Integral counterexamples to Gersten’s conjecture

1 · The essentials

These manuscripts ask whether a certain kind of mathematical information survives when a system of numbers is enlarged to allow fractions. The unrestricted prediction is that no information of this kind disappears. The manuscripts report two examples where something nonzero becomes zero after that enlargement. One example concerns information labeled degree five, and the other concerns degree three. Both examples use specially constructed systems with the same stated structural restrictions. They are presented as counterexamples to the unrestricted prediction, not as a rejection of every restricted version. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine transferring records into a larger filing system and finding that a meaningful entry is now indistinguishable from an empty one. The reported examples concern just such a loss of distinction, although the records here are mathematical classes rather than documents.

Why it matters

The manuscripts claim that unrestricted integral Gersten injectivity fails in degrees three and five. Their significance is specific: each supplies an example in which passing from a ring to its fraction field loses a nonzero integral K-theory class. This limits the scope in which an injectivity statement of this kind can hold.

2 · Going further

The precise question concerns the natural map $K_{n} (A) \to K_{n} (Frac A)$ from the integral K-theory of a ring to that of its fraction field. Injectivity would mean that a class mapping to zero was already zero. The main manuscript claims a two-dimensional ramified regular local ring $A$ of mixed characteristic $(0, 5)$ for which $ker (K_{5} (A) \to K_{5} (Frac A)) \neq = 0$ .

The companion manuscript claims an example with the same stated ring properties in degree three: $ker (K_{3} (A) \to K_{3} (Frac A)) \neq = 0$ . The repository summary describes these as two explicit rings with nonzero integral K-theory classes that vanish over their fraction fields. Thus the two manuscripts address the same injectivity question in different degrees.

The argument described in the abstracts is construction-based: produce a ring satisfying the specified hypotheses and a nonzero kernel for the relevant map. The supplied abstracts do not give the ring formulas, the classes, or the steps establishing their behavior. Both manuscripts present their examples as negative answers to the unrestricted integral form of Gersten’s conjecture; the supplied text offers no broader historical account or description of restricted cases.

Useful notions

Rings and fraction fields
Maps, kernels, and injectivity
Local rings and dimension
Mixed characteristic
Integral algebraic K-theory

What was known before / what is new

Before

The source identifies the target as Gersten’s conjecture in its unrestricted integral form. In the maps considered here, the proposed injectivity would rule out nonzero classes becoming zero over the fraction field. The supplied text does not describe earlier partial results or the conjecture’s history.

New

The main manuscript claims an integral degree-five counterexample, and the companion claims an integral degree-three counterexample. Both use two-dimensional ramified regular local rings of mixed characteristic $(0, 5)$ . The repository summary describes the two constructions as explicit.

Limits and assumptions

The reported examples concern integral K-theory in degrees three and five, for two-dimensional ramified regular local rings of mixed characteristic $(0, 5)$ . They target the unrestricted injectivity assertion; the supplied text gives no conclusions for other degrees, other characteristics, or additional restrictions. It also does not identify which restricted questions remain open. The abstracts omit the construction details and argument. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Fraction field: The field obtained from an integral domain by allowing division by its nonzero elements.
Injectivity: A map is injective when distinct inputs remain distinct outputs.
Kernel: The kernel of a group map consists of the elements that it sends to zero.
Integral K-theory: Here this means the K-theory groups $K_{n} (A)$ with integral coefficients, whose classes are compared with those over the fraction field.
Mixed characteristic $(0, 5)$: This means that the local ring has characteristic zero while its residue field has characteristic five.

Keywords : Gersten’s conjecture · Integral K-theory · Mixed characteristic

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic $(0, 5)$ have nonzero integral K-theory classes that vanish over their fraction fields.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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An integral counterexample to Gersten's conjecture

25/09/2026

To be verified

We construct a two-dimensional ramified regular local ring A of mixed characteristic $(0, 5)$ for which $K_{5} (A) \to K_{5} (Frac A)$ has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.

PDF GitHub folder

An integral degree-three Gersten counterexample

26/09/2026

To be verified

We construct a two-dimensional ramified regular local ring A in mixed characteristic $(0, 5)$ for which the integral map $K_{3} (A) \to K_{3} (Frac A)$ has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

PDF GitHub folder