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](https://doi.org/10.1016/j.aim.2010.11.004)* 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.



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> **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?
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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](https://math.stackexchange.com/a/1553851/10968).)



Graded and projective (but not bounded below) module that is not graded-projective?
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.


