 
Summary: RECASTING RESULTS IN EQUIVARIANT GEOMETRY
AFFINE COSETS, OBSERVABLE SUBGROUPS AND EXISTENCE OF GOOD QUOTIENTS
JAROD D. ALPER AND ROBERT W. EASTON
ABSTRACT. Using the language of stacks, we recast and generalize a selection of results in
equivariant geometry.
1. INTRODUCTION
When an algebraic group G acts on a variety X, there is a precise dictionary between
the Gequivariant geometry of X and the geometry of the quotient stack [X/G]. This is
typical of the strong interplay between equivariant geometry and algebraic stacks. In
deed, results (as well as their proofs) in the theory of algebraic stacks are often inspired
by analogous results in equivariant geometry. As the simplest stacks are quotient stacks,
they are fertile testing grounds for more general results. Conversely, algebraic stacks can
prove quite useful for proving results in equivariant geometry. The purpose of the present
paper is to provide some examples of this power, reproving and generalizing several the
orems in equivariant geometry via the language of algebraic stacks.
We begin in Section 2 by summarizing the relationship between the equivariant geom
etry of a scheme and the geometry of its corresponding quotient stack. In Section 3, we
review the classical notion of a good quotient and the more modern notion of a good
moduli space, and explore the relationship between them. As a result, we recover and
generalize [BB´S97, Thm. B]:
