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 $M$.

## 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

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

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

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 $D(A)$.

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