  The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper
bound for the multiplicity of any graded $k$-algebra as well as a lower bound
for Cohen-Macaulay algebras. In this note we extend this conjecture in several
directions. We discuss when these bounds are sharp, find a sharp lower bound in
case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes
of 3-space, and we propose an upper bound for finitely generated graded torsion
modules. We establish this bound for torsion modules whose codimension is at
most two.
