# Projective varieties are the zeroes of finitely many homogenous polynomials

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?

Thanks

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Hint: $<T>$ homogenous Ideal (generated by homogeneous Polynomials) $\Rightarrow$ for all $f\in <T>$ every homogeneous component $f_i$ of $f=f_0+f_1+f_2+...+f_n$ is also in $<T>$ – Blah Mar 12 '12 at 19:20

If you take a homogeneous ideal $I$ in $k[x_0,\cdots,x_n]$, then $I=\langle f_1,\cdots,f_m\rangle$, for some polynomials $f_i$. Now take the homogeneous components of each of the $f_i$ and you get a finite set of homogeneous polynomials generating $I$.
(note that $I$ is homogeneous if and only if the homogeneous components of all polynomials in $I$ are still in $I$)