Let $G\subseteq\mathrm{Gl}\_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine algebraic variety. Then, it appears to be well-known that the quotient $V/G$ has the structure of an algebraic variety such that the quotient map $\pi:V\to V/G$ is a morphism of varieties. However, I cannot find a proof for this statement. There is an abundance of proofs for the case where $G$ is finite, using the Reynolds operator and corollaries of Hilbert's basis theorem, but I would like to see a proof in the general case. Thanks very much in advance!


**Edit:** I forgot to mention that I assume $G$ to be reductive and the action on $V$ is regular. You are free to assume even more about $G$ if that allows you to provide a comprehensible reference for a proof.



Why can I divide an affine variety by the action of the general linear group?
If $G$ is reductive, try looking at [Fogarty, Kirwan, Mumford, Geometric Invariant Theory, p. 27](http://books.google.ie/books?id=dFlv3zn_2-gC&pg=PR3&lpg=PR3&dq=fogarty+kirwan&source=bl&ots=nR5HzpIUeY&sig=8Uxe2guLjSavpLeBdA1swNQctL8&hl=en&sa=X&ei=LJ-NT6SzL8fW0QX7k4XsDA&ved=0CCAQ6AEwAA#v=onepage&q=fogarty%2520kirwan&f=false)


