On Ramanujam's Theorem About Finite Dimensional Groups of Automorphisms

Ramanujam’s theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam’s theorem to arbitrary (not necessarily irreducible) varieties.