Subgroups of finite index and the fc-localization

In [4] Gabriel gave a general theory for constructing modules and rings of quotients, by which known localizations may be defined. The fc-localization of a finitely generated group algebras R = K G is defined through R-homomo rphisms I --> R from finite codimensional right ideals I of R. In case the group C is infinite land finitely gnerated) then R embeds in its localizati on Q(fc)(R) (denoted also Q(fc)(G) when K is fixed). We examine the relatio n between Q(fc)(G) and Q(fc)(H) when H is a subgroup of G of finite index. This is done through different embeddings. A peculiar and interesting prope rty of Q(fc)(G) is that it may lack the unique rank (UR) property (or invar iant basis number), e.g. when G is a virtually-free group. In this case the Leavitt numbers (mu (G), nu (G)) are of interest, as they are invariants o f the group. We show that if H ( G and the index of H in G is m then Q(fc)( H) has UR if and only if so does Q(fc)(G) When both are without UR then mu (G) less than or equal to mu (H) less than or equal to m mu (G), and nu (H) = d nu (G) for some divisor d of m.