Free extensions and Jordan type

JOURNAL OF ALGEBRA

卷:549

Article

Iarrobino, Anthony¹  Marques, Pedro Macias²  McDaniel, Chris³ 

[1] Northeastern Univ, Dept Math, Boston, MA 02115 USA

[2] Univ Evora, Ctr Invest Matemat & Aplicacoes, Inst Invest & Form Avancada, Dept Matemat,Escola Ciencias & Tecnol, Rua Romao Ramalho 59, P-7000671 Evora, Portugal

[3] Endicott Coll, Dept Math, Beverly, MA 01915 USA

关键词: Artinian algebra;    Coinvariant;    Deformation;    Free extension;    Hilbert function;    Invariant;    Jordan type;    Lefschetz property;    Tensor product;   

DOI  :  10.1016/j.jalgebra.2020.01.003

来源: Elsevier

【 摘 要 】

Free extensions of graded Artinian algebras were introduced by T. Harima and J. Watanabe, and were shown to preserve the strong Lefschetz property. The Jordan type of a multiplication map m by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for m. We show that a free extension C of the Artinian algebra A with fiber B is a deformation of the usual tensor product. This has consequences for the generic Jordan types of A, B and C: we show that the Jordan type of C is at least that of the usual tensor product in the dominance order (Theorem 2.5). In particular this gives a different proof of the T. Harima and J. Watanabe result concerning the strong Lefschetz property of a free extension. Examples illustrate that a non-strong-Lefschetz graded Gorenstein algebra A with non-unimodal Hilbert function may nevertheless have a non-homogeneous element with strong Lefschetz Jordan type, and may have an A-free extension that is strong Lefschetz. We apply these results to algebras of relative coinvariants of linear group actions on a polynomial ring. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free