Families of Artinian and one-dimensional algebras

JOURNAL OF ALGEBRA

卷:311

Article

Kleppe, Jan O.

关键词: parametrization;    Artinian algebra;    level algebra;    Gorenstein algebra;    Licci;    Hilbert scheme;    duality;    algebra (co)homology;    canonical module;    normal module;   

DOI  :  10.1016/j.jalgebra.2006.11.019

来源: Elsevier

【 摘 要 】

The purpose of this paper is to study families of Artinian or one-dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg(H)(R) be the scheme parametrizing graded quotients of R with Hilbert function H. Let B -> A be any graded surjection of quotients of R with Hilbert function H (B) = (1, h(1), . . . , h (j), . . .) and H (A), respectively. If dim A = 0 (respectively dim A = depth A = 1) and A is a truncation of B in the sense that H (A) = (1, h(1), . . . , h(j - 1), alpha, 0, 0, . . .) (respectively H-A = (1, h (1), . . . , h (j - 1), alpha, alpha, alpha, . . .)) for some alpha <= h (j), then we show there is a close relationship between GradAlg(H A) (R) and GradAlg(H B) (R) concerning e.g. smoothness and dimension at the points (A) and (B), respectively, provided B is a complete intersection or provided the Castelnuovo-Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to non-truncated Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg(H) (R), H = (1, 3, 6, 10, 14, 10, 6, 2), whose general elements are Artinian level algebras of type 2. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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