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TILTING up ITERATED TILTED ALGEBRAS 1. Introduction Throughout This Paper, We Let K Denote a Fixed Algebraically Closed Field. B

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TILTING up ITERATED TILTED ALGEBRAS 1. Introduction Throughout This Paper, We Let K Denote a Fixed Algebraically Closed Field. B PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 8, Pages 2223{2232 S 0002-9939(99)05230-2 Article electronically published on November 29, 1999 TILTING UP ITERATED TILTED ALGEBRAS IBRAHIM ASSEM, DIETER HAPPEL, AND SONIA TREPODE (Communicated by Ken Goodearl) Abstract. We show that, if A is a representation-finite iterated tilted algebra of euclidean type Q, then there exist a sequence of algebras A = A0;A1;A2;:::; (i) (i) Am, and a sequence of modules T ,where0≤ i<m,suchthateachT is Ai Ai (i) an APR-tilting Ai-module, or an APR-cotilting Ai-module, End T = Ai+1 Ai and Am is tilted representation-finite. 1. Introduction Throughout this paper, we let k denote a fixed algebraically closed field. By algebra is always meant a finite dimensional associative k-algebra with an identity, which we assume moreover to be basic and connected, and by module is meant a finitely generated right A-module. Tilting theory is by now an established tool in the representation theory of algebras. In particular, the classes of tilted and iterated tilted algebras, introduced by means of the tilting process, were very useful, for instance, in the classification of the self-injective algebras of polynomial growth (see [1], [7], [11]). We recall their definition. Let Q be a finite, connected quiver without oriented cycles and let kQ denote the path algebra of Q.AnalgebraA is said to be iterated tilted of type Q if there exists a going-up tilting series from A to kQ, that is, a sequence of algebras A = A0;A1;:::;Am = kQ and a sequence of tilting modules T (i),where0≤ i<m, such that End T (i) = A and each T (i) Ai Ai i+1 Ai is separating, that is, such that each indecomposable Ai-module M satisfies either (i) 1 (i) HomAi (T ;M)=0orExtAi (T ;M) = 0 (see [2]). This implies that, if Ai is representation-finite, then so is Aj for each 0 ≤ j ≤ i.Ifm ≤ 1, then A is said to be a tilted algebra of type Q. In order to study the representation-finite iterated tilted algebras of euclidean type,O.Rold´an (private communication) has formulated the following conjecture which would allow to reduce their study to that of the representation-finite tilted algebras of the same type. Let Q be a euclidean quiver, and A be a representation- finite iterated tilted algebra of type Q. The conjecture states that there exists a sequence of tilts A = A0;A1;:::;Am−1, Am = kQ as in the definition above, but with Am−1 representation-finite [1] (6.5). We show that the conjecture, in this form, is not true (see example (3.4) (a)), so that separating tilting modules do not suffice to reduce the study of iterated tilted algebras of euclidean type Received by the editors December 15, 1997 and, in revised form, September 10, 1998. 2000 Mathematics Subject Classification. Primary 16G60, 16G20. Key words and phrases. Representation-finite iterated tilted algebras of euclidean type, APR- tilting and cotilting modules, derived category. c 2000 American Mathematical Society 2223 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2224 IBRAHIM ASSEM, DIETER HAPPEL, AND SONIA TREPODE to that of representation-finite tilted algebras of the same type. The objective of this paper is to show that the use of both separating tilting and separating cotilting modules (even, of a particularly nice type) will achieve that goal. Namely, replacing the separating tilting modules by APR-tilting or APR-cotilting modules (for the definitions, see (2.1)), one obtains the following statement. Theorem. Let Q be a euclidean quiver without oriented cycles, and A be a repre- sentation-finite iterated tilted algebra of type Q. Then there exist a sequence of algebras A = A , A ;:::;A , and a sequence of modules T (i),where0 ≤ i<m, 0 1 m Ai such that each T (i) is an APR-tilting A -module, or an APR-cotilting A -module, Ai i i End T (i) = A and A is tilted representation-finite. Ai i+1 m Further, we show that this theorem is not true if one allows Q to be a wild quiver instead of a euclidean quiver (see Example (3.4) (b)). The case where the underlying graph of Q is the euclidean diagram A~ n was considered by the third author in [12], [13], using the classification of the tilted and iterated tilted algebras of type A~ n (see [3], [10]). The purpose of the present paper is to provide a proof for the general case that does not use any classification result. The paper is organised as follows. In section 2, after setting the notation and briefly recalling some definitions and results that will be needed in the sequel, we prove a necessary and sufficient condition for an iterated tilted algebra of euclidean type to be representation-finite, then a lemma on almost complete tilting modules over tame hereditary algebras. Section 3 consists of two lemmata followed by the proof of our theorem and ends with the aforementioned examples. 