Bounding the socles of powers of squarefree monomial ideals

Summary

Let S = K[x₁, . . . , x_n] 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 Soc(S/I k) of S/I k , i.e., u ∈ I k and uxi ∈ I k for 1 ≤ i ≤ n. We prove that if a monomial x^a₁ ¹ · · · x^a_n ⁿ belongs to Soc(S/I ^k), then aⁱ ≤ k −1 for all 1 ≤ i ≤ n. We then discuss squarefree monomial ideals I ⊂ S for which x^k-1_[−n]∈ Soc(S/I ^k), where x[_n] = x₁x₂ · · · x_n. Furthermore, we give a combinatorial characterization of finite graphs G on [n] = {1, . . . , n} for which depth S/(I_G)² =0, where I_G is the edge ideal of G.

The depth of powers of an ideal (especially, a monomial ideal) of the polynomial ring has been studied by many authors. In the present paper, we are interested in the socle of powers of a squarefree monomial ideal.

Let K be a field, S = K[x1, . . . , xn] the polynomial ring in n variables over K, and I ⊂ S a graded ideal. We denote by m = (x1, . . . , xn) the graded maximal ideal of S. An element f + I ∈ S/I is called a socle element of S/I if xi f ∈ I for i = 1, . . . , n. Thus f + I is a nonzero socle element of S/I if f ∈ I : m\ I . The set of socle elements Soc(S/I ) of S/I is called the socle of S/I . Notice that Soc(S/I ) is a K-vector space isomorphic to (I : m)/I . One has depth S/I = 0 if and only if Soc(S/I ) = {0}.