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Torsion pairs over \(n\)-hereditary rings. (English) Zbl 1433.18005
Given a nonnegative integer \(n\), one considers the class \(\mathcal{FP}_{n}\) of all (left) modules \(M\) over a ring \(R\) such that there exists an exact sequence \[ F_{n} \rightarrow F_{n-1}\rightarrow \dots \rightarrow F_{0} \rightarrow M \rightarrow 0 \] of \(R\)-modules, where \(F_{i}\) is finitely generated and projective for all \(i \in \{ 0, \dots, n\}\). This forms a decreasing sequence of subclasses of \(R\)-\(\operatorname{Mod}\). Note that \(\mathcal{FP}_{0}\) is the class of finitely generated \(R\)-modules.
In the article under review, the authors study homological properties associated to \(\mathcal{FP}_{n}\), for every nonnegative integer \(n\). More precisely, they introduce the notion of (left) \(n\)-hereditary ring (see Definition 3.1), which coincides with the usual notion of (left) hereditary ring if \(n = 0\) and of (left) semi-hereditary ring if \(n = 1\). One can also introduce the classes \(\mathcal{FP}_{n}\)-\(\operatorname{Inj}\) and \(\mathcal{FP}_{n}\)-\(\operatorname{Flat}\) of \(\operatorname{FP}_{n}\)-injective and \(\operatorname{FP}_{n}\)-flat \(R\)-modules, respectively, which coincide respectively with the classes of injective and flat modules if \(n=0\). The authors then prove that the new notion of \(n\)-hereditary ring satisfies some of the usual expected homological properties, involving \(\operatorname{FP}_{n}\)-injective and \(\operatorname{FP}_{n}\)-flat \(R\)-modules (see Theorem 4.6). They also study the notion of \(n\)-coherent ring, which coincides with the usual notion of coherence if \(n = 1\).
By their Theorem 4.6, the class \(\mathcal{FP}_{n}\)-\(\operatorname{Inj}\) (resp., \(\mathcal{FP}_{n}\)-\(\operatorname{Flat}\)) is closed under direct sums, extensions and quotients (resp., direct sums, extensions and submodules), provided \(R\) is \(n\)-hereditary. This allows them to consider the torsion pairs \((\mathcal{FP}_{n}\)-\(\operatorname{Inj},\mathcal{FP}_{n}\)-\(\operatorname{Inj}^{\perp})\) and \(({}^{\perp}\mathcal{FP}_{n}\)-\(\operatorname{Flat},\mathcal{FP}_{n}\)-\(\operatorname{Flat})\) in that case (see Theorem 5.3).
The article includes several illustrative examples, and in particular they construct a Bézout ring, which is \(2\)-hereditary but not \(1\)-hereditary, for which the previous torsion pairs for \(n=2\) are nontrivial (see Example 3.3, Proposition 5.6, and Propositions A.2–A.4).
MSC:
18E40 Torsion theories, radicals
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
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