Corrigendum to Complete homology and related dimensions of groups [J. Group Theory 12 (2009), 431–448]

We have realized that the statement and proof of Proposition 4.5 (2) of [5] is incorrect. Note that for a ring R, the following hold (cf. [4]):

1. A right R-module M is flat if and only if its character module

   M * := Hom ℤ ⁢ ( M , ℚ / ℤ )

   is an injective left R-module.

2. If R is right noetherian, then M is injective if and only if M * is flat.

By the argument of the proof of Proposition 4.5 (2) we can prove that the following holds:

Let G be a group such that ℤ ⁢ G is noetherian. Then sfli ⁡ G = silf ⁡ G .

Thus unfortunately, we cannot conclude that hd ¯ G = 0 if and only if G is locally finite under the assumption in Theorem 4.10 or Corollary 4.11 of [5]. However, Emmanouil proved in [2] that for any group G, hd ¯ G = 0 if and only if G is locally finite. In [1], Emmanouil showed that for any group G, spli ⁡ G = silp ⁡ G . Moreover, Emmanouil and Talelli proved in [3] that for any group G, silf ⁡ G = silp ⁡ G . Also, they gave an example of a group G such that sfli ⁡ G < spli ⁡ G < ∞ in [3]. Hence we conclude that there is an example of a group G with sfli ⁡ G < silf ⁡ G < ∞ .