Definition 92.18.2. Let \mathcal{C} be a site. Let \mathcal{A} \to \mathcal{B} be a homomorphism of sheaves of rings on \mathcal{C}. The cotangent complex L_{\mathcal{B}/\mathcal{A}} is the complex of \mathcal{B}-modules associated to the simplicial module
\Omega _{\mathcal{P}_\bullet /\mathcal{A}} \otimes _{\mathcal{P}_\bullet , \epsilon } \mathcal{B}
where \epsilon : \mathcal{P}_\bullet \to \mathcal{B} is the standard resolution of \mathcal{B} over \mathcal{A}. We usually think of L_{\mathcal{B}/\mathcal{A}} as an object of D(\mathcal{B}).
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