On Quotient Stacks

A natural problem in algebraic geometry is the formation of quotients. This is particularly important in the theory of moduli, where many moduli spaces are naturally constructed as quotients of parameter spaces by linear algebraic groups. Examples of quotient moduli spaces include moduli spaces of curves, stable maps and stable vector bundles (with fixed determinant). Unfortunately, the quotient of a scheme by a a group need not exist as a scheme, or even as an algebraic space. Moreover, even when a quotient exists as an algebraic space, the quotient morphism may not have expected properties. For example, if Z and G are smooth, then the morphism Z → Z/G need not be smooth. To overcome this difficulty we consider quotients as stacks, rather than schemes or algebraic spaces. If G is a flat group scheme acting on a space Z, then a quotient [Z/G] always exists as a stack, and the morphism Z → [Z/G] makes Z into a principal G-bundle over [Z/G]. Knowing that a stack has a presentation as a quotient [Z/G] can often make the stack easier to study. The reason is that the geometry of the stack can be viewed as G-equivariant geomtry on the space Z. For example, the Chow groups of a stack [Z/G] are simply the Gequivariant Chow groups of Z ([E-G]). In this way one can easily show that the smooth quotient stacks have an intersection ring. A natural question is to determine which stacks are quotient stacks. Observe that if F ≃ [Z/G] then the automorphism groups of objects over geometric points are linear algebraic groups. Thus, a trivial necessary condition is that automorphism groups of objects are linear algebraic groups. In particular, not all stacks can be quotients. For example, the classifying stack BE, where E is an elliptic curve, is obviously not a quotient stack in the sense above. (Of course BE is a quotient stack for a non-linear algebraic group.)