Quotient Prestack And Stack

Definition

Let \(G\to S\) be a smooth affine group scheme acting on a scheme \(U\) over \(S\).

(Quotient Prestack). The quotient prestack \([U/G]^{\text{pre}}\) of an action of a smooth affine group scheme \(G\to S\) on an \(S\)-scheme \(U\) is the category over \(\Sch/S\) whose

• objects are pairs \((T,u)\) where \(T\) is an \(S\)-scheme and \(u\in U(T)\)
• morphisms \((f,g): (T', u') \to (T, u)\) are morphisms \(f:T'\to T\) of \(S\)-schemes such that there is an element \(g\in G(T')\) satisfying \(f^*u = g\cdot u'\).
• The projection map is to \(\Sch/S\), where we send \((T,u)\) to \(T\) (forget the map to \(U\)).

In particular, if we choose \(T' = T\) but are given two different morphisms \(u,u':T\to U\), then an \(S\)-morphism \(f:T\to T\) is only a morphism in \([U/G]^{\text{pre}}(T)\) if there is a \(g\) so that \[f^*u = u\circ f = g\cdot u'.\] Note that the fiber category \([U/G]^{\text{pre}}(T)\) is identified with the quotient groupoid \([U(T)/G(T)]\).

(Quotient Stack). The quotient stack \([U/G]\) is the category over \(\Sch/S\) whose

• objects are diagrams

where \(P\to T\) is a principal \(G\) bundle and \(P\to U\) is a \(G\)-equivariant morphism of \(S\)-schemes

• morphisms \((T'\leftarrow P' \to U)\to (T\leftarrow P\to U)\) consist of a morphism \(f:T'\to T\) as \(S\)-schemes and a \(G\)-equivariant morphism \(P'\to P\) of schemes such that the diagram

is commutative and the left square is Cartesian.