Generalized Analytic Independence

On the number of Elements Independent with respect to an Ideal

“…We use an obvious localization argument. The ideal a has a reduced primary decomposition: Theorem 2 answers many questions raised in [1] and [11]. In particular, we can describe the rings in which sup a = ht a for all ideals.…”

d ‐sequences, local cohomology modules and generalized analytic independence

“…We shall write the complex C{si{x), M) as and, throughout, we shall use this notation without further comment. Note that, for all ieN, l/(x), is the expansion [9; 3.2] of the triangular subset {(x"\ ..., x?<): ctj e N for all j = 1,..., i} of A 1 .…”

“…Let a be a proper ideal of A. Following [1], a set of elements a 1 ,...,a n of A is called a-independent if every form feA[Xi,...,X n ] vanishing at a lt ..., a n has all its coefficients in a. It is well known (see, for example, [1; 3]) that if Xj,..., x n is an A-sequence in A, then x l ,...,…”