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Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is locally finite: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \bigoplus_{n \in \mathbb{Z}} P_n$ is graded-projective if there is a graded free module $F = \bigoplus A(l_i)$ (for some shifting parameters $l_i \in \mathbb{Z}$) and a graded module $Q$ such that $F \cong P \oplus Q$ as graded modules. We say that $P$ is bounded below if $P_n = 0$ for $n \ll 0$.

In section 2 of The structure of AS Gorenstein algebras by Minamoto and Mori, it is shown that if $P$ is a left $A$-module that is graded, projective (as an "ungraded" module), and bounded below, then it is graded-projective. Nakayama's lemma seems crucial to their methods, so I expect something to go wrong if $P$ is not bounded below.

Question: Is there a locally finite graded algebra $A$ and a graded left $A$-module $P$ that is projective, not bounded below, and not graded-projective?

If something indeed does go wrong, I would already expect it to occur in the case where $A$ is connected, meaning that $A_0 = k$. (This question arose earlier on Math StackExchange.)

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The answer to your question is no. More generally, the properties of being graded-projective or projective are equivalent in settings far more general than yours. See C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library 28, 1982.

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    $\begingroup$ Thanks for the reference! For those seeking an exact reference, see Corollary I.2.2. This indeed works for graded modules over rings graded by an arbitrary group. $\endgroup$ – Manny Reyes Jun 26 '17 at 14:20
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    $\begingroup$ Dear @Manny, sorry for not giving the exact reference - this was certainly not intended! $\endgroup$ – Fred Rohrer Jun 26 '17 at 14:42

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