The Stacks project

Definition 10.134.1. Let $R \to S$ be a ring map. The naive cotangent complex $\mathop{N\! L}\nolimits _{S/R}$ is the chain complex (10.134.0.2)

\[ \mathop{N\! L}\nolimits _{S/R} = \left(I/I^2 \longrightarrow \Omega _{R[S]/R} \otimes _{R[S]} S\right) \]

with $I/I^2$ placed in (homological) degree $1$ and $\Omega _{R[S]/R} \otimes _{R[S]} S$ placed in degree $0$. We will denote $H_1(L_{S/R}) = H_1(\mathop{N\! L}\nolimits _{S/R})$1 the homology in degree $1$.

[1] This module is sometimes denoted $\Gamma _{S/R}$ in the literature.

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