The Homotopy Obstructions in Complete Intersections

TL;DR

This paper proves the homotopy obstruction set coincides with Nori's obstruction, establishes its group structure, and links it to Euler class groups in the context of complete intersections over regular rings.

Contribution

It demonstrates the equivalence of the homotopy obstruction set with Nori's obstruction and connects it to Euler class groups, providing new structural insights.

Findings

The obstruction set $oldsymbol{ ext{pi}_0(Q_{2n})(A)}$ coincides with Nori's obstruction.

$oldsymbol{ ext{pi}_0(Q_{2n})(A)}$ has a natural group structure when $2n \\geq d+2$.

A surjective homomorphism from Euler class groups to the obstruction set is established and shown to be an isomorphism.

Abstract

Let $A$ be a regular ring over a field $k$, with $1/2\in k$ and dimension $d$. We discuss the Homotopy Conjecture of Madhav V. Nori, in the complete intersection case (meaning when the projective module in question if free, of rank at least 2). Recently, an obstruction set (sheaf) $\pi_0(Q_{2n})(A)$ was introduced [F] to detect when a surjective map $A^n\to I/I^2$ lifts to a surjective map $A^n\to I$. We establish that $\pi_0(Q_{2n})(A)$ coincides with the obstruction set of equivalence classes, originally suggested by Nori. We also establish that $\pi_0(Q_{2n})(A)$ has a natural groups structure, when $2n\geq d+2$. Further, we establish that, when $2n\geq d+2$, there is a surjective homomorphism $E^n(A) \to \pi_0(Q_{2n})(A) $, where $E^n(A)$ denotes the Euler class group defined by Bhatwadekar and Sridharan [BS2]. This homomorphism is an isomorphism, whenever triviality, in $\pi_0(Q_{2n})(A)$, of an orientation $(I, \omega_I), guarantees that $omega_I$ lifts to a surjective map $A^n\to I$. We also give a Quadratic version of Lindel's Theorem, on extendibility of projective modules.