  This work generalizes the short resolution given in Proc. Amer. Math. Soc.
\textbf{131}, 4, (2003), 1081--1091, to any affine semigroup. Moreover, a
characterization of Ap\'{e}ry sets is given. This characterization lets compute
Ap\'{e}ry sets of affine semigroups and the Frobenius number of a numerical
semigroup in a simple way. We also exhibit a new characterization of the
Cohen-Macaulay property for simplicial affine semigroups.
