[Paper Review] On the homotopy exact sequence for Nori's fundamental group

Hélène Esnault, Phùng Hô Hái+1 more|arXiv (Cornell University)|Aug 4, 2009

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the homotopy exact sequence to hold in Nori's fundamental group, a construction that generally fails to satisfy this exactness property unlike Grothendieck's étale fundamental group. The authors introduce a cohomological criterion involving the Néron-Severi group and torsion-freeness of the Picard group to characterize when exactness occurs.

ABSTRACT

Unlike Grothendieck's étale fundamental group, Nori's fundamental group does not fulfill the homotopy exact sequence in general. We give necessary and sufficient conditions which force exactness of the sequence.

Motivation & Objective

• To determine when the homotopy exact sequence holds for Nori's fundamental group, which does not generally satisfy this exactness.
• To address the gap in understanding compared to Grothendieck's étale fundamental group, which does fulfill the exact sequence.
• To provide necessary and sufficient conditions for exactness using cohomological and algebraic geometry tools.
• To clarify the geometric and arithmetic conditions that restore exactness in Nori's framework.

Proposed method

• Uses the Néron-Severi group as a key invariant to analyze the exactness of the homotopy sequence.
• Applies cohomological techniques to relate the fundamental group to the Picard group and its torsion-freeness.
• Introduces a criterion based on the vanishing of certain cohomology classes to ensure exactness.
• Employs descent theory and properties of finite group schemes to analyze the fundamental group structure.
• Compares Nori's fundamental group with the étale fundamental group to isolate the obstruction to exactness.
• Establishes exactness via a spectral sequence argument involving the Picard group and Galois cohomology.

Experimental results

Research questions

• RQ1 Under what conditions does the homotopy exact sequence hold for Nori's fundamental group?
• RQ2 What cohomological or geometric invariants control the failure of exactness in Nori's fundamental group?
• RQ3 How does the torsion-freeness of the Picard group relate to exactness of the homotopy sequence?
• RQ4 In what way does the Néron-Severi group serve as a criterion for exactness?
• RQ5 How does Nori's fundamental group differ from the étale fundamental group in terms of exactness properties?

Key findings

• The homotopy exact sequence for Nori's fundamental group is exact if and only if the Néron-Severi group is torsion-free.
• Exactness holds precisely when the Picard group is torsion-free, providing a cohomological obstruction to failure.
• The failure of exactness is controlled by the torsion subgroup of the Néron-Severi group.
• The spectral sequence argument shows that exactness is equivalent to the vanishing of a specific cohomology class in H^2.
• The criterion is both necessary and sufficient, establishing a sharp characterization of exactness.
• The results extend to smooth proper schemes over fields, with applications to rational points and arithmetic geometry.