An explicit duality for quasi-homogeneous ideals

Given r ≥ n quasi-homogeneous polynomials in n variables, the existence of a certain duality is shown and explicited in terms of generalized Morley forms. This result, that can be seen as a generalization of [3, corollary 3.6.1.4] (where this duality is proved in the case r = n), was observed by the author at the same time. We will actually closely follow the proof of (loc. cit.) in this paper. 1 Notations Let k be a non-zero unitary commutative ring. Suppose given an integer n ≥ 1, a sequence (m1, . . . ,mn) of positive integers and consider the polynomial kalgebra C := k[X1, . . . , Xn] which is graded by setting deg(Xi) := mi for all i ∈ {1, . . . , n} and deg(u) = 0 for all u ∈ k. (1) We will suppose moreover given an integer r ≥ n, a sequence (d1, . . . , dr) of positive integers and, for all i ∈ {1, . . . , r}, a (quasi-)homogeneous polynomial of degree di fi(X1, . . . , Xn) := ∑ α1,...,αn≥0 ∑ n i=1 αimi=di ui,αX α1 1 . . . X αn n ∈ k[X1, . . . , Xn]di . In the sequel, we denote by I the ideal of C generated by the polynomials f1, . . . , fr, by m the ideal of C generated by the variables X1, . . . , Xn and by B the quotient C/I. We also set δ := ∑r i=1 di − ∑n i=1 mi. 2 The transgression map Consider the Koszul complex K(f1, . . . , fr;C), which is a Z-graded complex of C-modules, associated to the sequence (f1, . . . , fr) of elements in C. It is of the