Preliminaries.

From now on, we restrict attention to noether filtrations which take only non-negative values, and all noether filtrations mentioned will be assumed to satisfy this condition. It follows that the symbol v(f-) will never occur, and we will therefore write v(f) in place of v(f+).

We now consider the following question. Suppose that A is a noetherian ring and that f is a noether filtration on A. Then it is natural to ask whether the integral closure f* of f is equivalent to f. Since f*(x) ≥ f(x) for all x, this is equivalent to the statement that there exists a constant K such that f(x) ≤ f*(x) ≤ f(x) + K for all x. An equivalent formulation is that u-kG(f) ⊇ G(f*), which in turn is equivalent to G(f*) being a finite G(f)-module, and hence implies that f* is also a noether filtration.

Note that the restriction to non-negative noether filtrations implies that f and f* are equivalent if and only if f* is a noether filtration. For in this case A0(f*) = A0(f) = A, and f* is a noether filtration if and only if G(f*) is finitely generated over A. But this is equivalent to G(f*) being finitely generated over G(f), and since G(f*) is an integral extension of G(f), this in turn is equivalent to G(f*) being a finite G(f)-module and hence equivalent to f.