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=Spec A carries an action of the multiplicative group 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\0 is smooth.

I am interested in (the derived category of) coherent sheaves on the quotient stack [U/G_m] or equivalently in 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

Ext^k(F,G) \simeq 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 G_m-invariants (degree 0) produces the desired Serre duality on [U/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.