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I am trying to find an example, such a example is asked in exercise 3.3 of the book Monomial Ideals (Herzog & Hibi).

I would like to find a graded ideal $I\subset k[x_1,\cdots,x_n]$ which is complete intersection, but $\mbox{In}_<(I)$ is not even Cohen-Macaulay (for some monomial order $<$).

I would be happy with any hint.

PS: I'm trying to find a example using Singular and Macaulay2, but until to now I got nothing .

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