Tilting objects in abelian categories and quasitilted rings

In the paper under review the authors, generalizing classical tilting theory and the theory of quasi-tilted algebras, define and investigate quasi-tilted rings and tilting objects in arbitrary abelian categories. In more detail, let A be an abelian category and T an object of A such that arbitrary small coproducts of copies of T exist in A. Then T is called a tilting object of A if: (α) T is self-small, that is, the functor HomA(T,−) preserves small coproducts of copies of T; (β) an object A∈A is generated by T if and only if Ext1A(T,A)=0; and (γ) A is the smallest full subcategory of A which contains all objects generated by T and is closed under subobjects. In this setting the authors prove the first main result of the paper, which gives a tilting theorem between A and the module category over the endomorphism ring of T. Then, based on work by D. Happel, I. Reiten and S. O. Smalø [Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88 pp.; MR1327209 (97j:16009)], the authors call a ring R right quasi-tilted provided that there is a split torsion pair in the category Mod-R of right R-modules such that the torsion-free class contains R and the projective dimension of any torsion-free object is at most one. For comparison we recall that an Artin algebra Λ is called quasi-tilted if the category mod-R of finitely generated modules contains a split torsion pair such that the torsion-free class contains Λ and the projective dimension of any torsion-free object is at most one. Then the second main result of the paper characterizes (right) quasi-tilted rings as the endomorphism rings of tilting objects in cocomplete hereditary abelian categories. Further, it is shown that the (right) global dimension of a (right) quasi-tilted ring is at most two and the injective dimension of any torsion module is at most one. Note that all of these results hold true for quasi-tilted Artin algebras. In this connection the authors state and discuss two open problems. The first one asks whether any ring of right global dimension at most two and such that each of its right modules is a direct sum of a module of injective dimension at most one and a module of projective dimension at most one is (right) quasi-tilted (this is true for quasi-tilted Artin algebras by results of Happel, Reiten and Smalø in [op. cit.]). The second problem asks whether a quasi-tilted Artin algebra is quasi-tilted as a ring. For an affirmative answer to the second problem we refer to a recent paper by E. Gregorio ["Every quasitilted algebra is a quasitilted ring'', J. Algebra, to appear]. The paper concludes with an example, further developed and analyzed in the recent paper by the authors and Gregorio [Colloq. Math. 104 (2006), no. 1, 151--156; MR2195804 (2007e:16033)], which shows that the class of right quasi-tilted rings properly extends the class of (right) tilted rings (defined as the endomorphism rings of finitely generated tilting modules over right hereditary rings). Finally, there is an appendix where the authors discuss the behavior of the functor Ext1(−,B) under direct sums in a cocomplete abelian category.