I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated in degree 1. Then $X=\mathrm{Spec} A$ carries an action of the multiplicative group $\mathbb{G}_m$, which is really what the grading means to me. Also, I want to assume that $X$ has a unique singularity at the `origin' 0 corresponding to the maximal ideal of positive elements of $A$, so that $U=X\setminus 0$ is smooth.

I am interested in (the derived category of) coherent sheaves on the quotient stack $[U/\mathbb{G}_m]$ or equivalently in $\mathbb{G}_m$-equivariant coherent sheaves on $U$. I'd like to have Serre duality in this category. I think one should be able to state this in the form

$\operatorname{Ext}^k(F,G) \simeq \operatorname{Ext}^{n-1}(G,F \otimes \omega_U)^*$

where $\omega_U$ is the canonical sheaf of $U$ and $*$ is the graded dual, so that taking $\mathbb{G}_m$-invariants (degree 0) produces the desired Serre duality on $[U/\mathbb{G}_m]$.

I am willing to assume the singularity of $X$ is Cohen-Macaulay or even Gorenstein. I think such a statement could be deduced from local duality if $A$ were local rather than graded. But I don't understand these things well enough to see right away if there is a graded version. Also, I'm not sure what reference to consult. It would be helpful to have both a geometric and an algebraic reference.