I am looking for a proof of this fact: Let $W=k[x_1,\dots,x_n]$ be a positively graded polynomial ring. Every graded free resolution of a finitely generated $W$-module $M$ is isomorphic to the direct sum of the minimal graded free resolution and a trivial complex, that is, a direct sum of complexes of the form $$0\to W(-p)\xrightarrow{1}W(-p)\to0.$$
I found a proof of almost the same result, where the positively graded ring $W$ is replaced by a quotient $R=S/I$, where $S$ is a standard graded polynomial ring and $I$ is a homogeneous ideal. But I cannot understand if I can deduce from this the result I need. Is it so?
Thank you very much in advance for your help!