Constructing Tilting Modules

We investigate the structure of (infinite dimensional) tilting modules over hereditary artin algebras. For connected algebras of infinite representation type with Grothendieck group of rank n, we prove that for each 0 ≤ i < n− 1, there is an infinite dimensional tilting module Ti with exactly i pairwise non-isomorphic indecomposable finite dimensional direct summands. We also show that any stone is a direct summand in a tilting module. In the final section, we give explicit constructions of infinite dimensional tilting modules over iterated one-point extensions. The study of finite dimensional tilting modules of projective dimension at most one over finite dimensional algebras was initiated by Brenner and Butler [11] and continued by Happel and Ringel [17]. Since then, many variations of this concept have been introduced and used successfully, for example: Tilting modules of higher projective dimension, tilting modules over rings, tilting complexes in derived categories, tilting objects in hereditary categories or in cluster categories. In this paper, we use the term tilting module as follows: Let R be a ring and T be a right R-module. Then T is a tilting module provided that (T1) p.dimT ≤ 1, (T2) ExtR(T, T ) = 0 for any set I, and (T3) there is a short exact sequence 0 → R → T0 → T1 → 0 where T0 and T1 are direct summands in a direct sum of (possibly infinitely many) copies of T . Equivalently, T is tilting if and only if Gen(T ) = {T}⊥, [13]. Here, Gen(T ) denotes the class of all homomorphic images of direct sums of copies of T , and, for a class of modules C, C⊥ = KerExtR(C,−) = {M ∈ Mod-R | ExtR(C,M) = 0 for all C ∈ C}. If T is a tilting module, then {T}⊥ is a torsion class in Mod-R, the tilting class generated by T . If T ′ is another tilting module, then T is said to be equivalent to T ′ if {T}⊥ = {T ′}⊥. Though our definition of a tilting module allows infinitely generated modules, there is an implicit finiteness property connected with tilting, recently proved by Bazzoni and Herbera in [9, Theorem 2.4]. Namely, any tilting module T is of finite type, that is, there exists a set S consisting of finitely presented modules of projective dimension at most 1 such that S⊥ = {T}⊥. Received by the editors November 29, 2005. 2000 Mathematics Subject Classification. Primary 16E30, 16G10, 16G30; Secondary 16D50, 18E40. This research was done during visits of the first author to Charles University, Prague, and of the second author to Heinrich Heine University, Düsseldorf, within the bilateral university exchange program. The second author was supported by grants GAUK 448/2004/B-MAT and MSM 0021620839. c ©2007 American Mathematical Society Reverts to public domain 28 years from publication