Happel, Dieter; Reiten, Idun; Smalø, Sverre O. Tilting in abelian categories and quasitilted algebras. (English) Zbl 0849.16011 Mem. Am. Math. Soc. 575, 88 p. (1996). The authors generalize tilting with respect to a tilting module of projective dimension at most one for an Artin algebra [see D. Happel and C. M. Ringel, Trans. Am. Math. Soc. 274, 399-443 (1982; Zbl 0503.16024)] to tilting in an abelian category in the following way. Let \(\mathcal A\) be a locally finite abelian \(R\)-category over a commutative Artin ring \(R\), \(({\mathcal T},{\mathcal F})\) a torsion pair in \(\mathcal A\) and \(T\) an object in \(\mathcal A\). Denote by \(\text{add }T\) the full subcategory of direct summands of finite direct sums of copies of \(T\). Then \(T\) is called a tilting object in \(\mathcal A\) if the following conditions are satisfied: (1) \(\mathcal T\) is a cogenerator for \(\mathcal A\); (2) \(\mathcal T\) coincides with the full subcategory of \(\mathcal A\) of epimorphic images of objects in \(\text{add }T\); (3) \(\text{Ext}^i(T,X)=0\) for \(X\in{\mathcal T}\) and \(i>0\); (4) if \(Z\in{\mathcal T}\) satisfies \(\text{Ext}^i(Z,X)=0\) for all \(X\in{\mathcal T}\) and \(i>0\), then \(Z\in\text{add }T\); (5) If \(\text{Ext}^i(T,X)=0\) for \(i\geq 0\) and all \(X\) in \(\mathcal A\) then \(X=0\). In case \(\mathcal A\) is moreover hereditary, the \(R\)-algebra \(A=\text{End}(T)^{\text{op}}\) is called quasitilted. The class of quasitilted algebras contains the tilted algebras and the canonical algebras [see C. M. Ringel, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)]. It is shown that for \(\mathcal A\) to have a tilting object it is necessary that the Grothendieck group is free of finite rank and that \(\mathcal A\) has almost split sequences.Next, homological characterizations of quasitilted algebras are given. The most important one is that quasitilted algebras can be characterized by the facts that their global dimension is at most 2 and any indecomposable module has projective dimension at most 1 or injective dimension at most 1. Moreover, an algebra \(\Lambda\) is quasitilted if and only if the category \(\mathcal R\) of indecomposable \(\Lambda\)-modules with all successors of injective dimension at most one contains all injective modules. \(\mathcal R\) and its dual subcategory \(\mathcal L\) give rise to torsion pairs which can be used to tilt to a hereditary category.The last chapter of the paper under review deals with methods of construction of quasitilted algebras. In particular the authors study the question when a one-point extension \(\Lambda=S[M]\) of a quasitilted algebra \(S\) over a field is again quasitilted and develop for this reason the concept of a module governing and dominating a class of modules. Now, for a hereditary algebra \(S\), \(S[M]\) is quasitilted if and only if \(M\) is not the middle of a short chain or \(M\) is indecomposable regular and governs the class of \(S\)-modules which have a non-zero nondirecting summand or is zero. In case \(S\) is tame hereditary and \(M\) is regular it follows that \(\Lambda\) is quasitilted if and only if \(M\) is simple regular. Reviewer: H.Meltzer (Chemnitz) Cited in 32 ReviewsCited in 282 Documents MSC: 16G10 Representations of associative Artinian rings 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16G20 Representations of quivers and partially ordered sets 18E10 Abelian categories, Grothendieck categories 16D90 Module categories in associative algebras 16E10 Homological dimension in associative algebras 16P10 Finite rings and finite-dimensional associative algebras 16S50 Endomorphism rings; matrix rings 18E30 Derived categories, triangulated categories (MSC2010) 18E40 Torsion theories, radicals 18G20 Homological dimension (category-theoretic aspects) Keywords:tilting torsion class; almost hereditary algebras; one point extensions; tilting modules; projective dimension; Artin algebras; Abelian categories; locally finite Abelian categories; commutative Artin rings; torsion pairs; direct summands; finite direct sums; tilting objects; cogenerators; quasitilted algebras; tilted algebras; canonical algebras; Grothendieck groups; almost split sequences; global dimension; indecomposable modules; injective dimension; injective modules; hereditary categories Citations:Zbl 0503.16024; Zbl 0546.16013 PDFBibTeX XMLCite \textit{D. Happel} et al., Tilting in abelian categories and quasitilted algebras. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0849.16011) Full Text: DOI