This post continues the series on local cohomology.

Let be a noetherian ring, an ideal. We are interested in the category of quasi-coherent sheaves on the complement of the closed subscheme cut out by . When for , then

and so is the category of modules over . When is not principal, the open subschemes are generally no longer affine, but understanding quasi-coherent sheaves on them is still of interest. For instance, we might be interested in studying vector bundles on projective space, which pull back to vector bundles on the complement . This is not affine once .

In order to do this, let’s adopt the notation

and let be the open imbedding. This induces a functor

which is right adjoint to the restriction functor . Since the composite is the identity on , we find by a formal argument that is fully faithful.

Fully faithful right adjoint functors have a name in category theory: they are *localization* functors. In other words, when one sees a fully faithful right adjoint , one should imagine that is obtained from by inverting various morphisms, say a collection . The category sits inside as the subcategory of -local objects: in other words, those objects such that turns morphisms in into isomorphisms. (more…)