Linearity Defects of Face Rings

Abstract

Let $S = K[x_1, ..., x_n ]$ be a polynomial ring over a field $K$, and $E = K < y_1, ..., y_n >$ 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 $ld_E(N) < \infty$ 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 $I_\Delta$ (resp. $J_\Delta$) is the squarefree monomial ideal of $S$ (resp. $E$) corresponding to a simplicial complex $\Delta$ on ${1, >..., n}$, then $ld_E(E/J_\Delta) = ld_S(S/I_\Delta)$. Moreover, except some extremal cases, $ld$ is a topological invariant of the Alexander dual $\Delta^\vee$ of $\Delta$. We also show that, when $n > 3$, $ld_E(E/J_\Delta) = n-2$ (this is the largest possible value) if and only if $\Delta$ is an $n$-gon.