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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: $$ homogenous Ideal (generated by homogeneous Polynomials) $\Rightarrow$ for all $f\in $ every homogeneous component $f_i$ of $f=f_0+f_1+f_2+...+f_n$ is also in $$2012-03-12

2 Answers 2