Let $k$ be a field. Given a (not necessarily commutative) $k$ graded ring $A$, M. Artin and J.J. Zhang introduced a notion of "noncommutative projective scheme" $Proj(A)$ in this paper. It is defined as a $k$-linear abelian category $$ Tails(A)=Gr(A)/Tor(A) $$ where $Gr(A)$ is the abelian category of right graded $A$-modules and $Tor(A)$ is its full subcategory consisting of torsion modules. (Their definition also contains "structure sheaf" and "shift functor", but we ignore them). When $A$ is commutative, $Tails(A)$ is equivalent to $Coh(Proj(A))$ by Serre's theorem.

Assume $A$ is very simple, say $$ A=k[x_{1},\dots,x_{n+1}]/(x_{i}x_{j}=q_{i,j}x_{j}x_{i})_{i< j}. $$ Since $A$ is AS regular algebra, it is known that $\mathrm{gl.dim}(Tails(A))=n$. I now would like to explicitly show this result;

Given a graded right $A$-module module, take a minimal graded injective (not projective, sorry) resolution of $M$ and show that the $k$th-term is torsion for $k>n$.

Is it possible? Thanks in advance.

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