nLab affine localization

A localization functor $Q^* : A\to B$ is affine if it has a right adjoint $Q_*$ (which is automatically fully faithful, i.e. the localized category if reflective), and which itself is having its own right adjoint $Q^!$. Therefore $Q^*\dashv Q_* \dashv Q^!$.

For example, every left Ore localization of rings $i: R\to S^{-1}R$, induces a flat affine localization functor $M\mapsto S^{-1}M = S^{-1}R\otimes_R M$.

A functor is an affine localization if it is an inverse image part of an affine morphism with full direct image functor. Here affine morphism means an adjoint triple like above with conservative direct image functor; the conservativeness of an exact additive functor among abelian categories is equivalent to its faithfulness. While the notion of affine localization is fixed, the notion of an affine morphism given here has variants depending on the categorical context.