Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

Abstract

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of $a$ is zero, while the map $K_2(A)\to K_2(F)$ is injective. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.