Hexagon

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.

Primarily AI-generated textHuman understanding: no parts

Oct 20, 2026PDF ↗View work →

hexagon:2610.00384v1

math.AC — Commutative Algebra

A counterexample to the Eisenbud–Mazur conjecture

Contributed by Chung Pang Mok, Yiqi Xu

We construct a height-five prime ideal $P$ in the six-variable formal power series ring $R$ over $\mathbb{Q}$ such that $P^{(2)}\not\subseteq\mathfrak{m}P$ , thus providing a prime-ideal counterexample to the Eisenbud--Mazur conjecture (with $\mathbb{Q}$ being the coefficient field). The ideal is the kernel of an explicit quadratic parametrization over a number field of degree $28$ over $\mathbb{Q}$ . A witness has order three, with leading term the determinant of a symmetric $3\times3$ matrix. We construct the witness by eliminating critical values to all orders and then straightening the resulting formal curve. Interpolation at the $28$ nodes of an arrangement of eight lines supplies the required lifting maps. A rational linear functional on third and fourth jets certifies nonmembership in $\mathfrak{m}P$ . The construction also yields an unmixed radical ideal with $28$ smooth branches over $\mathbb{C}$ , and provides a counterexample to the more general form of the Eisenbud--Mazur conjecture stated by More (with $\mathbb{C}$ being the coefficient field). Finally, the prime ideal counterexample descends to the localized polynomial ring $\mathbb{Q}[x_0,\ldots,x_5]_{(x_0,\ldots,x_5)}$ , where a polynomial witness with the same fourth jet exists. All finite identities used in the proof admit exact verification.

A mix of human-written and AI-generated textHuman understanding: all partsEisenbud–Mazur conjectureSymbolic powersVeronese surfacecritical valuesformal curves

Oct 20, 2026PDF ↗View work →

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Quillen's local parameter conjecture in dimension four

A counterexample to the Eisenbud–Mazur conjecture

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