A problem with Artin's Vanishing for torsion motivic homology

TL;DR

This paper discusses issues with Artin's Vanishing theorem for torsion motivic homology, highlighting gaps in previous proofs and proposing future directions to address these problems.

Contribution

It identifies gaps in existing proofs related to Artin's Vanishing for torsion motivic homology and suggests potential resolutions through standard motivic conjectures.

Findings

Identified gaps in proofs of key theorems.

Proposed reliance on motivic conjectures for future results.

Acknowledged the need for corrections in previous work.

Abstract

The paper is suspended. The reason: as was noted by prof. H. Esnault, Theorem 2.1.1 of the previous version (as well as the related Theorem 6.1.1 of http://arxiv.org/PS_cache/math/pdf/9908/9908037v2.pdf of D. Arapura and P. Sastry) is wrong unless one assumes H to be a generic hyperplane section. Hence the proofs of all results starting from 2.3 contain gaps. The author hopes to correct this (somehow) in a future version. At least, most of the results follow from certain "standard" motivic conjectures (see part 1 of Remark 3.2.4 in the previous version). If the author would not find a way to prove Theorems 2.3.1 and 2.3.2 (without 2.1.1), then in the next version of the preprint the results of section 4 will be deduced from certain conjectures; certainly this is not a very exiting result.