Odd-Dimensional Projective-Module Splitting

Definitions and mathematical statement

Let $k$ be a field of characteristic zero, let $X=\operatorname{Spec}A$ be a smooth affine $k$-variety of odd dimension $n\ge2$, and let $P$ be a finitely generated projective $A$-module of constant rank $n$. Thus the permitted dimensions are $n=3,5,7,\ldots$; the oddness and stated lower bound are both retained.

Let $c_n(P)$ be the top algebraic Chern class of the associated vector bundle. On the smooth $n$-dimensional variety, regard it as an element of \[ CH^n(X)=CH_0(X), \] the Chow group of zero-cycles modulo rational equivalence. Assume \[ c_n(P)=0\quad\text{in }CH_0(X). \] The question is whether there exists a finitely generated projective module $Q$ of rank $n-1$ such that \[ P\cong A\oplus Q\qquad\text{as }A\text{-modules}. \] Equivalently, does $P$ contain a unimodular element: an element $v\in P$ and an $A$-linear functional $\varphi:P\to A$ with $\varphi(v)=1$? This asks for an actual free rank-one direct summand, not only a stable decomposition or a trivial determinant.

The characteristic-zero field is not stated to be algebraically closed, and no such restriction is inserted. The module $Q$ need not be free. The Chern-class hypothesis is equality in the Chow group itself, not merely vanishing of a numerical degree or of a class after extending the base field. The usual finite-generation and constant-rank conventions for the module/vector-bundle correspondence are made explicit.

Short English statement

For a rank-$n$ projective module on a smooth affine variety of odd dimension $n$ over any characteristic-zero field, does vanishing of the top Chow-theoretic Chern class force an actual free rank-one summand? The complementary module may remain nonfree.

Sources

• Local UnsolvedMath registry, permanent ID 30000646, code catalogue entry, "Odd-Dimensional Projective-Module Splitting." Proposed year 2007; proposer unspecified.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01; https://doi.org/10.4171/owr/2007/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says spectrum, dimension, rank, Chern-class, Chow-group, and splitting notation was enclosed in balanced delimiters.
• Literature triage checked 2026-08-21: open. It generically describes an unresolved problem/conjecture/direction and reports no authoritative complete resolution from the supplied targeted check. No particular field classes or partial cases are given. Created 2026-08-25; updated 2026-09-16. Current status and source scope require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.