Comparing Hilbert depth of I with Hilbert depth of S/I

2024/03/25 by Bordianu, Andreea I., Cimpoeas, Mircea
#05A18 #06A07 #13C15 #13F20 #13P10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.17078

Abstract

Let I be a monomial ideal of S=K[x₁,…,x_n]. We show that the following are equivalent: (i) I is principal, (ii) hdepth(I)=n, (iii) hdepth(S/I)=n-1. Assuming that 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 then hdepth(I)≥ hdepth(S/I).

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