τ-Complexity and Tilting Modules

Background. Tilting theory plays an important role in the modern representation theory of algebras. Let A be a finite dimensional algebra over a field k and T a tilting A-module. It is well known that A and End A (T) are derived equivalent. The endomorphism algebra of a tilting module preserves many significant invariants, for example, the center of an algebra, the number of nonisomorphic simple modules, the Hochschild cohomology groups, and Cartan determinants. In particular, if T is a separating and splitting tilting Amodule (see the definition in Section 2), then End A (T)


Introduction
Background.Tilting theory plays an important role in the modern representation theory of algebras.Let  be a finite dimensional algebra over a field  and  a tilting -module.It is well known that  and End  () are derived equivalent.The endomorphism algebra of a tilting module preserves many significant invariants, for example, the center of an algebra, the number of nonisomorphic simple modules, the Hochschild cohomology groups, and Cartan determinants.In particular, if  is a separating and splitting tilting module (see the definition in Section 2), then End  () preserves representation dimension [1].
On the other hand, -complexity (see the definition in Section 2) is an important invariant in the representation theory of algebras.With -complexity, Bergh and Oppermann described the classification of hereditary algebras and studied the classification of cluster tilted algebra [2].
However, the precise value of -complexity of a given algebra is not known in general, and it is hard to compute even for small examples.One possible way is to compare complexities of "nicely" related algebras.
Question.Suppose  is the endomorphism algebra of a tilting module  over an algebra .What is the relationship between -complexities of  and ?
Note that in general  and  do not have the same complexities, since there are examples where  is representation finite while  is representation infinite.Our main result in this paper is the following theorem.

Theorem 1.
Let  be a tilting module over a finite dimensional -algebra , with  =   ().If  is separating and splitting, then    =    , where    ,    denote -complexity of  and , respectively.
Organization.This paper is organized as follows.In Section 2, we shall give the proof of our main result Theorem 1.In Section 3, we shall give two examples to illustrate our results.

Proof of the Main Theorem
Throughout this paper,  is an algebraically closed field,  is a finite dimensional -algebra.Denote by mod  the category of finitely generated left -modules, P(mod ) the full subcategory of mod  consisting of all projective objects in mod , and gl ⋅ dim  the global dimension of . fl Hom  (−, ) denotes the standard duality functor between mod  and mod  op .Given a left -module , add  denotes the full subcategory of mod  consisting of all direct summands of finite direct sums of copies of .Torsion Pair.A pair (T, F) of full subcategories of mod  is called a torsion pair, if the following conditions are satisfied: (1) Hom  (, ) = 0 for all  ∈ T,  ∈ F; (2) Hom  (, −)| F = 0 implies  ∈ T; (3) Hom  (−, )| T = 0 implies  ∈ F. A torsion pair (T, F) is called splitting if each indecomposable -module lies either in T or in F.
A module  is called a tilting module if the following three conditions are satisfied: 2 Advances in Mathematical Physics (3) there exists a short exact sequence: It is well known that   induces a torsion pair (T  , F  ) in mod , and a torsion pair (X  , Y  ) in mod . is said to be separating if the induced torsion pair (T  , F  ) in mod  is splitting and said to be splitting if the induced torsion pair (X  , Y  ) in mod  is splitting.
The following lemma is crucial in this paper.
(b) there exists a constant  > 0, such that dim   ≤  dim  .
(2) For any  ∈ F  , (a) there exists a constant Proof.The proof is similar to [3, Lemma 3.
(a) Assume that  is a torsion module.Then Ext 1  ( 2 , ) = 0 in the long exact sequence because  2 is in add  and  is torsion.We obtain the short exact sequence 0 → Hom  ( 2 , ) → Hom  ( 1 , ) → Hom  (, ) → 0. Noting that Hom  (, ) ≅ , we have dim   ≤  dim  Hom(, ) =  dim  .(b) For any finitely generated -module , we have dim   ≤ dim   ⋅ dim   [3, Remark 3.2], we set  = dim  , and then the assertion follows immediately. ( (a) Assume that  is torsion-free.In the long exact sequence ( * ) above, we now have Hom  ( 1 , ) = 0 since  is torsion-free.We thus have the short exact sequence 0 → , and then the assertion follows immediately.
Let  be a tilting -module,  = End  ().Denote by   ,   the Auslander-Reiten translation in mod  and mod , respectively.Let  ∈ mod ; the   -complexity of  is defined as follows: When no such  ∈ N exists, we say that   -complexity of  is infinite and write    () = ∞.And -complexity of the algebra  is defined to be the supreme of   -complexities of all the finitely generated -modules, which will be denoted by    .
Proof of Theorem 1.
Case 1 (0 ̸ =  ∈ X  ).In this case, there exists 0 ̸ =  ∈ F  such that  ≅ ().If  belongs to Case 1.1 in Step 1, then denote  0 by the minimal positive integer such that  Otherwise  belongs to Case 2.2 in Step 1; that is, there exists a positive integer  0 such that  (1)  is of finite representation type if and only if    = 0.
(2)  is of tame representation type if and only if    = 2.
(3)  is of wild representation type if and only if    = ∞.
Combining Theorem 1 and Proposition 3, we have the following corollary.

Corollary 4.
Let  be a finite dimensional hereditary algebra and  a separating and splitting -module, with  = End  ().If  is a hereditary algebra, then  and  are of the same representation type.Remark 5. Let  be a finite dimensional hereditary algebra and  a APR-tilting module.It is well known that  = End() is a hereditary algebra.Therefore, they are of the same representation type.

Two Examples
In this section, we shall give two examples to illustrate our result.