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Thomas Schick: The strong Bass conjecture for group elements of finite
order and for residually finite groups
Let k be an integral domain and ring and G a discrete group. Then the
Hattori-Stallings trace $\tau_P$ of a finitely generated projective
kG-module P is a function on the conjugacy classes of elements
of G with values in k.
The strong Bass conjecture asserts that $\tau_P(g)$ is zero if the
order of g is not invertible in k (with the
convention that $\infty$ is never invertible).
In this note, we prove the strong Bass conjecture for arbitrary
integral domains
$k$ and for group elements of
finite order. This is a generalization of the corresponding result of
Linnell for $k=\integers$ and (like Linnell's proof) is based on a
calculation of Cliff.
The establish the strong Bass conjecture (for all group elements) for
domains in which no prime
is invertible, if G is residually finite. If G is residually a
finite p-group for every prime p, then we prove the strong Bass
conjecture for $kG$ for every integral domain k.
Thomas Schick