Power series rings and projectivity

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A, m) is any local ring, then A[[X]] behaves like a projective module in the sense that Ext(A)(p)(A[[X]], M) = 0 for all m-adically complete A-modules. The latter result is shown more generally for any flat A-module B instead of A[[X]]. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings.