Quotient stack

In algebraic geometry, a quotient stack is a stack that generalizes the quotient of a scheme or a variety by a group. It is defined as follows. Let G be an affine flat group scheme over a scheme S and X a S-scheme on which G acts. Let $[X/G]$ be the category over S: an object over T is a principal G-bundle E →T (in etale topology) together with equivariant map E →X; an arrow from E →T to E' →T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps E →X and E' →X. It is a theorem of Deligne–Mumford that $[X/G]$ is an algebraic stack. If $X=S$ with trivial action of G, then $[S/G]$ is called the classifying stack of G (in analogy with the classifying space of G) and is usually denoted by BG.