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,[inline-graphic not available: see fulltext]) is any local ring, then A[[X]] behaves like a projective module in the sense that ExtpA(A[[X]], M)=0 for all [inline-graphic not available: see fulltext]-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.