Tilting and trivial extensions (Q1047900)

The authors generalize some results on the lifting of a tilting modules to a trivial extension by a bimodule, known for Artin algebras, to similar results for an arbitrary ring. The authors use the one-to-one correspondence between quasi-abelian categories and tiltings to show that the trivial extension of a quasi-abelian category corresponds to a trivial extension of the associated tilting. In particular, the main results are as follows: Theorem 1: Let \(\mathcal A\) be a quasi-abelian category with a fully exact additive endo-functor \(F\). Then the trivial extension of \(\mathcal A\) by \(F\) (in the sense of Fossum) is likewise quasi-abelian; the form of the corresponding tilting is also given. Theorem 2: Let \(_RT\) be a tilting module over a ring \(R\), and let \(_RP_R\) be a bimodule with \(P_R\) flat such that the endofunctor \(F_R:=P\otimes_R-\) of Mod\((R)\) respects the (Gen\(_RT\))-torsion part of \(R\)-modules. Then the Fossum extension of \(R\) by \(P\) tensored with \(T\) is a tilting module.