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Let $A = \bigoplus_{i = 0}^{\infty} A_i$ be a graded ring (for simplicity, we can take the grading over $\mathbb{N}$), and $M$ a graded $A$-module. In various occasions I have met the statement that $M$ is projective in the category of graded $A$-modules iff it is projective in the category of ungraded $A$-modules.

The direction "projective ungraded" $\implies$ "projective graded" is easy, and proved for instance here. But I don't see a reason why the implication "projective graded" $\implies$ "projective ungraded" should hold.

It seems this should be trivial to prove, and everyone seems just to assume it without mention, for instance here. I feel a bit silly with this question, but sometimes it is better to ask silly questions as well.

Silly
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1 Answers1

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If $M$ is projective in the category of graded modules, then it is a direct summand of a free graded module (i.e., a direct sum of copies of $A$, each shifted in degree). But a free graded module is free as an ungraded module, so $M$ is projective as an ungraded module.

Jeremy Rickard
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