[[Graded module]]
# Shifted graded module

Let $\mathfrak{A} \leq \mathfrak{B}$ be a [[submonoid]].
Let $R$ be a $\mathfrak{A}$-[[graded ring]]
and $M$ be $\mathfrak{A}$-[[graded module]] over $R$.
Given some $\beta \in \mathfrak{B}$,
we can define a $\beta$-**shifted gradation** on $M$ with the following relabelling
$$
\begin{align*}
V_{\alpha} \mapsto V_{\alpha + \beta}
\end{align*}
$$
where we consider $A_{\gamma} = 0$ for $\gamma \notin \mathfrak{A}$
and $V_{\gamma} = 0$ for $\gamma \notin \mathfrak{A} + \beta$. #m/thm/module 


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#state/tidy | #lang/en | #SemBr