Comparison theorems for deformation functors via invariant theory

We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.