Castelnuovo-Mumford regularity of a shifted module

Question

Let $R$ be a graded ring and $M$ be a graded $R$-module.

Regularity definition

$\operatorname{reg}(M)=\max\{j-i\mid \beta _{i,j}(M) \not=0\} $.

What is the relation between $\operatorname{reg}(M)$ and $\operatorname{reg}(M(-n))$, where $M(-n)_k=M_{k-n}$?

Answer 1

$$\beta_{i,j}(M)=\dim_K\operatorname{Tor}_R^i(K,M)_j$$ So $$\beta_{i,j}(M(-n))=\dim_K\operatorname{Tor}_R^i(K,M(-n))_j=\dim_K\operatorname{Tor}_R^i(K,M)_{j-n}=\beta_{i,j-n}(M).$$ Can you conclude from here?