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Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of positive degree that generate $S$ over $A$.

It then follows that $S$ is isomorphic to $k[x_1, ..., x_M]/I$ as rings for some ideal $I$. Does it follow that $I$ must be a homogeneous ideal of $k[x_1, ..., x_M]$? I am guessing it is the case, but I just wasn't sure. Thank you!

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Only if you correctly define "homogeneous". For example, take $R = k[X^2, X^3]$. Then $R$ is isomorphic to the quotient $k[s,t]/(s^3 - t^2)$, and $(s^3-t^2)$ is not a homogeneous ideal according to the usual definition.

However, it is still homogenous if we put a different grading on $k[s,t]$, namely we specify that $s$ has degree $2$ and $t$ has degree $3$. We should do this anyways, if we want $R$ to be a quotient as a graded ring, rather than just as a ring.

Given this definition, here is how to see that $I$ is homogeneous in general: Suppose that $f \in I$, and $f$ equals the graded sum $\sum_i f_i$. Then $\sum_i \overline{f_i} = \overline{f} = \overline{0}$. Since each $\overline{f_i}$ has different degree, we have $\overline{f_i} = \overline{0}$ for each $i$, so $f_i \in I$ for each $i$.

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  • $\begingroup$ Thank you for the answer! How do we know that each $\bar{f}_i$ has a different degree? $\endgroup$ – user211392 Mar 24 '15 at 18:20
  • $\begingroup$ @user211392 If $\pi: k[t_i] \to S$ is the quotient map, then we choose the grading on $k[t_i]$ precisely so that $\pi$ preserves the grading. Then if $f=\sum_i f_i$ is a sum of homogeneous elements of different degrees, it follows that $\pi(f) = \sum_i \pi (f_i)$ is a sum of homogeneous elements of different degrees. $\endgroup$ – Slade Mar 24 '15 at 19:31
  • $\begingroup$ @Slade can we conclude here that $I$ does not contain elements of degree $1$ ? $\endgroup$ – R. Singh Apr 25 '17 at 10:01
  • $\begingroup$ @R.Singh In the example I gave, $I=(s^3-t^2)$ is generated by an element $s^3-t^2$ of degree $6$, and therefore has no homogeneous elements of positive degree $\leq 5$. $\endgroup$ – Slade Apr 25 '17 at 12:07
  • $\begingroup$ @Slade I'm not saying for the particular example you've used in your answer. I'm asking it for a general $I$, as mentioned in the question. $\endgroup$ – R. Singh Apr 25 '17 at 13:16

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