Hilbert-Serre theorem for an additive coefficient function (${\sum }_{j\ge 0}\lambda (M_j)t^j=Q(t)/{\prod }_i(1-t^{d_i})$)

Let $S_0$ be a commutative Noetherian ring, let a graded $S$ be generated over it by finitely many positive-degree homogeneous elements, and let $M$ be a finitely generated graded module. For an integer-valued function $\lambda$ additive on exact sequences of finitely generated $S_0$-modules, its graded-piece series has the displayed form. The exact sequence for multiplication by the last variable gives $(1-t^d)P_M=P_{M/xM}-t^d{P}_{(0{:}_{M}x)}$. Those two modules are finite over the ring with one fewer variable, and induction proves rationality. Finite graded pieces and the finite-generation assumptions cannot simply be omitted.