SUBGROUPS OF FINITE INDEX AND THE fc-LOCALIZATION

Abstract

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 = KG is defined through R-homomorphisms I → R from finite codimensional right ideals I of R. In case the group G is infinite (and finitely gnerated) then R embeds inits localization Q _fc (R) (denoted also Q _fc (G) when K is fixed). We examine the relation 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 property of Q _fc (G) is that it may lack the unique rank (UR) property (or invariant basis number), e.g. when G is a virtually-free group. In this case the Leavitt numbers (μ(G), ν(G)) are of interest, as they are invariants of 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 μ(G) ≤ μ(H) ≤ mμ(G), and ν(H) = d ν(G) for some divisor d of m.

Acknowledgments