Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth
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Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth Huy T`ai H`a · Mengyao Sun Dedicated to Professor Ngˆo Viˆet Trung on the occasion of his sixtieth birthday Received: 27 May 2014 / Revised: 8 August 2014 / Accepted: 11 August 2014 / Published online: 16 January 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract In a recent work (Kaiser et al., J. Comb. Theory Ser. A 123, 239–251, 2014), Kaiser et al. provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs and, thus, give counterexamples to a conjecture of Francisco et al. (Discrete Math. 310, 2176–2182, 2010). The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties. Keywords Persistence · Non-increasing depth · Associated primes · Monomial ideals · Cover ideals · Critical graphs Mathematics Subject Classification (2010) 13C15 · 15P05 · 05C15 · 05C25 · 05C38
1 Introduction Let k be a field and let R = k[x1 , . . . , xn ] be a polynomial ring over k. Let I ⊆ R be a homogeneous ideal. It is known by Brodmann [3] that the set of associated primes of I s stabilizes for large s, that is, Ass(R/I s ) = Ass(R/I s+1 ) for all s 0. However, the behavior of these sets can be very strange for small values of s. The ideal I is said to have the persistence property if Ass(R/I s ) ⊆ Ass(R/I s+1 ) ∀ s ≥ 1.
H. T. H`a () · M. Sun Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA e-mail: [email protected]; URL http://www.math.tulane.edu/∼tai/ M. Sun e-mail: [email protected]
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It is also known by Brodmann [4] that depth(R/I s ) takes a constant value for large s. The behavior of depth(R/I s ), for small values of s, can also be very complicated. The ideal I is said to have non-increasing depth if depth(R/I s ) ≥ depth(R/I s+1 ) ∀ s ≥ 1. Associated primes and depth of powers of ideals have been extensively investigated in the literature (cf. [1, 6–8, 10, 11, 13–15, 17, 19–21]). Even for monomial ideals, it is difficult to classify which ideals possess the persistence property or non-increasing depth. In this case, when I is a monomial ideal, the two properties are related by the fact that I possesses the persistence property if all monomial localizations of I have non-increasing depth. Herzog and Hibi [11] gave an example where m = (x1 , . . . , xn ) ∈ Ass(R/I s ) for small even integers s (whence depth(R/I s ) = 0) and m ∈ Ass(R/I s ) for small odd integers s (whence depth(R/I s ) > 0). Squarefree monomial ideals behave considerably better than monomial ideals in general, and many classes of squarefree monomial ideals were shown to have the persistence property. For instance, edge ideals of grap
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