Niels FELD

math.KT — K-Theory and Homology

Universal test rings for rational Gersten injectivity

Contributed by Niels FELD

For each prime $p$, rational Gersten injectivity in a fixed degree for mixed-characteristic henselian discrete valuation rings of residue characteristic $p$ is equivalent to injectivity for $W(\overline{\mathbb{F}_p(t_1,t_2,\ldots)})$. For a finite bound $d$ on residue transcendence degree, the test ring is $W(\overline{\mathbb{F}_p(t_1,\ldots,t_d)})$. For these henselian DVRs, rational localization boundary images depend only on the residue field and are reflected by arbitrary residue-field extensions. A normalization by the absolute ramification index makes the resulting comparisons of kernels canonical and functorial. The proof uses regular approximation, finite transfers, and perfection. An appendix gives an alternative comparison based on the existential embedding theorem of Anscombe--Dittmann--Jahnke, including an example with nonnoetherian valuation rings. We also describe the additional boundary condition that induction between consecutive test rings would require.