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arxiv: 2509.07263 · v3 · pith:IKVBQM5Gnew · submitted 2025-09-08 · 🧮 math.AG · math.AC· math.KT

The nonexistence of sections of Stiefel varieties and stably free modules

Sebastian Gant This is my paper

Pith reviewed 2026-05-21 22:08 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.KT
keywords Stiefel varietiesstably free modulesmotivic stable homotopysections of morphismsvector bundlesJames splitting
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The pith

Certain projections between Stiefel varieties over a field have no sections for r at least 2.

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

The paper examines Raynaud's 1968 question of whether the natural projection p from V_{r+ℓ}(A^n) to V_r(A^n) always admits a section. It identifies triples (r, n, ℓ) with r ≥ 2 where no section exists. These cases produce stably free modules over the polynomial ring that lack a free summand of rank ℓ. The authors also build a splitting of V_2(A^n) in the motivic stable homotopy category that mirrors the classical James splitting of Stiefel manifolds.

Core claim

The projection p: V_{r+ℓ}(A^n) → V_r(A^n) admits no section for certain triples (r, n, ℓ) with r ≥ 2. This non-existence is established via K-theory or obstruction theory after constructing a splitting of V_2(A^n) in the motivic stable homotopy category over a field, analogous to James' topological splitting.

What carries the argument

The projection p between Stiefel varieties V_r(A^n) defined as the homogeneous space GL_n / GL_{n-r}, whose non-existence of sections is detected in the motivic setting, along with the motivic stable homotopy category splitting for the r=2 case.

Load-bearing premise

The base field admits a well-behaved motivic stable homotopy category in which the classical James splitting lifts and non-existence arguments via K-theory or obstruction theory apply without additional characteristic restrictions.

What would settle it

An explicit section for one of the triples (r, n, ℓ) where non-existence is claimed, or a direct computation of the K-theory obstruction class showing it vanishes instead of obstructing the section.

read the original abstract

Let $V_r(\mathbb{A}^n)$ denote the Stiefel variety ${\rm GL}_n/{\rm GL}_{n-r}$ over a field. There is a natural projection $p: V_{r+\ell}(\mathbb{A}^n) \to V_r(\mathbb{A}^n)$. The question of whether this projection admits a section was asked by M. Raynaud in 1968. We focus on the case of $r \ge 2$ and provide examples of triples $(r,n,\ell)$ for which a section does not exist. Our results produce examples of stably free modules that do not have free summands of a given rank. To this end, we also construct a splitting of $V_2(\mathbb{A}^n)$ in the motivic stable homotopy category over a field, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James.

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.

Referee Report

2 major / 2 minor

Summary. The paper addresses Raynaud's 1968 question by exhibiting triples (r, n, ℓ) with r ≥ 2 such that the projection p: V_{r+ℓ}(A^n) → V_r(A^n) admits no section over a field k. These yield examples of stably free modules over k[x_1, …, x_n] without free summands of prescribed rank. The proof proceeds by constructing a splitting of V_2(A^n) in the motivic stable homotopy category that lifts the classical James splitting, then combining it with K-theoretic or obstruction-theoretic invariants to detect the absence of sections.

Significance. If the motivic splitting and non-existence arguments are valid, the paper supplies explicit algebraic counterexamples to a long-standing question and produces new families of stably free modules with controlled splitting behavior. The construction of the motivic splitting itself is a technical contribution that bridges classical topology and A^1-homotopy theory.

