1966

[5] [5]

Hartshorne, Algebraic G eometry , GTM , vol

R. Hartshorne, Algebraic G eometry , GTM , vol. 52, Springer- V erlag, 1977

1977

[6] [6]

H.\ Kraft and M.\ Zaidenberg, Algebraically generated groups and their L ie algebras , J. Lond. Math. Soc. (2) 109 (2024), no. 2, Paper No. e12866, 39. 4704157

2024

[7] [7]

L. H. Rowen, Ring theory. V ol. II , Pure and Applied Mathematics, vol. 128, Academic Press, Inc., Boston, MA, 1988

1988

[8] [8]

Skutin, Maximal L ie algebras among locally nilpotent derivations , Mat

A. Skutin, Maximal L ie algebras among locally nilpotent derivations , Mat. Sb. 212 (2021), no. 2, 138--146. 4223965

2021

[9] [9]

Locally finite solvable Lie algebras of derivations

M. Zaidenberg, Locally finite solvable L ie algebras of derivations , arXiv:2604.02864v3, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026