Cousin complexes in motivic homotopy theory

A. Druzhinin, Håkon Kolderup, Paul Arne Østvær

Published 2024-02-16Version 1

We investigate Cousin (bi-)complexes in the setting of motives. Over essentially smooth local schemes, the columns of the Cousin bicomplex with coefficients in any stable motivic homotopy type are shown to be acyclic. On the other hand, we also construct a family of non-acyclic Cousin complexes over any positive dimensional base scheme. Our method of proof employs the notion of extended compactified framed correspondences. Three major motivations for this study are to further our understanding of strict homotopy invariance, motivic infinite loop spaces, and connectivity in stable motivic homotopy theory. As applications of our main results on motivic Cousin complexes, we generalize several fundamental results in these topics to finite dimensional base schemes.