Detecting completeness from Ext-vanishing

Frankild, Anders; Sather-Wagstaff, Sean

doi:10.1090/S0002-9939-08-09199-5

Detecting completeness from Ext-vanishing
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by Anders J. Frankild and Sean Sather-Wagstaff PDF

Proc. Amer. Math. Soc. 136 (2008), 2303-2312 Request permission

Abstract:

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let $R$ be a commutative noetherian ring and $\mathfrak {a}$ an ideal in the Jacobson radical of $R$. Let $\widehat {\mathfrak {Ra}}$ be the $\mathfrak {a}$-adic completion of $R$. If $M$ is a finitely generated $R$-module such that $\operatorname {Ext}^i_R( M)=0$ for all $i\neq 0$, then $M$ is $\mathfrak {a}$-adically complete.

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