Linearity Defects of Face Rings

2006/07/31 by Okazaki, Ryota, Yanagawa, Kohji
#13D25 (Secondary) #13F55 (Primary) 13D02 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0607780

Abstract

Let S = K[x₁, ..., x_n ] be a polynomial ring over a field K, and E = K < y₁, ..., y_n > an exterior algebra. The "linearity defect" ld_E(N) of a finitely generated graded E-module N measures how far N departs from "componentwise linear". It is known that ld_E(N) < ∞ for all N. But the value can be arbitrary large, while the similar invariant ld_S(M) for an S-module M is alway at most n. We show that if I_Δ (resp. J_Δ) is the squarefree monomial ideal of S (resp. E) corresponding to a simplicial complex Δ on 1, >..., n, then ld_E(E/J_Δ) = ld_S(S/I_Δ). Moreover, except some extremal cases, ld is a topological invariant of the Alexander dual Δ^\vee of Δ. We also show that, when n > 3, ld_E(E/J_Δ) = n-2 (this is the largest possible value) if and only if Δ is an n-gon.

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