Fix a noetherian local ring .
Let (for the derived category of , or preferably its higher-categorical analog). Let us define the local cohomology functor
We can think of this at a number of levels: for instance, it is the (derived) functor of the ordinary functor on -modules which sends an -module to its submodule
of -power torsion elements. From this point of view, we can think of the cohomology groups
as defining “cohomology with supports” for the pair with coefficients in the sheaf . I’ll try to elaborate more on this point of view later.
Notation: The derived categories in this post will use cohomological grading conventions, for simplicity.
Our first goal here is to describe the calculation (which is now quite formal) of in the regular case, in terms of dualizing objects. So, let’s suppose is regular local on, of dimension . In this case, each lives in the smaller perfect derived category , and we will use the duality in that category.
Namely, recall that we have a functor given by , which induces a duality on the perfect derived category of , as we saw yesterday.
Let . We saw in the previous post that is cohomologically concentrated in the degree , and it is a shift of the module : we saw that was the injective envelope of . The next result will reduce the computation of to an computation.
Theorem 5 (Local duality) If and is regular, then we have a canonical isomorphism in ,