Associated to any ring maps $A\to B\to C$ there is the distinguished triangle $$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\longrightarrow} \ $$ in $D(C)$. The cotangent complex $\mathbf{L}_{C/A}$ is (the value at $C$ of) the Quillen left derived functor of $$C\otimes_-\Omega^1_{-/A} \ : \ \text{simplicial }A\text{ rings over }C\ \longrightarrow \ \text{simplicial }C\text{ modules}$$ Note that $D(C)$ is the homotopy category of simplicial $C$ modules, by Dold-Kan.
Is there a deeper reason for the triangle, or is it something very special about $\mathbf{L}$? i.e. can we say anything about when a derived Quillen functor (say between stable model categories) admits a long exact sequence like this?
The question must be fairly subtle because already the result fails if we replace rings with general schemes in the above, except under certain conditions on the maps $X\to Y\to Z$.
