Corollary 2

Suppose that A is a graded ring with positive degrees, M a bounded below graded A-module, and the $x_i$ homogeneous elements of degree > 0 of A. Then the conditions (i) to (v) of the theorem are equivalent. The $x$-adic topology is indeed separated for every bounded below graded A-module N, since if $N_n = 0$ for $n < n_0$ one has $x^a N \subset \sum_{j \geq n_0 + a} N_j$ for all $a \geq 0$.