# Has any finitely generated torsion module over a polynomial ring a finite resolution of the following type?

Let $R$ be a commutative unital ring and $R[T]$ the polynomial ring in one variable. Moreover let $g\in R[T]$ be a polynomial (possibly monic, in particular no zero divisor) and let $M$ be a finitely generated module over $R[T]$ such that $g^i$ acts nilpotently on $M$.

Edit: I write possibly monic, since this might be an extra assumption. The original statement works for $g=T$, which is monic. It might or might not be that this condition is needed. Possibly not being a zero divisor is enough.

Is there always a finite resolution of $M$ be standard modules? Here a standard module is a $R[T]$ module of the form $P[T]/g^iP[T]$ for some projective $R$-module $P$.

Motivation: This is one step to prove a generalised version of the Fundamental Theorem for $K$-theory. For the usual Fundamental Theorem one has $g=T$. Here the statement is apparently true, but I don't see how, hence I can't really answer the more general question.

Edit: The Fundamental Theorem of $K$-Theory states that there is a (non-canonical) split exact sequence $$0\to K_n(R)\to K_n(R[T])\oplus K_n(R[T])\to K_n(R[T,T^{-1}])\to K_{n-1}(R)\to 0$$ If $R$ is regular the split injection $K_n(R)\to K_n(R[T])$ on the left is an isomorphism.

Idea: I thought I could just take a projective resolution of $M$ and mod out $g^i$. But projective modules over $R[T]$ are not necessarily of the form $P[T]$. See also my previous question

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 What does «possibly monic» mean? Isn't every polynomial possibly monic? :) – Mariano Suárez-Alvarez♦ May 29 '12 at 17:11 What is the "Fundamental Theorem of $K$-Theory" ? – Georges Elencwajg May 29 '12 at 17:14 I adressed your comments in the edits. – Simon Markett May 29 '12 at 17:30 IIRC, in the fundamental theorem, the splittings are canonical. – Martin Brandenburg May 29 '12 at 19:17 @MartinBrandenburg the statement in that form is taken from weibel's article in Alg K-thy and its applications (Bass et al.). Presumably the noncanonical part refers to the isomorphism of $K(\mathbb P_R^1)$ to the two copies of $K(R)$. – Simon Markett May 30 '12 at 8:55