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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.

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)
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