invertible bimodules on group algebras Are there any simple ways of constructing non-trivial invertible bimodules on discrete group algebras? (I.e. $A$-bimodules $E$ which have an inverse bimodule $E'$ with $E'\otimes_A E\cong A$ obeying the usual rules in algebraic $K$-theory.) By non-trivial I am hoping to have some $E$ which are not freely generated as left modules.
After trying and failing to construct some, I am wondering if such things exist at all. I am probably being silly here, any help would be appreciated. I may just not know enough ways to construct any sort of modules on group algebras...
 A: If $A$ is an algebra and $\sigma:A\to A$ is an algebra automorphism, there is an $A$-bimodule $A_\sigma$ which coincides with the regular $A$-bimodule $A$ as a left $A$-module, but such that when $m\in A_\sigma$ and $a\in A$ the right action is given by $$m\cdot a=m\sigma(a),$$ with the product on the right the multiplication of $A$.
You an easily check that $A_\sigma\otimes_AA_\tau\cong A_{\sigma\tau}$ and that $A_{\mathsf{id}_A}\cong A$ as bimodules. From this it follows easily that $A_\sigma$ is an invertible bimodule. Moreover, it is not difficult to show that if $\sigma$ is an inner automorphism of $A$, then $A_\sigma\cong A$ as bimodules. From this, we get an exact sequence $$1\to\operatorname{Inn}(A)\to\operatorname{Aut}(A)\to\operatorname{Pic}(A)$$ with $\operatorname{Inn}(A)$ denoting the group of inner automorphisms of $A$, $\operatorname{Aut}(A)$ the group of all automorphisms, and $\operatorname{Pic}(A)$ the group of isomorphism classes of invertible $A$-bimodules; the last map sends an automorphism $\sigma$ to the bimodule $A_\sigma$.
From this, you can see that whenever $\sigma$ is an automorphism of the group $G$, then you get an invertible $kG$-bimodule by constructing $kG_{\sigma}$. Moreover, if $\sigma$ is not an inner automorphism of $G$, then it does not induce an inner  automorphism of $kG$, and then the bimodule $kG_\sigma$ is a non-trivial invertible bimodule.
You can therefore pick any group with a non-inner automorphism and construct an example of what you want. Abelian groups give many examples of this. The symmetric group $S_6$ and many other noncommutative groups also do.

You want bimodules which are not free on the left. Consider then for example the ring of integers in a cyclotomic field $\mathbb Q(\zeta)$ obtained from adjoining a primitive $p$th root of unity to the field of rational numbers. Its ring of integers is $\mathbb Z[\zeta]$, and this is the group ring of the cyclic group of order $p$. Now elements of the class group of this ring provide (symmetric) invertible modules, and nontrival elements of the classgroup, invertible modules which are not free. The smallest example where this does give a nonfree bimodule is $p=23$, in which case the class group has order $3$.
