In particular, the sequence formula_43 is regular, and the Koszul complex is thus a projective resolution of formula_44formula_45 In this case, the th syzygy module is free of dimension one (generated by the product of all formula_46); the th syzygy module is thus the quotient of a free module of dimension by the submodule generated by formula_47 This quotient may not be a projective module, as otherwise, there would exist polynomials formula_48 such that formula_49 which is impossible (substituting the formula_50 by 0 in the latter equality provides ). This proves that the projective dimension of formula_51 is exactly .