Lemma 101.5.1. 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 101.3.1) 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 67.23.6. 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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