Submersive and unipotent group quotients among schemes of a countable type over a field $k$

Abstract:

An algebraic group G is called submersive if every quotient in affine schemes ${c^G}:{\text {Spec}}\;A \to {\text {Spec}}\;{A^G}$ which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if ${c^G}({\text {Spec}}\;A)$ is open in ${\text {Spec}}\;{A^G}$ or if some restrictions on the action of G on A are made, ${c^G}$ is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.