Define a projective variety to be a subspace $V \subset \mathbb P^n$ such that $V$ is the zero set of some set $T$ of homogenous polynomials in $k[x_0, \ldots , x_n]$. My book claims that "as with affine varieties, we can assume $T$ is finite.
I'm having trouble seeing why this is true. $Z(T) = Z(\langle T \rangle)$, and $\langle T \rangle $ is a homogeneous ideal since it's generated by homogenous polynomials. Since $k[x_0, \ldots , x_n]$ is Noetherian, $\langle T \rangle $ is finitely generated, but why can we take those generators to be homogeneous?