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.

To prove the converse, suppose ${f}$ is finitely presented and ${D^-(B) \rightarrow D^-(A)}$ is fully faithful. There is an adjunction:

$\displaystyle f^*, f_*: D^-(A) \rightleftarrows D^-(B)$

where ${f^* = \stackrel{\mathbb{L}}{\otimes}_A B}$ and ${f_*}$ is restriction. We are assuming that ${f_*}$ is fully faithful. By general nonsense, this implies that the adjunction maps

$\displaystyle f^* f_* \rightarrow \mathrm{Id}$

are isomorphisms in ${D^-(B)}$. That is, for any complex ${C_\bullet \in D^-(B)}$, one has

$\displaystyle C_\bullet \simeq C_\bullet \stackrel{\mathbb{L}}{\otimes}_A B.$

In particular, one has

$\displaystyle B \stackrel{\mathbb{L}}{\otimes}_A B \simeq B.$

Taking homology in degree zero, this gives ${B \otimes_A B \simeq B}$. Geometrically, if we write ${X = \mathrm{Spec} B }$ and ${Y = \mathrm{Spec} A}$, then this is saying that the map

$\displaystyle X \rightarrow X \times_Y X$

is an isomorphism: that is, ${X \rightarrow Y}$ is a monomorphism in the category of schemes.

But one has a little more. We can actually show that ${X \rightarrow Y}$ (or, equivalently, ${A \rightarrow B}$) 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.

$\displaystyle L_{B/A} \simeq 0.$

Since ${B \stackrel{\mathbb{L}}{\otimes}_A}$ acts as the identity on ${D^-(B)}$, it equates to showing that

$\displaystyle L_{B/A}\stackrel{\mathbb{L}}{\otimes}_A B \simeq 0.$

But this in turn is ${L_{B \stackrel{\mathbb{L}}{\otimes}_A B/B}}$ because the formation of the cotangent complex is compatible with derived base-change: that is, we should consider ${B \stackrel{\mathbb{L}}{\otimes} B}$ as a derived (e.g., simplicial) commutative ring and take its cotangent complex with respect to ${B}$. However, since ${B \stackrel{\mathbb{L}}{\otimes}_A B \simeq B}$, we conclude that ${L_{B/A} \stackrel{\mathbb{L}}{\otimes}_A B \simeq L_{B/A} \simeq 0}$. In other words, ${A \rightarrow B}$ is étale. This completes the proof.