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

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.)