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Consider a $\mathbb{Z}$-graded polynomial ring $R = k[x_1,\cdots,x_n]$ over a field $k$, where the elements $x_i$ are homogeneous (they may have negative degree). Let $I$ be a homogeneous ideal that is generated by a regular sequence (i.e., $R/I$ is a graded ring that and a complete intersection), say

$$ (f_1,f_2,\cdots,f_r), $$ where the $f_i$ are NOT necessarily homogeneous.

Question: Can I find a homogeneous regular sequence that generates the ideal?

Aside: I think that this is (probably?) known in the $\mathbb{N}$-graded setting, as any permutation of the regular sequence is still regular, but I'm not sure if this extends to the $\mathbb{Z}$-graded case.

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    $\begingroup$ For the $\mathbb N$-graded case the property holds. However, I'm not sure how you used that any permutation of a regular sequence is regular in order to prove it. $\endgroup$
    – user26857
    Commented Jul 27, 2019 at 20:10
  • $\begingroup$ I didnt really write out a proof, but the one I had in my mind was to just take the $d$ homogeneous generators of the ideal and check that it formed a regular sequence. I thought that the permutation property would come into play at some point because the construction is clearly symmetric. Maybe I'll try and write out a proof as soon as I get a chance.. $\endgroup$
    – kcnitin
    Commented Jul 28, 2019 at 0:57

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