Notice that $M/\mathfrak{m}M$ is an $R$-module of finite length, so $H^i_{\mathfrak{m} }(M/\mathfrak{m}M) = 0$ for all $i>0$ and $H^0_{\mathfrak{m} }(M/\mathfrak{m}M) = M/\mathfrak{m}M$. Recalling that CM regularity can be computed via local cohomology module (see Brodmann-Sharp: local cohomology), we have $$reg (M/\mathfrak{m}M) = \max { end (H^i_{\mathfrak{m} }(M/\mathfrak{m}M)) +i | i= 0,...,n }.$$ Here, consider a graded $R$-module $N = \oplus_iN_i$, we denote $end(N) = \sup {i | N_i \neq 0}$. Therefore $reg (M/\mathfrak{m}M) = end ((M/\mathfrak{m}M))$. $end ((M/\mathfrak{m}M))$ is equal to the maximal degree of generator of $M$ by graded Nakayama lemma.
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Notice that $M/\mathfrak{m}M$ is an $R$-module of finite length, so $H^i_{\mathfrak{m} }(M/\mathfrak{m}M) = 0$ for all $i>0$ and $H^0_{\mathfrak{m} }(M/\mathfrak{m}M) = M/\mathfrak{m}M$. Recalling that CM regularity can be computed via local cohomology module (see Brodmann-Sharp: local cohomology), we have $$reg (M/\mathfrak{m}M) = \max { end (H^i_{\mathfrak{m} }(M/\mathfrak{m}M)) +i | i= 0,...,n }.$$ Therefore $reg (M/\mathfrak{m}M) = end ((M/\mathfrak{m}M))$. $end ((M/\mathfrak{m}M))$ is equal to the maximal degree of generator of $M$ by graded Nakayama lemma. |
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