The Stacks project

91.19 The Atiyah class of a sheaf of modules

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules. Let $\mathcal{P}_\bullet \to \mathcal{B}$ be the standard resolution of $\mathcal{B}$ over $\mathcal{A}$ (Section 91.18). For every $n \geq 0$ consider the extension of principal parts

91.19.0.1
\begin{equation} \label{cotangent-equation-atiyah-extension} 0 \to \Omega _{\mathcal{P}_ n/\mathcal{A}} \otimes _{\mathcal{P}_ n} \mathcal{F} \to \mathcal{P}^1_{\mathcal{P}_ n/\mathcal{A}}(\mathcal{F}) \to \mathcal{F} \to 0 \end{equation}

see Modules on Sites, Lemma 18.34.6. The functoriality of this construction (Modules on Sites, Remark 18.34.7) tells us (91.19.0.1) is the degree $n$ part of a short exact sequence of simplicial $\mathcal{P}_\bullet $-modules (Cohomology on Sites, Section 21.41). Using the functor $L\pi _! : D(\mathcal{P}_\bullet ) \to D(\mathcal{B})$ of Cohomology on Sites, Remark 21.41.3 (here we use that $\mathcal{P}_\bullet \to \mathcal{A}$ is a resolution) we obtain a distinguished triangle

91.19.0.2
\begin{equation} \label{cotangent-equation-atiyah-general} L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{F} \to L\pi _!\left(\mathcal{P}^1_{\mathcal{P}_\bullet /\mathcal{A}}(\mathcal{F})\right) \to \mathcal{F} \to L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{F} [1] \end{equation}

in $D(\mathcal{B})$.

Definition 91.19.1. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules. The map $\mathcal{F} \to L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{F}[1]$ in (91.19.0.2) is called the Atiyah class of $\mathcal{F}$.


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