2. Preliminaries 2.1. Notation. Let A be a finite dimensional basic and connected k-algebra. We denote by QA the ordinary quiver of A,andby(QA)0 the set of points of QA.For apointa 2 (QA)0,wedenotebyS(a) the corresponding simple A-module, and by P (a) the projective cover of S(a). We use freely, and without further reference, facts about the module category mod A and the Auslander-Reiten translations τA =DTr −1 and τA =TrD,asin[5],[9]. An A-module T is called a tilting (or cotilting) module if pd TA ≤ 1(or ≤ 1 id TA 1, respectively), ExtA(T;T) = 0 and the number of isomorphism classes of indecomposable summands of T equals the number of points in QA. Given a tilting module TA, there exists a close connection between the representation theories of A and of End TA : this connection is known as tilting theory, and we refer the reader to [1], [7] for the details. A tilting module TA is called separating if, for each 1 indecomposable A-module M,wehaveeitherHomA(T;M)=0orExtA(T;M)=0. If TA is a separating tilting module, then the almost split sequences in mod A are totally determined by those in mod End TA. An example of separating tilting module is provided by the so-called APR-tilting modules:letP (a)A beL a simple −1 ⊕ projective module which is not injective; then the module τA P (a) ( P (b)) b=6 a is the corresponding APR-tilting module. Separating cotilting modules and APR-cotilting modules are defined dually. 2.2. The derived category. Let A be an algebra. We denote by Db(mod A)the derived category of bounded complexes of A-modules. The study of Db(mod A)is License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TILTING UP ITERATED TILTED ALGEBRAS 2225 Figure 1. especially fruitful if A is an iterated tilted algebra, as is seen from the following theorem [7] (IV.5.4), p. 176. Theorem. Let Q be a finite connected quiver without oriented cycles. An algebra A is iterated tilted of type Q if and only if there exists an equivalence of triangulated ∼ categories Db(mod A) = Db(mod kQ). Further, the structure of Db(mod kQ) is known [7] (I.5.5) p. 54. If, in partic- ular, Q is a euclidean quiver, then the Auslander-Reiten quiver of kQ consists of acomponentP containing the projective kQ-modules, a component Q containing R R the injective kQ-modules, and a stable tubular family =( λ)λ2P1(k) separat- ing P from Q (see [6], [9]). The quiver of Db(mod kQ) may then be visualised as shown in Figure 1, where each P[i](orQ[i]; R[i]) is a copy of P (or Q; R, respec- tively) indexed by i 2 Z. The components C[i] are referred to as the transjective components,andtheR[i]astheregular components. Let A be an iterated tilted algebra of type Q. The image of the module AA under the composition of the embedding of mod A inside Db(mod A) with the triangle- b ∼ b • equivalence F : D (mod A) = D (mod kQ) is a tilting complex T = F (AA)(in 2 the sense of [8]) which we call the standardL complex. For a point a (QA)0,we write T •(a)=FP(a). Thus, T • = T •(a). a2(QA)0 2.3. Proposition. Let A be an iterated tilted algebra of euclidean type Q,andlet F : Db(mod A) ! Db(mod kQ) be a triangle-equivalence. Then A is representation- finite if and only if there exist two points a =6 b in QA, and two integers ia =6 ib • • such that T (a) 2C[ia] and T (b) 2C[ib]. Proof. Since the sufficiency is just part (a) of [7] (IV.7.1), p. 188, we only show the necessity. We assume that the stated condition does not hold, and prove that this implies that A is representation-infinite. • We first notice that there exists at least one point a 2 (QA)0 such that T (a) lies in a transjective component : indeed, if this is not the case, then the indecomposable summands T •(x) of the standard complex T • all belong to the regular components, which would imply that the vectors dimT •(x) are linearly dependent, a contradic- b tion to the fact that they form a basis in K0(D (mod kQ)), by [7] (III.1.2), p.
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