Let $A=\bigoplus_{n=0}^{\infty} A_n$ be a Noetherian graded ring, in which case $A$ is generated as an algebra over $A_0$ by elements $x_1,\dots,x_s$ of degrees $k_1,\dots,k_s$. Let $\lambda$ be an additive (with respect to exact sequences) function on the class of all finitely generated $A_0$-modules, taking values in $\mathbb{Z}$. Then given a graded finitely generated $A$-module $M=\bigoplus_{n=0}^{\infty} M_n$, we define the Poincare series of $M$ with respect to $\lambda$: $P(M,t) := \sum_{n=0}^{\infty} \lambda(M_n) t^n$. Then Proposition 11.1 in Atiyah and MacDonald says that $P(M,t) = \frac{f(t)}{\prod_{i=1}^s(1-t^{k_i})}$, where $f(t)$ is a polynomial with integer coefficients. We can subsequently define $d(M)$ to be the order of the pole of $P(M,t)$ at $t=1$.
In Proposition 11.3 the authors claim that if $x \in A$ is a non-zero divisor on $M$, then $d(M/xM) = d(M)-1$. I can see the statement if $\lambda$ is taking values in $\mathbb{Z}^+$. Do you think the authors really have this in mind or is this statement true for $\lambda$ possibly taking negative values as well?
Edit I: Let me elaborate a little bit on where the issue is. Let $x \in A$ be a non-zero divisor of $M$ of degree $k$. Then we have an exact sequence \begin{align} 0 \rightarrow M_n \stackrel{x}{\rightarrow} M_{n+k} \rightarrow L_{n+k} \rightarrow 0 \, \, \, \, \, \, (1). \end{align} Note that the graded module $L=\bigoplus_{n=0}^{\infty} L_n$ is precisely equal to $M/xM$. From the above exact sequence we get \begin{align} \lambda(M_n) - \lambda(M_{n+k}) + \lambda(L_{n+k}) = 0\, \, \, \, \, \, (2). \end{align} Multiplying $(2)$ by $t^{n+k}$ and summing over $n$ we get \begin{align} P(M,t) = \frac{P(L,t) + g(t)}{1-t^k}. \end{align} Let us write $P(L,t) = \frac{f(t)}{(1-t)^{d(L)} h(t)}$, where $f(1)h(1) \neq 0$. Then \begin{align} P(M,t) = \frac{f(t) + (1-t)^{d(L)} h(t) g(t)}{(1-t^k)(1-t)^{d(L)} h(t)} \, \, \, \, \, \, (3). \end{align} If $d(L)>0$, then everything is fine: (3) gives $d(M)=d(L)+1$. But now suppose that $d(L)=0$. This means that \begin{align} P(M,t) = \frac{f(t) + h(t) g(t)}{(1-t^k)h(t)} \, \, \, \, \, \, (4). \end{align} But now it may be the case that $f(1)+h(1)g(1)=0$ and so $d(M)\le d(L)=0$.
Edit II: The link in MO provided by darij gives three conditions that make Proposition 11.3 well-behaved for $\lambda$ not restricted to non-negative values. It is worth mentioning that according to my understanding, if we replace $\lambda$ with the length function, then Proposition 11.3 becomes true as long as $M \neq 0$.
