On the number of generators of ideals in polynomial rings

TL;DR

This paper introduces an obstruction theory for generators of ideals in polynomial rings over certain fields and proves the obstruction vanishes in polynomial rings over infinite perfect fields, resolving a longstanding conjecture.

Contribution

It develops a new obstruction framework linking ideal generators to homotopy classes and proves the obstruction vanishes in polynomial rings over infinite perfect fields, confirming Murthy's conjecture.

Findings

Obstruction vanishes for polynomial rings over infinite perfect fields.

Provides a homotopy-theoretic criterion for lifting generators.

Resolves an old conjecture of Murthy.

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.