I am considering the following situation. A$A$ is a finitely generated ring over a field K$K$ with non-negative grading and A_0=K$A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated in degree 1. Then X=Spec A$X=\mathrm{Spec} A$ carries an action of the multiplicative group G_m$\mathbb{G}_m$, which is really what the grading means to me. Also, I want to assume that X$X$ has a unique singularity at the `origin' 0 corresponding to the maximal ideal of positive elements of A$A$, so that U=X\0$U=X\setminus 0$ is smooth.
I am interested in (the derived category of) coherent sheaves on the quotient stack [U/G_m]$[U/\mathbb{G}_m]$ or equivalently in G_m$\mathbb{G}_m$-equivariant coherent sheaves on U$U$. I'd like to have Serre duality in this category. I think one should be able to state this in the form
Ext^k(F,G) \simeq Ext^{n-1}(G,F \otimes \omega_U)*$\operatorname{Ext}^k(F,G) \simeq \operatorname{Ext}^{n-1}(G,F \otimes \omega_U)^*$
where \omega_U$\omega_U$ is the canonical sheaf of U$U$ and *$*$ is the graded dual, so that taking G_m$\mathbb{G}_m$-invariants (degree 0) produces the desired Serre duality on [U/G_m]$[U/\mathbb{G}_m]$.
I am willing to assume the singularity of X$X$ is Cohen-Macaulay or even Gorenstein. I think such a statement could be deduced from local duality if A$A$ 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.
