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On the derived category of a finite-dimensional algebra. (English) Zbl 0626.16008

Let A be a finite dimensional associative k-algebra over an algebraically closed field k. Denote by \(D^ b(A)\) the derived category of bounded complexes over the category mod A of finite dimensional left A-modules. Two triangulated categories C and C’ are said to be triangle-equivalent if there is an additive functor \(F: C\to C'\) which commutes (up to isomorphism) with the translation functors and sends triangles to triangles. The main objective of this paper is the study of \(D^ b(A)\) for A of finite global dimension. Among many results the following ones are obtained: (1) Let M be an A-module such that \(Ext^ i_ A(M,M)=0\) for \(i>0\) and suppose that there is an exact sequence \(0\to_ AA\to M^ 0-...\to M^ s\to 0\) with \(M^ i\) from add M for \(0\leq i\leq s\). Let \(B=End M\) and suppose that gl.dim B is finite. Then \(D^ b(A)\) and \(D^ b(B)\) are triangle-equivalent. In particular, this holds if \((A,_ AM_ B,B)\) is a tilting triple; (2) If \(D^ b(A)\) and \(D^ b(B)\) are triangle-equivalent then A has finite global dimension iff so has B; (3) If gl.dim A is finite and \(D^ b(A)\) and \(D^ b(B)\) are triangle- equivalent then the Grothendieck groups \(K_ 0(A)\) and \(K_ 0(B)\) are isometric; (4) \(D^ b(A)\) has Auslander-Reiten triangles; (5) For hereditary algebras H, the quiver of \(D^ b(H)\) is described; (6) For a Dynkin quiver \({\vec \Delta}\), \(D^ b(k{\vec \Delta})\) is triangle- equivalent to \(D^ b(A)\) iff A is simply connected and the groups \(K_ 0(k{\vec \Delta})\) and \(K_ 0(A)\) are isometric iff A is an iterated tilted algebra of type \({\vec \Delta}\); (7) If A has finite global dimension, then \(D^ b(A)\) is triangle-equivalent to the stable module category mod \(\hat A\) of the repetitive algebra \(\hat A\) of A. Recently it was shown by D. Happel, J. Rickard and H. Schofield [a preprint “Piecewise hereditary algebras” is available] that, for an arbitrary quiver \({\vec \Delta}\) without oriented cycles, \(D^ b(A)\) is triangle-equivalent to \(D^ b(k{\vec \Delta})\) iff A is an iterated tilted algebra of type \({\vec \Delta}\). The equivalence (7) allows to apply the methods of the representation theory of algebras to the study of \(D^ b(A)\). Using this equivalence, I. Assem and the reviewer described recently all derived categories \(D^ b(A)\) for which every cycle of indecomposable complexes lies entirely in a tube [Algebras with cycle-finite derived categories, Math. Ann. (to appear; Zbl 0617.16017)].

MSC:

16Gxx Representation theory of associative rings and algebras
16D90 Module categories in associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
16E20 Grothendieck groups, \(K\)-theory, etc.
16E10 Homological dimension in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
16D50 Injective modules, self-injective associative rings
16Exx Homological methods in associative algebras

Citations:

Zbl 0617.16017
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