Projective modules over affine threefolds: a simpler case

TL;DR

This paper proves that in a smooth affine threefold over an algebraic closure of a finite field, two rank 2 projective modules with trivial determinant are isomorphic if they share a common surjection onto a height 2 ideal.

Contribution

It establishes a criterion for isomorphism of projective modules over affine threefolds based on their surjections onto a common ideal, simplifying previous understanding.

Findings

Isomorphism characterized by common surjection onto a height 2 ideal

Applicable for smooth affine algebras over algebraic closures of finite fields

Provides a simpler case for projective module classification

Abstract

Let $p\neq 2$, and let $R$ be a smooth affine algebra of dimension $3$ over $\overline{F}_p$ and $P, Q$ be projective $R$-modules of rank $2$, each with trivial determinant. We prove: $P$ is isomorphic to $Q$ if and only if there is an ideal $J\subset R$ of height $2$ such that both $P$ and $Q$ map onto $J$.