What is the injective hull of a graded polynomial ring (Graded category of modules)?

Question

Let $R=K[x]$ be usual graded ring (where $K$ is a field).

What is the injective hull of $R$?

Answer 1

I assume that $X$ is in degree $1$ (otherwise the claim is still true, but if $X$ has arbitrary degree $d\neq 0$ then below one has $K[X,X^{-1}]\text{-grMod}\cong K\text{-Vect}^{\oplus |d|}$ through the assignment $M\mapsto M_0\oplus\ldots\oplus M_{|d|-1}$ instead of $M\mapsto M_0$). The localization $K[X,X^{-1}]$ is naturally a graded module over $K[X]$.

Claim: $K[X]\to K[X,X^{-1}]$ is an injective hull in $K[X]\text{-grMod}$.

Proof: It's similar to the ungraded case: Given a graded $K[X]$-module $M$, one has $$(\ddagger)\quad\text{Hom}_{K[X]\text{-GrMod}}(M,K[X,X^{-1}])\cong\text{Hom}_{K[X,X^{-1}]\text{-grMod}}(M[X^{-1}],K[X,X^{-1}])\cong\text{Hom}_{K\text{-Vect}}(M[X^{-1}]_0,K),$$ the latter isomorphism since $K[X,X^{-1}]\text{-grMod}\cong K\text{-Vect}$ via $N\mapsto N_0$ (if $N\in K[X,X^{-1}]\text{-grMod}$, then multiplication by $X^n$ is a $K$-isomorphism between $N_0$ and $N_{n}$ for all $n\in{\mathbb Z}$, so $N_0\otimes_K K[X,X^{-1}]\to N$ is an isomorphism; conversely, given a $K$-vector space $V$, we have $(V\otimes_K K[X,X^{-1}])_0\cong V$). The right hand side in $(\ddagger)$ is exact in $M$, so $K[X,X^{-1}]$ is injective in $K[X,X^{-1}]\text{-grMod}$. As $K[X]\rightarrowtail K[X,X^{-1}]$ is essential, the claim follows.