Generation of modules and transcendence degree of zero-cycles

TL;DR

This paper constructs a specific algebraic example demonstrating that certain projective modules cannot be generated by fewer than a specific number of elements, thereby providing a counterexample to a potential generalization.

Contribution

It presents a new example of a regular algebra over complex numbers with a projective module that defies minimal generation expectations, extending Swan's earlier work.

Findings

Counterexample to minimal generator conjecture

Extension of Swan's real field example to complex numbers

Demonstrates limitations in module generation over regular algebras

Abstract

We construct an example of a regular algebra over $\mathbb C$ of dimension $d$ and a rank $r$ projective module over it which is not generated by $d+r-1$ elements. This strengthens an example by Swan over the field of real numbers.