Power series rings and projectivity

Abstract

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 $(A,{\mathfrak m})$ is local, then $A[[X]]$ behaves like a projective module in the sense that $Ext^p_A(A[[X]], M)=0$ for all ${\mathfrak m}$-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.