  We show that the property of a standard graded algebra R being Cohen-Macaulay
is characterized by the existence of a pure Cohen-Macaulay R-module
corresponding to any degree sequence of length at most depth(R). We also give a
relation in terms of graded Betti numbers, called the Herzog-Kuhl equations,
for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) -
depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing
all graded Cohen-Macaulay R-modules.
