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Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence x of elements of degree 1 so that R/x is zero dimensional?

I tend to believe that it holds, but could not find a reference. Maybe some extra condition should be imposed?

flag	asked Oct 26 2010 at 4:26 Li Li 95●4

Graham Leuschke · Accepted Answer · 2010-10-26 10:56:04Z UTC

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You want to assume also that the residue field is infinite. Then a minimal reduction of the irrelevant ideal will be a system of parameters, each linear, and will be a maximal regular sequence.

answered Oct 26 2010 at 10:56

Graham Leuschke
3,646●1●10●20

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And over a finite field $k$, consider the union $V$ of all hyperplans of $\mathbb P^n_k$ and let $R$ be the obvious corresponding homogeneous algebra. Then $R$ is Cohen-Macaulay, but all elements of degree 1 are zero divisors. – Qing Liu Oct 26 2010 at 12:27

Thanks to Graham and Qing! – Li Li Oct 26 2010 at 13:39

Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring

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