On a Theorem of Macaulay on Colons of Ideals∗
نویسندگان
چکیده
Let R be a polynomial ring over a field of dimension n. Let m be the maximal homogeneous ideal of R. Let I be a complete intersection homogeneous ideal of R minimally generated by polynomials of degrees d1, d2, . . . , dn. F. S. Macaulay showed that for all integers i = 0, 1, . . . , δ + 1, I : m = I + m where δ = d1 + d2 + . . .+ dn − n. We provide a modern proof of a generalization of this result to standard graded Gorenstein rings. In §86 of his monograph [M] of 1916, F. S. Macaulay proved the following Theorem 1.1. Let R be the polynomial ring k[X1, X2, . . . , Xn] over a field k. Let I be a height n ideal of R generated by homogeneous polynomials f1, f2, . . . , fn of degrees d1, d2, . . . , dn respectively. Set m = (X1, X2, . . . , Xn)R and δ = d1 + d2 + . . . + dn − n. Then for all integers i = 0, 1, . . . , δ + 1,
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