We review a way of constructing moduli spaces via geometric invariant theory. Once a classification problem is packaged into a moduli functor, a fine moduli space is a scheme representing that functor. When it does not exist, we can sometimes find a universal approximation: a coarse moduli space. In order to construct a moduli space it is often the case that we need to take the quotient of an affine algebraic group action. Since the orbit space as a locally ringed space need not be a scheme, we need a different notion of a quotient. We define a categorical quotient for affine algebraic group actions on affine and quasiprojective schemes and show that it always exists, when the group and scheme are sufficiently nice. In order to do that we need to introduce a notion of stability and throw out unstable points. The Hilbert-Mumford criterion gives a numerical criterion for the computation of stability.
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