This post continues the series on local cohomology.

Let {A} be a noetherian ring, {\mathfrak{a} \subset A} an ideal. We are interested in the category {\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))} of quasi-coherent sheaves on the complement of the closed subscheme cut out by {\mathfrak{a}}. When {\mathfrak{a} = (f)} for {f \in A}, then

\displaystyle \mathrm{Spec} A \setminus V(\mathfrak{a}) = \mathrm{Spec} A_f,

and so {\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))} is the category of modules over {A_f}. When {\mathfrak{a}} is not principal, the open subschemes {\mathrm{Spec} A \setminus V(\mathfrak{a})} 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 {\mathbb{A}^{n+1} \setminus \left\{(0, 0, \dots, 0)\right\}}. This is not affine once {n > 0}.

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

\displaystyle X' = \mathrm{Spec} A \setminus V(\mathfrak{a}) , \quad X = \mathrm{Spec} A,

and let {i: X' \rightarrow X} be the open imbedding. This induces a functor

\displaystyle i_* : \mathrm{QCoh}(X') \rightarrow \mathrm{QCoh}(X)

which is right adjoint to the restriction functor {i^* : \mathrm{QCoh}(X) \rightarrow \mathrm{QCoh}(X')}. Since the composite {i^* i_* } is the identity on {\mathrm{QCoh}(X')}, we find by a formal argument that {i_*} 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 {\mathcal{C} \rightarrow \mathcal{D}}, one should imagine that {\mathcal{C}} is obtained from {\mathcal{D}} by inverting various morphisms, say a collection {S}. The category {\mathcal{C}} sits inside {\mathcal{D}} as the subcategory of {S}-local objects: in other words, those objects {x} such that {\hom(\cdot, x)} turns morphisms in {S} into isomorphisms. (more…)