An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules

Pub Date : 2020-12-25 DOI:10.1090/conm/773/15542

Sabine El Khoury, Manoj Kummini, H. Srinivasan

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引用次数: 0

Abstract

Let R R be a polynomial ring over a field and M = ⨁ n M n M= \bigoplus _n M_n be a finitely generated graded R R -module, minimally generated by homogeneous elements of degree zero with a graded R R -minimal free resolution F \mathbf {F} . A Cohen-Macaulay module M M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 e_1 in terms of the shifts in the graded resolution of M M . When M = R / I M = R/I , a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

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Gorenstein代数和模的第一希尔伯特系数的上界

设R R是一个域上的多项式环，M= φ n M= φ n M= \bigo + _n M n是一个有限生成的梯度R R -模，最小由零次齐次元素生成，具有梯度R R -最小自由分辨率F \mathbf {F}。当梯度分辨率对称时，Cohen-Macaulay模M M为Gorenstein模。我们给出了第一希尔伯特系数e1e_1的上界，这是根据M M的梯度分辨率的偏移。当M = R/I M = R/I为Gorenstein代数时，该界与[ES09]在准纯分辨的Gorenstein代数中得到的界一致。对于较高的系数，我们推测出一个类似的界。

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