The Stacks project

Lemma 96.15.1. Let $S$ be a scheme. Fix a $1$-morphism $F : \mathcal{X} \longrightarrow \mathcal{Y}$ of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is a stack in groupoids and the inclusion functor

\[ \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \]

is representable and an open immersion.

Proof. Let $\Xi = (U, Z, y, x, \alpha )$ be an object of $\mathcal{H}_ d$. It follows from the remark following (96.15.0.1) that the pullback of $\Xi $ by $U' \to U$ belongs to $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ if and only if the base change of $x_\alpha $ is unramified and a local complete intersection morphism. Note that $Z \to U$ is finite locally free (hence flat, locally of finite presentation and universally closed) and that $X_ y \to U$ is flat and locally of finite presentation by our assumption on $F$. Then More on Morphisms of Spaces, Lemmas 75.49.1 and 75.49.7 imply exists an open subscheme $W \subset U$ such that a morphism $U' \to U$ factors through $W$ if and only if the base change of $x_\alpha $ via $U' \to U$ is unramified and a local complete intersection morphism. This implies that

\[ (\mathit{Sch}/U)_{fppf} \times _{\Xi , \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \]

is representable by $W$. Hence the final statement of the lemma holds. The first statement (that $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is a stack in groupoids) follows from this and Algebraic Stacks, Lemma 93.15.5. $\square$


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