I know my question is very imprecise. I am trying to understand Tate-Farrell cohomology of the infinite Lie group $S^1$ (say, with coefficients in $\mathbb C$). I would expect that the answer is something like the space of Laurent polynomials $\mathbb C[t^{-1},t]$. Is there any geometric intuition for this? What would be the meaning of multiplication by $t$? What is the meaning of the completion $\mathbb C((t))$?



Tate-Farrell cohomology of a circle
**Comment:** Farrell-Tate cohomology as defined in Brown's book "Cohomology of Groups" 
requires the group to be of finite virtual cohomological dimension (i.e. the group has a finite index subgroup which has a finite projective resolution). But $S^1$ doesn't have finite virtual cohomological dimension because it has finite subgroups of arbitrary order. 


There is a generalization of Farrell-Tate cohomology for arbitrary groups due to Benson-Carlson/Mislin, that is usually called "complete cohomoloy" or "complete Tate cohomology". I don't know if that cohomology has been computed for $S^1$ (most computations are done for discrete groups). 


