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…)