Definition 22.11.3. Let $R$ be a ring. Let $A$ be a $\mathbf{Z}$-graded $R$-algebra.

Given a right graded $A$-module $M$ we define the

*$k$th shifted $A$-module*$M[k]$ as the same as a right $A$-module but with grading $(M[k])^ n = M^{n + k}$.Given a left graded $A$-module $M$ we define the

*$k$th shifted $A$-module*$M[k]$ as the module with grading $(M[k])^ n = M^{n + k}$ and multiplication $A^ n \times (M[k])^ m \to (M[k])^{n + m}$ equal to $(-1)^{nk}$ times the given multiplication $A^ n \times M^{m + k} \to M^{n + m + k}$.

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