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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.

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Perhaps $Ext^{n-k}$ is meant? –  Martin Brandenburg Feb 26 '11 at 11:19

2 Answers 2

Fabio Nironi wrote a paper on Serre Duality on Deligne-Mumford stacks, http://arxiv.org/abs/0811.1955.

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Theres graded local duality which works just like local duality; however, it requires that $A_0$ is a field. I've had some luck making things work when A_0 is not a field, but then the local duality becomes more derived. Specifically, Matlis duality, which is an exact functor in the graded local case, gets replaced by $\operatorname{RHom}_{A_0}(-,A_0)$, the derived graded hom over $A_0$ into $A_0$. Then, at least in some cases I've looked at, the local cohomology $R\tau$ satifies the equation

$$R\tau(M) = \operatorname{RHom}_{A_0}(\operatorname{RHom}_A(M,A),A_0)\[d\](l)$$

This should be true when the ring $A$ is 'relatively Gorenstein over $A_0$', which means that $\operatorname{Ext}^i_A(A_0,A)$ is non-zero for a single $i=d$, and $\operatorname{Ext}^d_A(A_0,A)=A(l)$.

As for translating this to Serre duality, there should be an exact triangle $R\tau(M) \to M \to R\Gamma(M)\to$ relating the local cohomology to the derived global sections.

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