# Quasi-cyclic modules and coregular sequences

## TL;DR

This paper introduces the concepts of coregular sequences and quasi-cyclic modules to analyze set-theoretic complete intersections, providing new characterizations and necessary conditions for curves in projective three-space.

## Contribution

It develops a generalized theory of coregular sequences for modules beyond finitely generated cases and applies it to characterize set-theoretic complete intersections.

## Key findings

- Modules of codepth at least two are quasi-cyclic.
- Provides necessary conditions for curves to be set-theoretic complete intersections.
- Offers a framework to potentially identify counterexamples in projective three-space.

## Abstract

We develop the theory of coregular sequences and codepth for modules that need not be finitely generated or artinian over a Noetherian ring. We use this theory to give a new version of a theorem of Hellus characterizing set-theoretic complete intersections in terms of local cohomology modules. We also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. We then focus our attention on curves in projective three-space and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in projective three-space is a set-theoretic complete intersection