Contemporary Mathematics Volume 229, 1998 POSTPROJECTIVE PARTITIONS FOR TILTING TORSION PAIRS IBRAHIM ASSEM AND FLAVIO ULHOA COELHO Postprojective partitions for the module category mod A of an artin algebra A (or for an additive subcategory of mod A) were introduced by Auslander and Smal0 in [2], under the name of preprojective partitions. The situation we consider is the following: let TA be a tilting module and B = End TA, we wish to describe these partitions for each of the classes of the torsion pair induced by T in mod B, especially when A is hereditary (thus B is tilted). In doing so, we use essentially the relative postprojective partitions induced by the tilting functors, as defined by Betzler in [4]. We need some notation. Let (A, T, B) be a tilting triple and denote by (T, F) and (X, Y) the torsion pairs induced by TA in mod A and mod B, respectively [1]. We use sometimes the notation (X,-) and 1 (X,-) for the functors HomA(X,-) and Ext1 (X,-), respectively. We letS denote the additive subcategory of mod B generated by T .B 1 Hom A (T, DA) (where D denotes the standard duality between mod A and mod A0 P) and, for an indecomposable module X EX, we let ns(X) be the length of a shortest chain of irreducible morphisms from a module in S to X if such a path exists, and infinity otherwise. We prove the following theorem. THEOREM. Let (A, T, B) be a tilting triple, then: (a) The functors Hom A (T,-) and - ® B T induce quasi-inverse category equiv- alences between P~T,-)(T) and Pn(Y) for each n, and the functors Ext1(T,-) and Torf (-, T) induce quasi-inverse category equivalences between P ~ (T,-) (F) and Pn(X) for each n. (b) IfTA is a splitting tilting module, then Pn(Y) = Pn( mod B) n Y for each n. (c) If A is hereditary, then Pn(X) ={X E Xlns(X) = n} for each n. This paper consists of two sections, the first being devoted to preparatory results, and the second to the proof of our theorem. 1. Relative postprojective partitions Throughout this work, A denotes an artin algebra, and mod A the category of finitely generated right A-modules. We assume that C is an additive subcat- egory of mod A, with an epiclass £ (in the sense of [4]) such that C admits an £-postprojective partition. For an £-postprojective module X, lying in the class P~, we write 1r£(X) = n, and call n the £-level of X. For an A-module M, we denote by add M the additive subcategory of mod A consisting of the direct sums 1991 Mathematics Subject Classification. 16G20, 16G70, 16G99. (*) This work was started when the second author was visiting the first at the University of Sherbrooke, Quebec. The authors gratefully acknowledge partial support from NSERC of Canada and CNPq of Brazil. © 1998 American Mathematical Society 1
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