Definition 17.1.1.
For \(A \in \Ring\text{,}\) the cotangent complex is the functor \(L_{\bullet/A}\colon \Ring_A \to D(A)\) taking obtained by taking the left derived functor of the functor \(\Poly_A \to D(A)\) given by \(B \mapsto \Omega^1_{B/A}[0]\text{.}\) It is straightforward to check that in fact \(L_{B/A} \in D^{\leq 0}(B)\text{.}\)
Note that it also makes sense to talk about \(L_{B/A}\) when \(B\) is itself a simplicial object in \(\Ring_A\text{.}\) This will be useful when stating the base change property in Proposition 17.1.2.