I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:

A finitely presented morphism of rings $A \to B$ is

- Formally etale iff $L_{B/A}$ (this denotes the cotangent complex) is homotopy-equivalent to zero
- Formally smooth iff $\Omega_{B/A}$ is projective and $L_{B/A}$ is homotopy-equivalent to it (i.e. acyclic outside degree zero).
- (It's also true, and elementary (not in the paper), that formally unramified iff $\Omega_{B/A} = 0$.)

I've heard that these results are true even without finitely presented hypotheses. In fact, I understand that the fpqc localness of projectivity is what one uses to show that formal smoothness is, in fact, a local property (cf. 2 above). I also know how to show that if $A \to B$ is formally smooth, then the differentials are a projective $B$-module (take a quotient by some polynomial ring, $B = C/I$, and show that the sequence $I/I^2 \to \Omega_{C/A} \otimes_C B \to \Omega_{B/A} \to 0$ is actually split exact). In fact, one can show that if $C$ is a formally smooth $A$-algebra and $B = C/I$, then $B$ is smooth iff the conormal sequence above is split exact.

So I am guessing that there is an extension of the conormal sequence to the cotangent complex, and if this works will prove 2 without finitely presented hypotheses (and thus 1 as well). How does this "long exact sequence" work?