The Stacks project

Lemma 90.18.8. Let $\mathcal{D}$ be a site. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be homomorphisms of sheaves of rings on $\mathcal{D}$. There is a canonical distinguished triangle

\[ L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C} \to L_{\mathcal{C}/\mathcal{A}} \to L_{\mathcal{C}/\mathcal{B}} \to L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C}[1] \]

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

Proof. We will use the method described in Remarks 90.7.5 and 90.7.6 to construct the triangle; we will freely use the results mentioned there. As in those remarks we first construct the triangle in case $\mathcal{B} \to \mathcal{C}$ is an injective map of sheaves of rings. In this case we set

  1. $\mathcal{P}_\bullet $ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$,

  2. $\mathcal{Q}_\bullet $ is the standard resolution of $\mathcal{C}$ over $\mathcal{A}$,

  3. $\mathcal{R}_\bullet $ is the standard resolution of $\mathcal{C}$ over $\mathcal{B}$,

  4. $\mathcal{S}_\bullet $ is the standard resolution of $\mathcal{B}$ over $\mathcal{B}$,

  5. $\overline{\mathcal{Q}}_\bullet = \mathcal{Q}_\bullet \otimes _{\mathcal{P}_\bullet } \mathcal{B}$, and

  6. $\overline{\mathcal{R}}_\bullet = \mathcal{R}_\bullet \otimes _{\mathcal{S}_\bullet } \mathcal{B}$.

The distinguished triangle is the distinguished triangle associated to the short exact sequence of simplicial $\mathcal{C}$-modules

\[ 0 \to \Omega _{\mathcal{P}_\bullet /\mathcal{A}} \otimes _{\mathcal{P}_\bullet } \mathcal{C} \to \Omega _{\mathcal{Q}_\bullet /\mathcal{A}} \otimes _{\mathcal{Q}_\bullet } \mathcal{C} \to \Omega _{\overline{\mathcal{Q}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{Q}}_\bullet } \mathcal{C} \to 0 \]

The first two terms are equal to the first two terms of the triangle of the statement of the lemma. The identification of the last term with $L_{\mathcal{C}/\mathcal{B}}$ uses the quasi-isomorphisms of complexes

\[ L_{\mathcal{C}/\mathcal{B}} = \Omega _{\mathcal{R}_\bullet /\mathcal{B}} \otimes _{\mathcal{R}_\bullet } \mathcal{C} \longrightarrow \Omega _{\overline{\mathcal{R}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{R}}_\bullet } \mathcal{C} \longleftarrow \Omega _{\overline{\mathcal{Q}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{Q}}_\bullet } \mathcal{C} \]

All the constructions used above can first be done on the level of presheaves and then sheafified. Hence to prove sequences are exact, or that map are quasi-isomorphisms it suffices to prove the corresponding statement for the ring maps $\mathcal{A}(U) \to \mathcal{B}(U) \to \mathcal{C}(U)$ which are known. This finishes the proof in the case that $\mathcal{B} \to \mathcal{C}$ is injective.

In general, we reduce to the case where $\mathcal{B} \to \mathcal{C}$ is injective by replacing $\mathcal{C}$ by $\mathcal{B} \times \mathcal{C}$ if necessary. This is possible by the argument given in Remark 90.7.5 by Lemma 90.18.7. $\square$


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