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 $Ex{t}_A^p(A[[X]],M)=0$ 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.

Mainly thanks to remarks and pointers by L.L.Avramov and S.Iyengar, we added further context and references. To appear in Manuscripta Mathematica. 7 pages