Lemma 92.8.7 (08UN)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 92: The Cotangent Complex
Section 92.8: Localization and étale ring maps
Lemma 92.8.7 (cite)

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Lemma 92.8.7. Let $A \to B$ be a local ring homomorphism of local rings. Let $A^ h \to B^ h$ , resp. $A^{sh} \to B^{sh}$ be the induced maps of henselizations, resp. strict henselizations. Then

$L_{B^ h/A^ h} = L_{B^ h/A} = L_{B/A} \otimes _ B^\mathbf {L} B^ h \quad \text{resp.}\quad L_{B^{sh}/A^{sh}} = L_{B^{sh}/A} = L_{B/A} \otimes _ B^\mathbf {L} B^{sh}$

in $D(B^ h)$ , resp. $D(B^{sh})$ .

Proof. The complexes $L_{A^ h/A}$ , $L_{A^{sh}/A}$ , $L_{B^ h/B}$ , and $L_{B^{sh}/B}$ are all zero by Lemma 92.8.4. Using the fundamental distinguished triangle (92.7.0.1) for $A \to B \to B^ h$ we obtain $L_{B^ h/A} = L_{B/A} \otimes _ B^\mathbf {L} B^ h$ . Using the fundamental triangle for $A \to A^ h \to B^ h$ we obtain $L_{B^ h/A^ h} = L_{B^ h/A}$ . Similarly for strict henselizations. $\square$

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