On the number of generators of ideals in polynomial rings

Abstract

Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation $\sum x_iy_i=z(1-z)$. We associate with the pair $(I,\omega_I)$ an obstruction in the set of homomorphisms $\mathrm{Hom}_{\mathbb{A}^1}(\mathrm{Spec}(R),Q_{2n})$ up to naive homotopy whose vanishing is sufficient for $\omega_I$ to lift to a surjection $R^n\to I$. Subsequently, we prove that the obstruction vanishes in case $R=k[T_1,\ldots,T_m]$ for $m\in \mathbb{N}$ where $k$ is an infinite perfect field having characteristic different from $2$ thus resolving an old conjecture of M. P. Murthy.