I am currently reading Li Huishi's book "Zariskian filtrations". I am confused by a lemma (p. 34, Lemma 2 of section 4.3 in Chapter 1) in the book. It says:
Let $S = \oplus_{n \in \mathbb{Z}}S_n$ be any $\mathbb{Z}$-graded ring and let $T$ be a regular homogeneous element of degree 1 in $S$. Then $(1-T)S \cap S_n = 0, n \in \mathbb{Z}$ and if $\overline{M}$ is a $T$-torsionfree graded $S$-module (i.e. $Tu = 0$ implies $u = 0$ for any element $u \in \overline{M}$) then $(1-T)\overline{M}\cap\overline{M}_n = 0$ for all $n \in \mathbb{Z}$.
For the proof, the book says "This can be checked in a straightforward way." In fact, it seems that it should not be so difficult to prove this lemma, but I did not manage to prove it. It even seems to me that I came up with a counterexample:
Take $S = k[X,X^{-1}]$ to be the ring of all Laurent polynomials over a field $k$ with the obvious grading ($S_n$ contains all expressions of the form $aX^n$, where $a \in k$). It seems to me that we can take $s = (\cdots,2X^{n-1},X^n,X^{n+1},\cdots)$ and that then $(1-X)s = -X^n \in (1-X)S \cap S_n$.
Am I wrong or is there a mistake in the book?
Thank you very much for your help!