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

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