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Let $M$, $N$ be $R$-graded modules, say: $$M= \bigoplus_{i\in\mathbb Z} M_{i}, N= \bigoplus_{j\in\mathbb Z} N_{j}.$$
Then $$\operatorname{Hom}(M,N) \cong \prod_{i\in\mathbb Z} \bigoplus_{j\in\mathbb Z}\operatorname{Hom}(M_{i},N_{j})$$ and if $\phi:M\to N $ then it can be decomposed as $$\phi= \prod_{i \in Z} \bigoplus_{j\in Z}\phi_{j}^{i}$$ with $\phi^{i}_{j}:M_{i}\to N_{j}$

I have tried a lot to prove it and to define $\phi^{i}_{j}$ but without any success :(

Any response will be greatly appreciated.

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    Let $\imath_i: M_i \to M$ the inclusion and $\pi_j: N \to N_j$ to projection. Then $\phi_j^i = \pi_j \circ \phi \circ \imath_i$.2012-03-27

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