On the relation type of systems of parameters
Let \(A\) be a commutative ring. A relation on a sequence of elements \(x_1, \dots, x_n\) of \(A\) is a form of positive degree in \(S = A[U_1, \dots, U_n]\) that yields 0 when evaluated at \(x_1, \dots, x_n\). Let \(R_j\) denote the ideal of \(S\) generated by the relations on \(x_1, \dots, x_n\) of degree at most \(j\). The least integer (possibly infinite) \(t\) such that \(R_t=\bigcup_j R_j\) is called the relation type of the ideal \((x_1, \dots, x_n)\). In section two the following theorem is proven: Let \(B\) be a Buchsbaum local ring, \(I\) an ideal and \(A = B/I\). Then \(J^n \cap I = J^{n - k} (J^k \cap I)\) for any ideal \(J\) with \(JA\) being a parameter ideal of \(A\) and for \(n \geq k\) if and only if the relation type of parameter ideals of \(A\) is uniformly bounded by \(k\). In section three the main theorem is given, that is: Let \((A,m)\) be an equidimensional local ring of dimension \(d\) which possesses a dualizing complex. Then there is a certain power \(I\) of \(\prod_{i < d} \text{Ann} H^i_m (A)\) such that if \(x_1, \dots, x_d\) and \(y_1, \dots, y_d\) are systems of parameters with \(x_i - y_i\) in \(I\) for \(i = 1, \dots, d\), the relation type of \((x_1, \dots, x_d)\) and that of \((y_1, \dots, y_d)\) are the same. Using this theorem, the uniform boundedness of the relation type of systems of parameters for certain local rings is shown, e.g., for a local ring with finite local cohomology and with finite residue field.
uniform boundedness of the relation type
scientific article; zbMATH DE number 783926