Complete intersection ideals and a question of Nori

TL;DR

This thesis addresses a question of Nori by proving a result on lifting surjections from modules over polynomial rings to ideals in smooth affine domains, advancing understanding of algebraic geometry and ideal theory.

Contribution

It provides a positive answer to Nori's question on lifting surjections in the context of smooth affine domains over infinite perfect fields.

Findings

Established conditions for lifting surjections from modules to ideals.

Proved the existence of a surjection with prescribed properties.

Extended the understanding of complete intersection ideals.

Abstract

This is my PhD thesis from 2004 under Prof. S.M. Bhatwadekar. Here we answer a question of Nori and prove the following result. Let $A$ be a smooth affine domain of dimension $d$ over an infinite perfect field. Let $I$ be an ideal of $A[T]$ of height $n$ such that $2n \geq d+3$. Given surjections $\phi:(A[T]/I)^n \rightarrow I/I^2$ and $\rho :A^n \rightarrow I(0)$ such that $\phi(0)=\rho \otimes A/I(0)$, then there exist a surjection $\Phi :A[T]^n \rightarrow I$ such that $\Phi(0)=\rho$ and $\Phi \otimes A/I =\phi$. This is a joint work with S.M. Bhatwadekar.