A derived characterization of open immersions

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.

hyh Says:

June 4, 2012 at 8:22 pm

Nice post. The assumption just gives that the homotopy push out of $B\leftarrow A\rightarrow B$ is equal to $B$ , so the cotangent complex vanish, right?

BTW, I’m a little concerned about the ambiguity of $\otems^L$ , from the beginning they are derived tensor product of modules, and later they becomes total derived tensor product of $A$ -algebras! I’m not 100% sure I get it right. But this secret changing of meaning may confuse people. I don’t have a better suggestion for notation though.

June 4, 2012 at 8:24 pm

Sorry, repost, ignore the first one.

Nice post. The assumption just gives that the homotopy push out of $B\leftarrow A\rightarrow B$ is equal to $B$ , so the cotangent complex vanishes, right?

BTW, I’m a little concerned about the ambiguity of $\otimes^{L}$ , from the beginning they are derived tensor product of $A-$modules, and later they becomes total derived tensor product of $A$ -algebras! I’m not 100% sure I get it right. But this secret changing of meaning may confuse people. I don’t have a better suggestion for notation though.

Akhil Mathew Says:

June 4, 2012 at 8:44 pm
Yes, more or less: I think another way of saying this is that there aren’t many “homotopy monomorphisms” in (affine) derived schemes. (A map $X\to Y$ is a homotopy monomorphism if it satisfies the following derived notion of being a monomorphism: $X \to X \times_Y X$ (the latter being the homotopy fibered product) is a weak equivalence.) By contrast, there are lots of ordinary monomorphisms in schemes (e.g. closed immersions).

Regarding your concern, the point is that the forgetful functor from (derived) $A$ -algebras to (derived) $A$ -modules sends tensor products to tensor products. In the algebra language you can compute $B \stackrel{\mathbb{L}}{\otimes}_A B$ by taking a simplicial resolution $P_\bullet \to B$ of $A$ -algebras (where $P_\bullet$ is a cofibrant simplicial $A$ -algebra) and the answer is $P_\bullet \otimes_A B$ . But a simplicial cofibrant resolution of $B$ in algebras corresponds (via Dold-Kan or the Moore complex functor) to a projective resolution of $B$ as a chain complex. So it’s really the derived tensor product of modules as well as algebras.

I’ve been rather sloppy with notation because the foundations are anyway so varied — simplicial rings, dg rings (in char. 0), and a plethora of models of spectra.

June 5, 2012 at 1:37 pm

BTW, are there similar characterization of closed immersions? (Also I suppose this characterization works for schemes?)

Akhil Mathew Says:

June 5, 2012 at 2:06 pm
Not to my knowledge, though there is a nice functorial criterion: a proper monomorphism is a closed immersion. You can check properness via the valuative criterion plus a direct limit argument to check locally of finite presentation (i.e., whether homming into them commutes with direct limits of rings). This should work for schemes as well, though I haven’t checked it (there would be presumably some subtleties, e.g. in which derived category to use, etc.).

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