I’d like to discuss today a category-theoretic characterization of Zariski open immersions of rings, which I learned from Toen-Vezzosi’s article.

Theorem 1If is a finitely presented morphism of commutative rings, then is an open immersion if and only if the restriction functor between derived categories is fully faithful.

Toen and Vezzosi use this to *define* a Zariski open immersion in the derived context, but I’d like to work out carefully what this means in the classical sense. If one has an open immersion (for instance, a localization ), then the pull-back on derived categories is fully faithful: in other words, the composite of push-forward and pull-back is the identity. (more…)