Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic

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Mochizuki, Local Gersten's conjecture for regular system of parameters (arXiv:1503.07966v8) evidence for this event

Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic evidence for this event

Theorem 1.1 (degree five): with $V$ the integers of the unramified degree-eight extension of $\mathbb Q_5$ and $A=(V[x,y]/(5+x^4+y^4))_{(5,x,y)}$, $A$ is a two-dimensional regular local domain and $\ker(K_5(A)\to K_5(\mathrm{Frac}\,A))\ne0$; the class also gives generic noninjectivity with $\mathbb Z/5$ coefficients. Companion Theorem 1.1 (degree three): the local ring of a cyclotomic quintic model over $\mathbb Z_5[\zeta_5]$ at a special point has a nonzero class in $\ker(K_3(A)\to K_3(\mathrm{Frac}\,A))$ not divisible by 5. Not shown: the order of these classes, whether they survive rationally (the rational Gersten conjecture is untouched), anything about unramified or smooth-over-DVR rings, or the least degree of failure. parent of this event

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Companion: An integral degree-three Gersten counterexample evidence for this event

VibeMathed record: Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic evidence for this event

Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic parent of this event

Feld, Finite-coefficient Gersten injectivity fails in ramified mixed characteristic (arXiv:2608.05005) evidence for this event