Change of grading used in the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry

Question

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here.

Let $T=\bigoplus_{\underline n\in \mathbb Z^r}T_{\underline n}$ be an $r$-graded ring defined over a local ring. Let $S=\bigoplus_{n_j=0}T_{\underline n}$ and $\mathfrak M$ be maximal homogeneous ideal of $S.$

I do not understand how $T_{\mathfrak M}$ can be considered as $\mathbb Z$-graded ring defined over the local ring $S_{\mathfrak M}.$

If I consider $S=\bigoplus_{n_i,n_j=0}T_{\underline n}$ and $\mathfrak M$ is maximal homogeneous ideal of $S,$ then also the statement is true?

Please explain the change of grading used in the proof.

Answer 1

If you consider $S$ as you propose, that is $\bigoplus\limits_{n_i,n_j=0}T_{\underline{n}}$, then $T$ is a bi-graded ring over $S$: $$T_{p,q}=\bigoplus\limits_{n_i=p,n_j=q}T_{\underline{n}}.$$ By the choice the author makes, they just ensure to isolate the grading in one direction, thus making it a $\mathbb{Z}$-graded ring.