Comparing Hilbert depth of I with Hilbert depth of S/I
Abstract
Let S = K[x1,...,xn] be the ring of polynomials over a field K and let I be a monomial ideal of S. We prove that the following are equivalent: (i) I is principal, (ii) hdepth(I) = n, (iii) hdepth(S/I) = n − 1.
If I is squarefree, we prove that if hdepth(S/I) ≤ 3 or n ≤ 5, then hdepth(I) ≥ hdepth(S/I) + 1. Also, we prove that if hdepth(S/I) ≤ 5 or n ≤ 7, then hdepth(I) ≥ hdepth(S/I).
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