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Let $k$ be an algebraically closed field, and $I\subset k[x_0,\cdots,x_n]$ a homogeneous ideal of height $r$. If $I$ can be generated by $r$ elements, can we pick $r$ homogeneous elements in $I$ that generate $I$?

Background: Let $Y=V_+(I)$ be the corresponding closed subscheme. $Y$ is called a complete intersection if $I$ can be generated by $r$ elements. But it sounds weird to me, since it doesn't ask the generators to be homogeneous, which is not convenient to consider their corresponding hypersurfaces. So I guess we may choose $r$ homogeneous generators for $I$.

Could you provide some help?(prove it or give an counterexample) Thanks!

Richard
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  • Dear Richard, in both this and your other question posted recently, you've left out quite a bit of context and have posted essentially just the statement of the problem. This is a bit disappointing after you showed some good effort at avoiding this in our previous interaction. Please make some edits to improve your posts. – KReiser Dec 10 '21 at 10:39
  • @KReiser Thanks for pointing out. I have already edited them. By the way, I saw you posted an answer to this question but I havn't read it in detail since not convenient. Do you have any ideas on this problem? I'd like to learn it if you are willing to share your ideas. – Richard Dec 10 '21 at 11:44
  • (for some $j$) $I$ (assumed to be prime) will be the homogeneization of $J$ the height $r$ ideal of $k[x_1,\ldots,x_n]$ obtained by sending $x_j$ to $1$. That $J$ is generated by $r$ polynomials, its homogeneization $I'$ is generated by $r$ homogeneous polynomials, $V(I')=V(I)$, and it remains to check that $I'$ is prime. – reuns Dec 10 '21 at 13:30
  • Thank you for improving your wishing. I've restored my answer below. – KReiser Dec 10 '21 at 17:22

1 Answers1

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Suppose $I$ is generated by $r$ elements. Let $\{f_s\}_{s\in S}$ be a minimal system of homogeneous generators for $I$. Any relation $\sum g_sf_s=0$ among the $f_s$ must have $g_s\in(x_0,\cdots,x_n)$: if some $g_{s_0}$ has a nonzero constant term, then looking at the $\deg f_{s_0}$ component of the equation, we see that $f_{s_0}$ is in the ideal generated by the other $f_s$ involved. Now define $I\to k^{\oplus S}$ by $\sum g_sf_s=(g_s(0))_{s\in S}$: this is surjective and does not depend on the representation of an element as $\sum g_sf_s$ by the previous sentence. This implies that $k^{\oplus S}$ can be generated by $r$ elements, or $r\geq |S|$. Since every generating set of $I$ must have at least $\operatorname{ht} Y$ elements, we see that $I$ can indeed be generated by $r$ homogeneous elements.

KReiser
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