The Stacks project

Lemma 76.21.2. 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 $\mathop{N\! L}\nolimits _{X/Y}|_{U_{\acute{e}tale}} = \mathop{N\! L}\nolimits _{U/V}$ in $D(\mathcal{O}_ U)$.

Proof. Formation of the naive cotangent complex commutes with pullback (Modules on Sites, Lemma 18.35.3) 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 $q_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \mathcal{O}_{V_{\acute{e}tale}}$ by Properties of Spaces, Lemma 66.26.1. Tracing through the definitions we conclude that $\mathop{N\! L}\nolimits _{X/Y}|_{U_{\acute{e}tale}} = \mathop{N\! L}\nolimits _{U/V}$. $\square$

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