ag.algebraic geometry - Graded or stacky Serre duality

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 $Rtau$ satifies the equation

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

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 $Rtau(M) to M to RGamma(M)to$ relating the local cohomology to the derived global sections.

• Next ag.algebraic geometry - Why are non-singleton covering families often ignored?
• Previous dg.differential geometry - Principal Bundle Connection Correspondence for two descriptions of the $mathbb{CP}^2$