Algebraic stacks and their inertia stacks

An algebraic stack (or Artin stack) over $S$ is a stack in groupoids $\mathcal{X}$ over $(Sch\mathrm{/}S{)}_{fppf}$ (Descent data, prestacks and stacks in groupoids over the fppf site, Categories fibred in groupoids over a site) such that

1. the diagonal 1-morphism $\mathrm{\Delta }\text{\hspace{0.17em}⁣}:\mathcal{X}\to \mathcal{X}{\times }_S\mathcal{X}$ is representable by algebraic spaces (Morphisms representable by algebraic spaces), and
2. there exist an $S$-scheme $U$ and a 1-morphism ${\mathcal{S}}_U\to \mathcal{X}$ from the stack in setoids of $U$ which is representable by algebraic spaces, surjective and smooth (Smooth morphism of schemes).

Such a pair $(U,{\mathcal{S}}_U\to \mathcal{X})$ is called a presentation of $\mathcal{X}$. The algebraic stack is Deligne-Mumford when a presentation with ${\mathcal{S}}_U\to \mathcal{X}$ etale (Étale morphism of schemes) can be chosen. Under the inherited Axiom of Choice for represented-sheaf descent (The Axiom of Choice, Descent data, prestacks and stacks in groupoids over the fppf site), a scheme, viewed as a stack in setoids, is an algebraic stack with its identity as presentation. An algebraic space is an algebraic stack using any of its etale scheme covers as presentation; the identity is not a scheme presentation when the algebraic space is not a scheme.

The inertia stack ${\mathcal{I}}_{\mathcal{X}}$ is the category fibred in groupoids whose fibre category over $T$ has as objects the pairs $(x,\alpha )$ with $x$ an object of ${\mathcal{X}}_T$ and $\alpha \in {\mathrm{Aut}}_{{\mathcal{X}}_T}(x)$. For $f:T^{\mathrm{\prime }}\to T$, a morphism from $(y,\beta )$ over $T^{\mathrm{\prime }}$ to $(x,\alpha )$ over $T$ is an arrow $\gamma :y\to x$ of $\mathcal{X}$ over $f$ satisfying $\gamma \beta =\alpha \gamma$. Equivalently it is a vertical isomorphism $y\to f^{\ast }x$ intertwining $\beta$ with $f^{\ast }\alpha$. Composition is composition of these arrows; cartesian uniqueness supplies the pullback identifications. The projection ${\mathcal{I}}_{\mathcal{X}}\to \mathcal{X}$, $(x,\alpha )\mapsto x$, is a 1-morphism over $(Sch\mathrm{/}S{)}_{fppf}$.

For a 1-morphism $\mathcal{X}\to \mathcal{Y}$ of stacks in groupoids, the relative inertia ${\mathcal{I}}_{\mathcal{X}\mathrm{/}\mathcal{Y}}$ imposes the additional condition that the automorphism $\alpha$ map to the identity automorphism of the image of $x$ in ${\mathcal{Y}}_T$; the projection ${\mathcal{I}}_{\mathcal{X}\mathrm{/}\mathcal{Y}}\to \mathcal{X}$ is again a 1-morphism. Both ${\mathcal{I}}_{\mathcal{X}}$ and ${\mathcal{I}}_{\mathcal{X}\mathrm{/}\mathcal{Y}}$ are stacks in groupoids whenever $\mathcal{X}$ and $\mathcal{Y}$ are (Descent data, prestacks and stacks in groupoids over the fppf site), since automorphism data satisfies effective descent in groupoids.