math.KT — K-Theory and Homology

Quillen's local parameter conjecture in dimension four

Contributed by Yunpeng Li

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 $\mathrm{Sp}_4(R[1/r])$ 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 $R[1/r]$. 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.