# The Stacks Project

## Tag: 07BN

This tag has label algebra-definition-naive-cotangent-complex and it points to

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Definition 9.126.1. Let $R \to S$ be a ring map. The naive cotangent complex $\mathop{N\!L}\nolimits_{S/R}$ is the chain complex (9.126.0.2) $$\mathop{N\!L}\nolimits_{S/R} = \left(I/I^2 \longrightarrow \Omega_{R[S]/R} \otimes_{R[S]} S\right)$$ 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 $H_1(L_{S/R}) = H_1(\mathop{N\!L}\nolimits_{S/R})$\footnote{This module is sometimes denoted $\Gamma_{S/R}$ in the literature.} the homology in degree $1$.

\begin{definition}
\label{definition-naive-cotangent-complex}
Let $R \to S$ be a ring map. The {\it naive cotangent complex}
$\NL_{S/R}$ is the chain complex (\ref{equation-naive-cotangent-complex})
$$\NL_{S/R} = \left(I/I^2 \longrightarrow \Omega_{R[S]/R} \otimes_{R[S]} S\right)$$
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
$H_1(L_{S/R}) = H_1(\NL_{S/R})$\footnote{This module is sometimes
denoted $\Gamma_{S/R}$ in the literature.} the homology in degree $1$.
\end{definition}


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