Linearity Defects of Face Rings

Let S = K[x1, . . . , xn] be a polynomial ring over a field K, and E = ∧ 〈y1, . . . , yn〉 an exterior algebra. The linearity defect ldE(N) of a finitely generated graded E-module N measures how far N departs from “componentwise linear”. It is known that ldE(N) < ∞ for all N . But the value can be arbitrary large, while the similar invariant ldS(M) for an S-module M is always at most n. We will show that if I∆ (resp. J∆) is the squarefree monomial ideal of S (resp. E) corresponding to a simplicial complex ∆ ⊂ 2, then ldE(E/J∆) = ldS(S/I∆). Moreover, except some extremal cases, ldE(E/J∆) is a topological invariant of the geometric realization |∆∨| of the Alexander dual ∆ of ∆. We also show that, when n ≥ 4, ldE(E/J∆) = n − 2 (this is the largest possible value) if and only if ∆ is an n-gon.