  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
