The Stacks project

Lemma 100.5.1. Let $\mathcal{X}$ be an algebraic stack. Then the inertia stack $\mathcal{I}_\mathcal {X}$ is an algebraic stack as well. The morphism

\[ \mathcal{I}_\mathcal {X} \longrightarrow \mathcal{X} \]

is representable by algebraic spaces and locally of finite type. More generally, let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then the relative inertia $\mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an algebraic stack and the morphism

\[ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \longrightarrow \mathcal{X} \]

is representable by algebraic spaces and locally of finite type.

Proof. By Categories, Lemma 4.34.1 there are equivalences

\[ \mathcal{I}_\mathcal {X} \to \mathcal{X} \times _{\Delta , \mathcal{X} \times _ S \mathcal{X}, \Delta } \mathcal{X} \quad \text{and}\quad \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X} \times _{\Delta , \mathcal{X} \times _\mathcal {Y} \mathcal{X}, \Delta } \mathcal{X} \]

which shows that the inertia stacks are algebraic stacks. Let $T \to \mathcal{X}$ be a morphism given by the object $x$ of the fibre category of $\mathcal{X}$ over $T$. Then we get a $2$-fibre product square

\[ \xymatrix{ \mathit{Isom}_\mathcal {X}(x, x) \ar[d] \ar[r] & \mathcal{I}_\mathcal {X} \ar[d] \\ T \ar[r]^ x & \mathcal{X} } \]

This follows immediately from the definition of $\mathcal{I}_\mathcal {X}$. Since $\mathit{Isom}_\mathcal {X}(x, x)$ is always an algebraic space locally of finite type over $T$ (see Lemma 100.3.1) we conclude that $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is representable by algebraic spaces and locally of finite type. Finally, for the relative inertia we get

\[ \vcenter { \xymatrix{ \mathit{Isom}_\mathcal {X}(x, x) \ar[d] & K \ar[l] \ar[d] \ar[r] & \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[d] \\ \mathit{Isom}_\mathcal {Y}(f(x), f(x)) & T \ar[l]_-e \ar[r]^ x & \mathcal{X} } } \]

with both squares $2$-fibre products. This follows from Categories, Lemma 4.34.3. The left vertical arrow is a morphism of algebraic spaces locally of finite type over $T$, and hence is locally of finite type, see Morphisms of Spaces, Lemma 66.23.6. Thus $K$ is an algebraic space and $K \to T$ is locally of finite type. This proves the assertion on the relative inertia. $\square$


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