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We know that:

  • if a ring is Cohen-Macaulay, then it is unmixed. But the converse is not true.
  • if a ring is Cohen-Macaulay, then its $h$-vector is positive. The converse is not true.

Then I expect to find unmixed ideals that have positive $h$-vector, but they are not CM. Now the question is:

Is it possible to find an example of a not unmixed ideal which has positive $h$-vector?

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I solved my question starting from the simplicial complex $$\Delta = \{\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_4\},\{x_2,x_3\},\{x_2,x_4\},x_5\}$$

It is not unmixed but $h=(1,3,1)$, since $f=(1,5,5)$.

So, the ideal that I was looking for is

$$I_{\Delta} = (x_1x_2x_3, x_1x_2x_4,x_3x_4, x_1x_5,x_2x_5,x_3x_5, x_4x_5) \subseteq \mathbb{K}[x_1, \dots, x_5]$$

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