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 ${f: A \rightarrow B}$ is a finitely presented morphism of commutative rings, then ${\mathrm{Spec} B \rightarrow \mathrm{Spec} A}$ is an open immersion if and only if the restriction functor ${D^-(B) \rightarrow D^-(A)}$ 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 ${f: A \rightarrow B}$ (for instance, a localization ${A \rightarrow A_f}$), 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…)

Here are a bunch of notes I wrote up over winter break. The notes are intended to cover Grothendieck’s argument for Zariski’s Main Theorem (the quasi-finite version). It contains as subsections various blog posts I’ve done here, but also a fair bit of additional material. For instance, they cover some of the basic properties of unramified and étale morphisms of rings. It turns out that to prove things about them, you need ZMT in some form.  I wrote some of this up here too.

As written, the notes are still incomplete. Many arguments (such as the use of fpqc descent and the filtered colimit argument) are currently sketched. Someday I may expand these notes to be more complete; right now I don’t think I have the time. Still, I think the basic outline of what happens is present.