Linearity defects of face rings

JOURNAL OF ALGEBRA

卷:314

Article

Okazaki, Ryota ; Yanagawa, Kohji

关键词: Stanley-Reisner ring;    exterior face ring;    linearity defect;    weakly Koszul module;    componentwise linear;    sequentially Cohen-Macaulay;    squarefree module;   

DOI  :  10.1016/j.jalgebra.2007.02.049

来源: Elsevier

【 摘 要 】

Let S = K [x(1), . . .(,) x(n)] be a polynomial ring over a field K, and E = boolean AND < y(1), . . ., y(n)> an exterior algebra. The linearity defect Id(E)(N) of a finitely generated graded E-module N measures how far N departs from componentwise linear. It is known that Id(E)(N) < infinity for all N. But the value can be arbitrary large, while the similar invariant Id(S)(M) for an S-module M is always at most n. We will show that if I-Delta (resp. J(Delta)) is the squarefree monomial ideal of S (resp. E) corresponding to a simplicial complex Delta subset of 2({1, . . .,n}), then Id(E)(E/J(Delta)) = Id(S)(S/I-Delta). Moreover, except some extremal cases, Id(E)(E/J(Delta)) is a topological invariant of the geometric realization vertical bar Delta(boolean OR)vertical bar of the Alexander dual Delta(boolean OR) of Delta. We also show that, when n >= 4, Id(E)(E/J(Delta)) = n - 2 (this is the largest possible value) if and only if Delta is an n-gon. (c) 2007 Elsevier Inc. All rights reserved.

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