  Monomial ideals which are generic with respect to either their generators or
irreducible components have minimal free resolutions derived from simplicial
complexes. For a generic monomial ideal, the associated primes satisfy a
saturated chain condition, and the Cohen-Macaulay property implies shellability
for both the Scarf complex and the Stanley-Reisner complex. Reverse
lexicographic initial ideals of generic lattice ideals are generic.
Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in
the cogeneric case the Cohen-Macaulay type is greater than or equal to the
number of irreducible components. Methods of proof include Alexander duality
and Stanley's theory of local h-vectors.
