If $\Gamma$ is a finitely generated group. Consider the representation $\mathrm{Rep}(\Gamma, \mathrm{SL}_2(\mathbb{C} )):=\mathrm{Hom}(\Gamma, \mathrm{SL}_2(\mathbb{C} ))$.
How can we show $\mathrm{Rep}(\Gamma, \mathrm{SL}_2(\mathbb{C} ))$ is an affine algebraic set? I know we can get the polynomials from the relations between the generators, but I do not know how to write them down.
Also if $\Gamma$ is not finitely generated, Is the representation space an affine algebraic set? If not, what makes it work in the finitely generated case?
Could someone help me, please?