Linearity Defects of Face Rings
| dc.creator | Okazaki, Ryota | |
| dc.creator | Yanagawa, Kohji | |
| dc.date | 2006-07-31 | |
| dc.date | 2007-03-06 | |
| dc.date.accessioned | 2026-07-07T07:50:07Z | |
| dc.date.available | 2026-07-07T07:50:07Z | |
| dc.description | 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 $I_Δ$ (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. | |
| dc.description | 19pages. Section 5 is largely revised; particularly, the proof of Theorem 5.1 is simplified. To appear in J. Algebra | |
| dc.identifier | https://arxiv.org/abs/math/0607780 | |
| dc.identifier | http://arxiv.org/abs/math/0607780 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/125062 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13F55 (Primary) 13D02, 13D25 (Secondary) | |
| dc.title | Linearity Defects of Face Rings | |
| dc.type | text |