This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.
131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.