The Stacks project

91.17 The Atiyah class of a module

Let $A \to B$ be a ring map. Let $M$ be a $B$-module. Let $P \to B$ be an object of $\mathcal{C}_{B/A}$ (Section 91.4). Consider the extension of principal parts

\[ 0 \to \Omega _{P/A} \otimes _ P M \to P^1_{P/A}(M) \to M \to 0 \]

see Algebra, Lemma 10.133.6. This sequence is functorial in $P$ by Algebra, Remark 10.133.7. Thus we obtain a short exact sequence of sheaves of $\mathcal{O}$-modules

\[ 0 \to \Omega _{\mathcal{O}/\underline{A}} \otimes _\mathcal {O} \underline{M} \to P^1_{\mathcal{O}/\underline{A}}(M) \to \underline{M} \to 0 \]

on $\mathcal{C}_{B/A}$. We have $L\pi _!(\Omega _{\mathcal{O}/\underline{A}} \otimes _\mathcal {O} \underline{M}) = L_{B/A} \otimes _ B M = L_{B/A} \otimes _ B^\mathbf {L} M$ by Lemma 91.4.2 and the flatness of the terms of $L_{B/A}$. We have $L\pi _!(\underline{M}) = M$ by Lemma 91.4.4. Thus a distinguished triangle

91.17.0.1
\begin{equation} \label{cotangent-equation-atiyah} L_{B/A} \otimes _ B^\mathbf {L} M \to L\pi _!\left(P^1_{\mathcal{O}/\underline{A}}(M)\right) \to M \to L_{B/A} \otimes _ B^\mathbf {L} M [1] \end{equation}

in $D(B)$. Here we use Cohomology on Sites, Remark 21.39.13 to get a distinguished triangle in $D(B)$ and not just in $D(A)$.

Definition 91.17.1. Let $A \to B$ be a ring map. Let $M$ be a $B$-module. The map $M \to L_{B/A} \otimes _ B^\mathbf {L} M[1]$ in (91.17.0.1) is called the Atiyah class of $M$.


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