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Quillen's local parameter conjecture in dimension four

Primarily AI-generated textHuman understanding: no partsmath.KT — K-Theory and Homologymath.AC — Commutative Algebra

Contributed by Yunpeng Li ↗

Version 1 / Oct 20, 2026/ arXiv: 2610.13469 / CC BY 4.0

Abstract

Let RR be a Noetherian regular local ring of mixed characteristic and dimension four, and let rr be a member of a regular system of parameters. We prove that every finitely generated projective module over R[1/r]R[1/r] is free. We also prove that Sp4(R[1/r])\mathrm{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]R[1/r]. We establish these results using integral degree-two algebraic KK-theory, supported hermitian KK-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.

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  1. v1Submitted by Yunpeng LiInitial depositCurrentOct 20, 2026