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Symmetric decomposition of the associated graded algebra of an Artinian Gorenstein algebra
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.jpaa.2020.106496
Anthony Iarrobino , Pedro Macias Marques

Abstract We study the symmetric subquotient decomposition of the associated graded algebras A ⁎ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra A. This decomposition arises from the stratification of A ⁎ by a sequence of ideals A ⁎ = C A ( 0 ) ⊃ C A ( 1 ) ⊃ ⋯ whose successive quotients Q ( a ) = C ( a ) / C ( a + 1 ) are reflexive A ⁎ modules. These were introduced by the first author [46] , [47] , developed in the Memoir [48] , and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms. For us a Gorenstein sequence is an integer sequence H occurring as the Hilbert function H = H ( A ) for an AG algebra A, that is not necessarily homogeneous. Such a Hilbert function H ( A ) is the sum of symmetric non-negative sequences H A ( a ) = H ( Q A ( a ) ) , each having center of symmetry ( j − a ) / 2 where j is the socle degree of A: we call these the symmetry conditions, and the decomposition D ( A ) = ( H A ( 0 ) , H A ( 1 ) , … ) the symmetric decomposition of H ( A ) (Theorem 1.4). We here study which sequences may occur as the summands H A ( a ) : in particular we construct in a systematic way examples of AG algebras A for which H A ( a ) can have interior zeroes, as H A ( a ) = ( 0 , s , 0 , … , 0 , s , 0 ) . We also study the symmetric decomposition sets D ( A ) , and in particular determine which sequences H A ( a ) can be non-zero when the dual generator is linear in a subset of the variables (Theorem 4.1). Several groups have studied “exotic summands” of the Macaulay dual generator F: these are summands that involve more successive variables than would be expected from the symmetric decomposition of the Hilbert function H ( A ) . Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no “exotic” summands (Theorem 2.7). We apply this to Gorenstein algebras that are connected sums (Section 2.4). We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.

中文翻译:

Artinian Gorenstein 代数的相关分级代数的对称分解

摘要 我们研究了非齐次交换 Artinian Gorenstein (AG) 代数 A 的相关分级代数 A ⁎ 的对称子商分解。 这种分解源于 A ⁎ 通过一系列理想 A ⁎ = CA ( 0 ) ⊃ CA ( 1 ) ⊃ ⋯ 其连续商 Q ( a ) = C ( a ) / C ( a + 1 ) 是自反 A ⁎ 模。这些是由第一作者 [46] 、 [47] 介绍的,在回忆录 [48] 中开发,最近被几个小组使用,特别是那些对短 Gorenstein 代数和方案长度(仙人掌等级)感兴趣的小组的形式。对我们来说,Gorenstein 序列是一个整数序列 H,它作为 AG 代数 A 的 Hilbert 函数 H = H ( A ) 出现,这不一定是齐次的。这样的希尔伯特函数 H ( A ) 是对称非负序列 HA ( a ) = H ( QA ( a ) ) 的总和,每个都有对称中心 ( j − a ) / 2 其中 j 是 A 的底度:我们称这些为对称条件,分解 D ( A ) = ( HA ( 0 ) , HA ( 1 ) , … ) H ( A ) 的对称分解(定理 1.4)。我们在这里研究哪些序列可能作为被加数 HA ( a ) 出现:特别是我们以系统的方式构建了 AG 代数 A 的示例,其中 HA ( a ) 可以具有内部零点,如 HA ( a ) = ( 0 , s , 0 , … , 0 , s , 0 ) 。我们还研究了对称分解集 D ( A ) ,特别是确定当对偶生成器在变量子集中是线性的时哪些序列 HA ( a ) 可以是非零的(定理 4.1)。几个小组研究了麦考利对偶生成器 F 的“奇异被加数”:这些被加数涉及比希尔伯特函数 H ( A ) 的对称分解所预期的更多的连续变量。研究这些,我们回想起没有“奇异”被加数的 AG 代数的麦考利对偶生成器的范式(定理 2.7)。我们将此应用于连接和的 Gorenstein 代数(第 2.4 节)。我们给出了许多例子和反例,并以一些关于对称分解的开放性问题作为结论。我们将此应用于连接和的 Gorenstein 代数(第 2.4 节)。我们给出了许多例子和反例,并以一些关于对称分解的开放性问题作为结论。我们将此应用于连接和的 Gorenstein 代数(第 2.4 节)。我们给出了许多例子和反例,并以一些关于对称分解的开放性问题作为结论。
更新日期:2021-03-01
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