Stanley depth and symbolic powers of monomial ideals

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The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal I and every pair of integers k, s≥ 1, the inequalities sdepth (S/I(ks)) ≤ sdepth (S/I⁽ˢ⁾) and sdepth (I(ks)) ≤ sdepth (I⁽ˢ⁾) hold. If moreover I is unmixed of height d, then we show that for every integer k≥1, sdepth(I⁽ᵏ⁺ᵈ⁾)≤ sdepth(I⁽ᵏ⁾) and sdepth(S/I⁽ᵏ⁺ᵈ⁾)≤ sdepth(S/I⁽ᵏ⁾). Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.