Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous elements), and I am just assuming that a torsion free graded module is injective if and only if it is graded divisible, generalizing from the respective statement about ungraded modules. On the otherhand, the duality $(-)^*$ of $k[x]$-modules with $M^* = \hom_k(M, k)$ is contravariant exact, and thus sends injective modules to projective ones. But $(k[x,x^{-1}])^* = k[x, x^{-1}]$ is not projective as ungraded module, and thus also not projective as graded module, so $k[x, x^{-1}]$ is not injective.
What am I doing wrong here?
