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Morita equivalence of categories of graded modules. (Russian) Zbl 0629.16021

The result of K. Ohtake [Theorem 2.5 from J. Algebra 79, 169-205 (1982; Zbl 0499.18010)] is carried over to graded rings and modules. For \({\mathcal A}\subseteq gr-S\), \(U\in gr-S\) and \(R=HOM_ S(U,U)\) the pair \[ {\mathcal A}\rightleftarrows^{H=HOM_ S(U,-)}_{T=-\otimes_ RU}gr-R \] of functors is considered with natural transformations \(\Phi: TH\to 1_{{\mathcal A}}\) and \(\Psi:1_{gr-R}\to HT\), which define preradicals \(t\) and \(r\) by the rules: \(t(A)=Im\Phi_ A\), \(r(M)=Ker\Psi_ M\). Under weak restrictions on \(U\) it is proved the equivalence of the conditions: 1) \(U\) is \(t\)-projective; 2) \(\Phi_ A\) is a \(t\)-colocalization for every \(A\in {\mathcal A}\); 3) \(\Psi_ M\) isa \(r\)-localization for every \(M\in gr- R\). Hence it is obtained a graded version of Morita’s theorem on the equivalence of categories of modules.
Reviewer: A.Kashu

MSC:

16D90 Module categories in associative algebras
16W50 Graded rings and modules (associative rings and algebras)
18E35 Localization of categories, calculus of fractions
16P50 Localization and associative Noetherian rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18E40 Torsion theories, radicals

Citations:

Zbl 0499.18010
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