Bounding the socles of powers of squarefree monomial ideals

الأصل · EN

Let S=K[x₁,,xₙ] be the polynomial ring in n variables over a field K and I⊂ S a squarefree monomial ideal. In the present paper we are interested in the monomials u ∈ S belonging to the socle (S/Iᵏ) of S/Iᵏ, i.e., u ∈ Iᵏ and uxᵢ ∈ Iᵏ for 1 ≤ i ≤ n. We prove that if a monomial x₁ᵃ¹ xₙᵃⁿ belongs to (S/Iᵏ), then aᵢ≤ k-1 for all 1 ≤ i ≤ n. We then discuss squarefree monomial ideals I ⊂ S for which x[ₙ]ᵏ⁻¹ ∈ (S/Iᵏ), where x[ₙ] = x₁x₂ xₙ. Furthermore, we give a combinatorial characterization of finite graphs G on [n] = {1,, n} for which S/(IG)²=0, where IG is the edge ideal of G.