arxiv: 2604.09476 · v1 · submitted 2026-04-10 · 🧮 math.KT · math.AC

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Improved injective stability for relative K₁Sp-groups

Kuntal Chakraborty, Sourjya Banerjee

Pith reviewed 2026-05-10 15:53 UTC · model grok-4.3

classification 🧮 math.KT math.AC

keywords relative K-theoryK1-groupssymplectic K-theorystability theoremsVorst theoremKaroubi periodicitysmooth affine algebrasinjective stability

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The pith

Relative versions of Vorst's theorem and Karoubi periodicity improve injective stability bounds for relative K1 and K1Sp groups.

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

The authors prove that for a regular k-spot R and an ideal I such that R/I is regular, the group of invertible matrices over R[X] relative to I[X] coincides with the elementary matrices for sufficiently large matrix size. They define a relative version of the symplectic elementary Witt group and place it in a relative Karoubi periodicity exact sequence. These results are then used to strengthen the known thresholds at which the stabilization maps for relative linear K1 and symplectic K1Sp groups become injective when the base is a smooth affine algebra over suitable fields.

Core claim

The paper shows that a relative Vorst theorem holds under the stated regularity conditions, allowing the identification GL(R[X], I[X]) = E(R[X], I[X]), and that a relative symplectic Witt group fits into a periodicity sequence, which together yield improved injective stability for the relative K1Sp groups of smooth affine algebras.

What carries the argument

The relative symplectic elementary Witt group and the relative Karoubi periodicity sequence, which transfer stability information from the linear to the symplectic setting.

If this is right

The injective stability range for relative K1-groups of smooth affine algebras improves over previous bounds.
Analogous improvement occurs for the relative symplectic K1Sp-groups.
The results hold for various base fields where the classical theorems apply in the relative case.
These improvements facilitate better computations of K-groups for polynomial rings over regular rings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework might allow similar relative stability results for other classical groups like orthogonal groups.
Testing the bounds on explicit low-dimensional examples such as polynomial rings over fields could confirm the improvements.
Extensions to higher-dimensional polynomial rings or non-affine schemes may follow similar patterns.

Load-bearing premise

The ring R is a regular k-spot and R/I is regular, with the base field k allowing the classical Vorst and Karoubi theorems to hold relatively.

What would settle it

An explicit counterexample where for some n below the improved bound, there exists a matrix in GL_n(R[X], I[X]) that is not elementary would disprove the claimed improvement in stability.

read the original abstract

We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields.

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

They prove relative versions of Vorst's theorem and the Karoubi sequence for the symplectic Witt group, then combine them for modestly sharper injective stability bounds on relative K1 and K1Sp.

read the letter

The punchline is that Banerjee and Chakraborty extend Vorst's theorem to the relative setting and introduce a relative symplectic elementary Witt group that sits inside a relative Karoubi periodicity sequence. They then use both to improve the known injective stability ranges for the relative linear and symplectic K1-groups of smooth affine algebras over suitable base fields. The regularity conditions on R and R/I stay the same as in the classical statements, so the extensions look like direct adaptations rather than big leaps. What the paper does well is carry out these relative constructions without adding extra hypotheses or circular steps. The relative Vorst result equates invertible and elementary matrices over R[X] with respect to I, and the relative Witt group is defined so that the periodicity sequence holds. Once those are in place, tightening the stability bounds follows in the usual way. The work stays inside the standard toolkit for algebraic K-theory and applies it to the relative case for smooth algebras, which is useful for anyone who needs explicit ranges for computations. The soft spots are proportionate to the claim. The improvements are incremental, not a reorganization of the theory, and the gains in the bounds are probably small enough that only specialists will notice. The area is narrow, so the paper will mainly interest people already working on stability questions for K1 and K1Sp. Nothing in the abstract or the stress-test note points to a load-bearing gap, but the precise new bounds and the details of the relative Witt group definition would need checking in the full proofs. This paper is for algebraic K-theorists who care about concrete stability results and relative versions of classical theorems. A reader in that subfield gets value from the sharper numbers for their own calculations on affine algebras. It deserves a serious referee because the constructions are new, the approach is grounded, and the results are falsifiable. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a relative version of Vorst's theorem asserting that, for a regular k-spot R and ideal I with R/I regular, the group of invertible matrices over R[X] congruent to the identity modulo I equals the corresponding elementary subgroup. It defines a relative symplectic elementary Witt group and embeds it into a relative Karoubi periodicity sequence. These two results are combined to sharpen the known injective stability ranges for the relative linear K_1 and symplectic K_1Sp groups of smooth affine algebras over suitable base fields.

Significance. If the derivations are correct, the work supplies strictly better stability bounds for relative K-groups than those currently in the literature. This is a concrete advance in algebraic K-theory, as improved ranges directly affect computability and vanishing results for K_1 of affine rings. The paper employs only standard tools (relative Vorst and relative Karoubi periodicity) without introducing free parameters or circular reductions, and the regularity hypotheses are precisely those under which the classical statements are known to hold.

minor comments (3)

[Abstract and Introduction] The abstract refers to 'various base fields' without enumeration; the introduction or statement of the main theorem should list the precise fields (e.g., infinite fields, fields of characteristic not 2, etc.) for which the improved bounds are claimed.
[Section 1] Notation for the relative groups (e.g., K_1(R,I) versus K_1Sp(R,I)) should be fixed early and used consistently; minor inconsistencies in the use of 'relative' versus 'absolute' subscripts appear in the preliminary sections.
[Section 3] The statement of the relative Karoubi sequence would benefit from an explicit diagram or exact sequence display rather than a purely verbal description.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the main results on the relative Vorst theorem and the relative Karoubi sequence leading to improved stability bounds. Since no specific major comments were raised, we have no points to address individually at this stage.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper states that it proves a relative Vorst theorem for R[X] w.r.t. I (with R regular k-spot and R/I regular), introduces a relative symplectic elementary Witt group, and shows it fits a relative Karoubi periodicity sequence; these are then combined to tighten injective stability bounds for relative K1 and K1Sp. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the central results are presented as new proofs extending classical Vorst and Karoubi theorems under standard regularity hypotheses, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard domain assumptions of algebraic K-theory: regularity of the base ring R and of R/I, together with the known classical Vorst and Karoubi results over fields. No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption R is a regular k-spot and R/I is regular
Explicitly required for the relative Vorst theorem to hold, as stated in the abstract.
standard math Classical Vorst theorem and Karoubi periodicity hold over the base field k
The relative versions are built on top of these known results.

pith-pipeline@v0.9.0 · 5397 in / 1314 out tokens · 24418 ms · 2026-05-10T15:53:40.684155+00:00 · methodology

discussion (0)

Reference graph

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