Lemma 92.26.2 (08VE)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 92: The Cotangent Complex
Section 92.26: The cotangent complex of a morphism of algebraic spaces
Lemma 92.26.2 (cite)

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Lemma 92.26.2. Let $S$ be a scheme. Consider a commutative diagram

$\xymatrix{ U \ar[d]_ p \ar[r]_ g & V \ar[d]^ q \\ X \ar[r]^ f & Y }$

of algebraic spaces over $S$ with $p$ and $q$ étale. Then there is a canonical identification $L_{X/Y}|_{U_{\acute{e}tale}} = L_{U/V}$ in $D(\mathcal{O}_ U)$ .

Proof. Formation of the cotangent complex commutes with pullback (Lemma 92.18.3) and we have $p_{small}^{-1}\mathcal{O}_ X = \mathcal{O}_ U$ and $g_{small}^{-1}\mathcal{O}_{V_{\acute{e}tale}} = p_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}$ because $q_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \mathcal{O}_{V_{\acute{e}tale}}$ (Properties of Spaces, Lemma 66.26.1). Tracing through the definitions we conclude that $L_{X/Y}|_{U_{\acute{e}tale}} = L_{U/V}$ . $\square$

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