Power series rings and projectivity

2005/09/08 by Ragnar-Olaf Buchweitz, Buchweitz, R. -O., Hubert Flenner +1
Mathematics · #13C10 #13D03 #13D07 #18G05 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0509180

openalex publication_date 2005/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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