I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff (2)$ the following things has been stated.
Set $\mathfrak M_1^+=(\mathfrak{m} \bigoplus_{p >0} T_{(p,0)})T, \mathfrak M_2^+=(\mathfrak{m} \bigoplus_{q >0} T_{(0,q)})T, \mathfrak{M}^+=\mathfrak M_1^+\cap \mathfrak M_2^+, \mathfrak{M}_1^+ + \mathfrak{M}_2^+=\mathfrak{M}.$
"Since $T=\bigoplus_{q\geq 0}T_{.,q}=\bigoplus_{p\geq 0}T_{p,.}$ it follows that $$[H^i_{\mathfrak M_1^+}(M)]_{p,q}=[H^i_{\mathfrak M_{.,0}}(T_{.,q})]_p \hspace{1cm} and \hspace{1cm} [H^i_{\mathfrak M_2^+}(M)]_{p,q}=[H^i_{\mathfrak M_{0,.}}(T_{p,.})]_q$$ for all $p,q\in \mathbb{Z}$ and $i\geq 0$" (I think this is by change of grading but I don't know how).
I also did not understand how to get the following equalities.
"Therefore we obtain for all $i\geq 0$ that $$[H^i_{\mathfrak M_1^+}(M)]_{p,q}=0 \hspace{.2cm}if\hspace{.2cm} q<0 \hspace{.2cm} and \hspace{.2cm} [H^i_{\mathfrak M_2^+}(M)]_{p,q}=0 \hspace{.2cm}if \hspace{.2cm} p<0 ."$$
