The paper constructs a nonzero mod-3 K_2 class on a ramified regular local ring that dies in the fraction field, disproving finite-coefficient Gersten injectivity in this setting.
Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology
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abstract
We prove that the motivic cohomology of mixed characteristic schemes, introduced in our previous work, satisfies various expected properties of motivic cohomology, including a motivic refinement of Weibel's vanishing in algebraic $K$-theory, the projective bundle formula, a comparison to Milnor $K$-theory, and a universal characterisation in terms of pro cdh descent. These results extend those of Elmanto--Morrow to schemes which are not necessarily defined over a field.
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Finite-coefficient Gersten injectivity fails in ramified mixed characteristic
The paper constructs a nonzero mod-3 K_2 class on a ramified regular local ring that dies in the fraction field, disproving finite-coefficient Gersten injectivity in this setting.