Constructing tilting modules

Abstract

Abstract. 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 T i 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) Ext 1 R (T, T (I) ) = 0 for any set I, and (T3) there is a short exact sequence 0 → R → T 0 → T 1 → 0 where T 0 and T 1 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,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 toThough 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 } ⊥ .

Tilting Theory: A Gift of Representation Theory to Mathematics

“…Any finitely-generated large partial tilting module of projective dimension at most 1 is a tilting module [6, Theorem 1]. Now Property (Ã) and [13,Lemma 11.2] imply that any large partial tilting module over a semiperfect ring, say again T, is sincere [24], that is, it has the property that HomðP; T Þ 6 ¼ 0 for every projective module P 6 ¼ 0. Consequently, a module T of finite length is sincere if every simple module is a composition factor of T [3,32].…”

Infinite dimensional tilting theory

“…For the case of a hereditary artin algebra, see also [62]. We are going to phrase this problem in terms of tilting modules.…”

On partial tilting modules and right bounded complexes

“…We also refer to [19, Section II] for the importance of "subcategories of module categories" in Tilting Theory. In some sense, the big role played by bounded complexes in the sequel describes an "implicit finiteness property" [15,Introduction] of tilting objects. For the relationship between modules "of finite type" and tilting modules, see, for instance, [3,6,7,22] and the other papers quoted in [23].…”