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A construction of strong tilting modules. (English) Zbl 1344.18001

Let \(\Lambda\) be a basic Artin algebra. Denote by \(\mathrm{mod }{A}\) the category of finitely generated left \(\Lambda\)-modules. Suppose that \(T\) is a tilting module and denote \(\mathrm{End}_{\Lambda}(T)^{\mathrm{op}}\) by \(\Gamma\). The classical tilting functor \(\mathrm{Hom}(T,-):\mathrm{mod }{\Lambda}\longrightarrow \mathrm{mod }{\Lambda}\) induces an equivalence between \(T^{\perp}\) and \({}^{\perp}DT\), and that the tilting functor takes \(T\) to \(\Gamma\), \(D\Lambda\) to \(DT\). The main result of the paper under review is the following: For any module \(T'\) in \(T^{\perp}\) is (co)tilting if and only if \(\mathrm{Hom}(T,T')\) is (co)tilting in \({}^{\perp}DT\).
Then the author uses this to show that if there is a tilting \(\Gamma\)-module \(U\) in \({}^{\perp}DT\) such that \(\mathscr{P}^{\infty}({}^{\perp}DT)=\breve{\mathrm{add} U}\), then the preimage of \(U\) in \(T^{\perp}\) is a strong tilting \(\Lambda\)-module. This generalizes a result of A. Frisk and V. Mazorchuk [Proc. Lond. Math. Soc. (3) 92, No. 1, 29–61 (2006; Zbl 1158.16009)] for standardly stratified algebras. Moreover, it gives a sufficient condition for the little finitistic dimension of \(\Lambda\) to be finite.

MSC:

18A05 Definitions and generalizations in theory of categories
16G10 Representations of associative Artinian rings
16E10 Homological dimension in associative algebras

Citations:

Zbl 1158.16009
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References:

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