= Hilbert-Serre theorem for an additive coefficient function
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{title2=$\sum_{j\ge0}\lambda(M_j)t^j=Q(t)/\prod_i(1-t^{d_i})$}
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 $(1-t^d)P_M=P_{M/xM}-t^dP_{(0:_Mx)}$. 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.
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