Note
Adjacent-height vanishing for motivic ring spectra
Abstract
We prove that -vanishing implies -vanishing for normed motivic spectra over any scheme on which the chosen prime is invertible. The main step is to carry a nilpotence relation in a finite derived quotient through the geometric norm. After extending coefficients, we construct sections of the norms of the Bockstein boundary maps. These show that a coefficient vanishing in the source quotient has norm zero in the target quotient. Applying coefficient projections then decreases the nilpotence exponent. Over separably closed fields, Bott inversion and Hahn’s theorem complete the vanishing argument. Dualizability of the finite quotients over their coefficient algebras allows us to descend to arbitrary schemes. We also construct counterexamples at the residue characteristic and explain the counterexamples over .