Mathematics

The goal of this talk is to construct the quotient stack by a smooth group scheme and to see that it is an Artin stack. The talk will be divided into four parts. First, we define the algebraic space over a fixed scheme S. Algebraic spaces are sheaves on the big étale site Sch/S with the representable diagonal morphism and an étale covering. Then, we give the original definition of the stack. Stacks are categories fibered in groupoid over a site for which every covering is of effective descent. We also provide a necessary and sufficient condition for a category fibered in groupoid to be a stack. Third, we define the Artin stack. Artin stacks are stacks over the big étale site Sch/S with the representable diagonal morphism and an smooth covering. Here, algebraic spaces play a role to describe property of morphisms of stacks. Finally, given a smooth group scheme G acting on a scheme X over S, we construct the quotient [X/G] and check that [X/G] satisfies the all conditions to be a stack, moreover an Artin stack. If time allows, we will provide some examples of quotient stacks.