ON THE COMPLEMENTARITY OF TWO QUASI-TILTING TRIPLES

Abstract

Let R and S be two rings. Each category equivalence between a torsion class of left (right) R-modules and a torsion-free class of left (right) S-modules is represented by a left (right) quasi-tilting triple. Suppose we have a pair of equivalences T ⇄ Y and X F between the torsion class T of R-modules and the torsion-free class Y of S-modules and between the torsion class X of S-modules and the torsion-free class F of R-modules. Denote by (R, V, S) and (S, U, R) the quasi-tilting triples representing these equivalences. We say that (R, V, S) and (S, U, R) are complementary if T, F) and X, Y) are torsion theories in R-Mod and S-Mod, respectively. We find necessary and sufficient conditions on the bimodules RVS and SUR to have the complementarity of (R, V, S) and (S, U, R).