Quillen's local parameter conjecture in dimension four
Abstract
Let R be a Noetherian regular local ring of mixed characteristic and dimension four, and let r be a member of a regular system of parameters. We prove that every finitely generated projective module over R[1/r] is free. We also prove that Sp4(R[1/r]) acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor--Witt sheaf and the alternating Witt group of R[1/r]. We establish these results using integral degree-two algebraic K-theory, supported hermitian K-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor--Witt multiplication and yields the required symplectic frames.
Related
- On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four
- The stability conjecture for the Bernoulli problem in dimension four
- The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
- Quillen's finiteness conjecture for the cotangent complex in characteristic two
- Fiber-Connectivity of Almost-Toric Systems in Four-Dimension
BibTeX
@misc{li2026quillens,
title = {{Quillen's local parameter conjecture in dimension four}},
author = {Yunpeng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13469},
eprint = {2610.13469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13469},
}
RIS
TY - UNPB
TI - Quillen's local parameter conjecture in dimension four
AU - Yunpeng Li
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13469
UR - https://arxiv.org/abs/2610.13469
AB - Let R be a Noetherian regular local ring of mixed characteristic and dimension four, and let r be a member of a regular system of parameters. We prove that every finitely generated projective module over R[1/r] is free. We also prove that Sp_4(R[1/r]) acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor--Witt sheaf and the alternating Witt group of R[1/r]. We establish these results using integral degree-two algebraic K-theory, supported hermitian K-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor--Witt multiplication and yields the required symplectic frames.
ER -