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On modules with self Tor vanishing. (English) Zbl 1457.13034

The Auslander-Reiten conjecture states that a finitely generated module \(M\) over an Artin algebra \(\Lambda\) satisfying \(\mathrm{Ext}_{\Lambda}^i(M,\Lambda)=\mathrm{Ext}_{\Lambda}^i(M,M)=0\) for all \(i>0\) must be projective. Recently, some similar questions for commutative rings were considered in terms of vanishing of \(\mathrm{Tor}\). In this context, the notion of \(\mathrm{Tor}\)-persistent rings emerged.
A commutative ring \(R\) is \(\mathrm{Tor}\)-persistent if every finitely generated \(R\)-module \(M\) such that \(\mathrm{Tor}^R(M,M)\) is bounded (meaning that there is some positive integer \(i_0\) such that \(\mathrm{Tor}_i^R(M,M)=0\) for all \(i\geq i_0\)) has finite projective dimension.
It is known that a commutative noetherian ring is \(\mathrm{Tor}\)-persistent if and only if all of its localizations at maximal ideals are \(\mathrm{Tor}\)-persistent. For that reason, in this paper the authors consider commutative noetherian local rings, and they prove that if \(R\) is such a ring with the maximal ideal \(\mathfrak m\), then the following four conditions are equivalent:
(i)
\(R\) is \(\mathrm{Tor}\)-persistent;
(ii)
the completion \(\widehat R\) is \(\mathrm{Tor}\)-persistent;
(iii)
the ring of formal power series \(R[[X_1,\ldots,X_n]]\) is \(\mathrm{Tor}\)-persistent;
(iv)
the localization \(R[[X_1,\ldots,X_n]]_{(\mathfrak m,X_1,\ldots,X_n)}\) is \(\mathrm{Tor}\)-persistent.

MSC:

13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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References:

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