major comments (2)
  1. [Introduction and §2] Setup and main statements (Introduction and §2): the results are asserted for an arbitrary field k, yet the motivic stable homotopy category is invoked for the splitting of V_2(A^n) and for the obstruction classes that detect algebraic sections. Standard references (Morel–Voevodsky, Ayoub) require k perfect or char 0 for A^1-invariance, proper base change, and representability of the relevant homotopy sheaves; these hypotheses are load-bearing for both the splitting construction and the translation from homotopy classes to module-theoretic statements. The manuscript must either restrict the base field or supply a reference showing the constructions hold unconditionally.
  2. [Motivic splitting section] Motivic splitting construction (presumably §3 or §4): the claim that the constructed splitting detects non-sections via K-theory or obstruction theory needs an explicit comparison map or diagram showing how a hypothetical algebraic section would produce a null-homotopy in the motivic category; without this, the passage from the motivic splitting to the non-existence of sections remains schematic.
minor comments (2)
  1. The notation V_r(A^n) = GL_n / GL_{n-r} is introduced but a one-sentence reminder of its geometric interpretation as the variety of r-frames in A^n would aid readers outside algebraic K-theory.
  2. Citations to James' classical splitting and to the relevant motivic homotopy references should be given with precise theorem numbers rather than general pointers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment of its significance. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Introduction and §2] Setup and main statements (Introduction and §2): the results are asserted for an arbitrary field k, yet the motivic stable homotopy category is invoked for the splitting of V_2(A^n) and for the obstruction classes that detect algebraic sections. Standard references (Morel–Voevodsky, Ayoub) require k perfect or char 0 for A^1-invariance, proper base change, and representability of the relevant homotopy sheaves; these hypotheses are load-bearing for both the splitting construction and the translation from homotopy classes to module-theoretic statements. The manuscript must either restrict the base field or supply a reference showing the constructions hold unconditionally.

    Authors: We acknowledge that the standard foundations of the motivic stable homotopy category, as developed in Morel–Voevodsky and Ayoub, require the base field to be perfect (or of characteristic zero) to guarantee A^1-invariance, proper base change, and the representability of the relevant homotopy sheaves. Although our statements were phrased for an arbitrary field k, the constructions of the motivic splitting and the obstruction-theoretic arguments do rely on these properties. We will therefore revise the manuscript to explicitly restrict all statements and proofs to the case where k is a perfect field. This hypothesis is standard in the literature and preserves the interest of the examples, which include algebraically closed fields and finite fields. revision: yes

  2. Referee: [Motivic splitting section] Motivic splitting construction (presumably §3 or §4): the claim that the constructed splitting detects non-sections via K-theory or obstruction theory needs an explicit comparison map or diagram showing how a hypothetical algebraic section would produce a null-homotopy in the motivic category; without this, the passage from the motivic splitting to the non-existence of sections remains schematic.

    Authors: We agree that the logical link between the motivic splitting and the non-existence of algebraic sections should be made fully explicit. In the revised manuscript we will insert a commutative diagram (or a detailed step-by-step comparison) in the relevant section that shows how the existence of a section of the projection V_{r+ℓ}(A^n) → V_r(A^n) would induce a morphism in the motivic stable homotopy category whose composition with the constructed splitting yields a null-homotopy, contradicting the non-vanishing of the chosen K-theoretic or obstruction-theoretic invariant. This addition will render the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a splitting of V_2(A^n) directly in the motivic stable homotopy category, presented as analogous to the external classical James splitting. Non-existence results for sections of the projection p: V_{r+ℓ}(A^n) → V_r(A^n) (r ≥ 2) are obtained by combining this splitting with K-theoretic invariants or obstruction theory, yielding examples of stably free modules. These steps are explicit constructions and proofs over a field rather than self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or claims reduce by construction to the paper's own inputs; the central claims retain independent content against external benchmarks such as classical topology and standard motivic homotopy references.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of the general linear group, the motivic stable homotopy category over a field, and classical results in algebraic K-theory. No free parameters or invented entities are introduced in the abstract; the splitting construction and non-existence arguments rest on domain assumptions about the base field and the homotopy category.

axioms (2)
  • domain assumption The motivic stable homotopy category over the base field admits a stable splitting of V_2(A^n) analogous to the topological James splitting.
    Invoked when the authors construct the splitting of V_2(A^n).
  • domain assumption Non-existence of sections can be detected via obstruction theory or K-theoretic invariants that are well-defined for the given triples.
    Used to produce the examples of non-sections and the corresponding module statements.

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