COLLOQUIUM MATHEMATICUM VOL. 115 2009 NO. 2 LIMITS OF TILTING MODULES BY CLEZIO A. BRAGA (Cascavel) and FL ´ AVIO U. COELHO (S˜ ao Paulo) Abstract. We study the problem of when a direct limit of tilting modules is still a tilting module. Tilting theory first appeared in the context of finitely generated modules over artin algebras [12, 18] (see also [5]). Due to its success in this setting, several generalizations were considered. In this work we shall investigate when a direct limit of tilting modules is still a tilting module. The motivation for the construction of such direct limits was inspired by the work of Buan and Solberg [13] who established conditions for an inverse limit of finitely generated cotilting modules to be still a cotilting module. Unfortunately, their proof cannot be dualized to tilting modules. In this paper, we shall use the notion of special preenvelope to prove a similar result for tilting modules (see 1.4 for definitions). Let R be a ring with unity. We say that a (not necessarily finitely gener- ated) R-module T is tilting provided: pd T< ∞; Ext i R (T,T (I ) ) = 0 for each i ≥ 1 and all sets I ; and there exists an exact sequence 0 → R f 0 −→ T 0 f 1 −→··· fr −→ T r → 0 with T i ∈ Add T for each 0 ≤ i ≤ r (see [1, 2]). Our main result is as follows (see Section 1 for further definitions). Theorem. Let R be a ring and {T i } i∈N be a sequence of tilting modules such that Add T i = Add T j if i = j , T i+1 ∈ (T i ) ⊥ and pd T i ≤ n. Then there exists another sequence { T i } i∈N of tilting modules with Add T i = Add T i , T i+1 ∈ ( T i ) ⊥ and pd T i ≤ n for some n ≥ 1. This latter sequence is a direct system of monomorphisms such that T = lim −→ i∈N T i is a tilting module in Mod R and pd T ≤ n +1. In [11], we apply this result to the class of tilted algebras to construct infinitely generated tilting modules. The paper is organized as follows. After some preliminaries in Sections 1 and 2, we prove the above result in Section 3. 2000 Mathematics Subject Classification : 16D90, 16E30, 16G10. Key words and phrases : tilting modules, direct limits. DOI: 10.4064/cm115-2-6 [207] c  Instytut Matematyczny PAN, 2009