Definition 18.35.1. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The naive cotangent complex $\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}$ is the chain complex (18.35.0.2)

$\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} = \left(\mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega _{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{B}]} \mathcal{B}\right)$

with $\mathcal{I}/\mathcal{I}^2$ placed in degree $-1$ and $\Omega _{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{B}]} \mathcal{B}$ placed in degree $0$.

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