Integral counterexamples to Gersten’s conjecture

The explanation

Start with a ring of numbers and pass to its fractions. Take the integers and allow every denominator not divisible by 5. A number like 2 or 3/7 can be inverted in this ring. It stays a nonzero fraction when you pass to all of the rational numbers. Nothing is lost on the way.

K-theory attaches a sequence of groups K₀, K₁, K₂ and so on to every ring. In low degrees they are familiar. For rings like this one, K₁ is essentially the group of invertible elements. Higher groups record subtler information about matrices over the ring. Gersten's conjecture says that, for well-behaved rings, nothing is lost in any degree when passing to fractions. A class that dies in the fraction field must already be zeroSource 1. Well-behaved here means regular local: the ring describes the neighborhood of a smooth point.

Now look at the paper's ring. Take the ring of integers of an extension of the 5-adic numbers of degree eight. Add two variables x and y, impose 5 + x⁴ + y⁴ = 0, and look near the point where 5, x and y vanishSource 2. In this ring, 5 equals −x⁴ − y⁴. So 5 lies deep inside the ideal generated by x and y, in its fourth powerSource 3. Rings where the prime 5 is this small are called ramified. This one is regular, but it is not smooth over its baseSource 4.

The papers claim Gersten's conjecture fails for such rings. In this ring, some nonzero class in K₅ becomes zero in the fraction fieldSource 5. A second paper builds a different ramified ring, from a degree-five curve, with the same failure in K₃Source 6. Both rings have mixed characteristic (0, 5). That means the ring contains the integers, but 5 becomes zero at the special point.

Much was known before. Quillen formulated the conjecture and proved it for rings built from finite-type algebras over a fieldSource 7. Panin proved the full case where ring and point share a characteristicSource 8. In mixed characteristic, several positive results need smoothness over a base, or the unramified conditionSource 9. They also often allow only finite coefficients, such as working mod 5. The paper's ring is excluded from the unramified results, since 5 lies in the fourth power of the maximal idealSource 10.

The single obstacle is integrality. Feld had found a failure with coefficients mod 3 at a ramified ringSource 11. But a class mod 3 need not come from an honest integral class that diesSource 12. The K₅ paper lifts a class from arithmetic to integral K₅. That needs a vanishing fact about integral K₄Source 13. It then detects the class by reducing mod 5, using an elliptic curve hidden in the quartic equation.

The result has limits. The detection does not find the order of the integral classSource 14. It does not say whether the class survives when one allows rational coefficientsSource 14. So the rational version of the conjecture stays open. Neither paper uses the other, and neither finds the smallest degree where failure happensSource 15.

How sure is this? There is no Lean formalization for these results in the openai/math release, so no computer has checked the proofs. The arguments rely on deep arithmetic K-theory. No specialist review of either paper has been published.

What it doesn't show. This does not settle Gersten's conjecture with rational coefficients, or for unramified rings.