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 1 If 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.
To prove the converse, suppose is finitely presented and is fully faithful. There is an adjunction:
where and is restriction. We are assuming that is fully faithful. By general nonsense, this implies that the adjunction maps
are isomorphisms in . That is, for any complex , one has
In particular, one has
Taking homology in degree zero, this gives . Geometrically, if we write and , then this is saying that the map
is an isomorphism: that is, is a monomorphism in the category of schemes.
But one has a little more. We can actually show that (or, equivalently, ) is étale, and now a general result of Grothendieck tells us that an étale radicial morphism (e.g., an étale monomorphism) is an open immersion. How can we check étaleness? We have to show that the cotangent complex vanishes, i.e.
Since acts as the identity on , it equates to showing that
But this in turn is because the formation of the cotangent complex is compatible with derived base-change: that is, we should consider as a derived (e.g., simplicial) commutative ring and take its cotangent complex with respect to . However, since , we conclude that . In other words, is étale. This completes the proof.