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If $M,N$ are graded modules over a graded ring $R$, then is $\operatorname{Hom}_{R}(M,N)$ also a graded module and how?

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If by $\text{Hom}_R$ you mean graded homomorphisms (those that preserve the grading), then no. However, there is a "graded Hom" where the $i^{th}$ graded component consists of homomorphisms which raise degree by $i$, and the zeroth graded component of the graded Hom is the ordinary Hom. The general keyword here is internal Hom.

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    If the domain is f.g. then there is a natural grading on the hom, so in that case the answer is yes, actually. If the domain is free of infinite rank, for example, there is no grading giving the obviou degree to the elements one wants.2015-10-16