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Integral counterexamples to Gersten’s conjecture

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.

Manuscripts in this family · 2

An integral counterexample to Gersten's conjecture

Sep 25, 2026·An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf

We construct a two-dimensional ramified regular local ring A of mixed characteristic

(0,5)

for which

K_5(A)\to K_5(\mathop{\mathrm{Frac}}\nolimits A)

has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.

Full abstract

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An integral degree-three Gersten counterexample

Sep 26, 2026·paper.pdf

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(\mathop{\mathrm{Frac}}\nolimits A)

has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

Full abstract

PDF on GitHub LaTeX source