  If A is a bialgebra over a field k and M, N are either left-right
Yetter-Drinfel'd modules or left-right Hopf modules over A, we construct
deformation cohomologies H^*(M,N) as total cohomologies of certain double
complexes Y(M,N) and C(M,N), respectively. In both cases, H^1(M,N) is
isomorphic to the group of equivalence classes of extensions of M by N in the
corresponding category. In the Yetter-Drinfel'd case, H^*(k,k) is just the
Gerstenhaber-Schack cohomology of the bialgebra A. If M, N are Hopf bimodules
we construct a natural subbicomplex of the above C(M,N), yielding a cohomology
theory for Hopf bimodules, similar to the one recently introduced by R.
Taillefer.
