The Stacks project

Lemma 81.4.3. Let $S$ be a scheme. Let $i : Y \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $y \in |Y|$ with image $x \in |X|$. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The following are equivalent

  1. $\mathcal{G}$ has length $d$ at $y$, and

  2. $i_*\mathcal{G}$ has length $d$ at $x$.

Proof. Choose an étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$. Set $V = Y \times _ X U$. Denote $g : V \to Y$ and $j : V \to U$ the projections. Then $j : V \to U$ is a closed immersion and there is a unique point $v \in V$ mapping to $y \in |Y|$ and $u \in U$ (use Properties of Spaces, Lemma 65.4.3 and Spaces, Lemma 64.12.3). We have $j_*(\mathcal{G}|_ V) = (i_*\mathcal{G})|_ U$ as modules on the scheme $V$ and $j_*$ the “usual” pushforward of modules for the morphism of schemes $j$, see discussion surrounding Cohomology of Spaces, Equation ( In this way we reduce to the case of schemes: if $i : Y \to X$ is a closed immersion of schemes, then

\[ (i_*\mathcal{G})_ x = \mathcal{G}_ y \]

as modules over $\mathcal{O}_{X, x}$ where the module structure on the right hand side is given by the surjection $i_ y^\sharp : \mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$. Thus equality by Algebra, Lemma 10.52.5. $\square$

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