The Homotopy Obstructions in Complete Intersections

Let $A$ be a regular ring over a field $k$, with $1\mathrm{/}2\in k$ 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 $A^n\to I\mathrm{/}{I}^2$ 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 $2n\ge d+2$. Further, we establish that, when $2n\ge d+2$, 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 $(I,{\omega }_I),guaranteesthat$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.