On triangulated orbit categories.

*(English)*Zbl 1086.18006Let \(T\) be an additive category and \(F:T\rightarrow T\) an automorphism. The orbit category \(T/F\) has the same objects as \(T\) and \(\text{Hom}_{T/F}(X,Y):=\oplus_{n\in\mathbb Z} \text{Hom}_T(X,F^nY)\). There is a canonical functor \(\pi:T\rightarrow T/F\) being the identity on objects and sending a homomorphism to the same homomorphism, putting the components of \(\text{Hom}_T(X,F^nY)\) to \(0\) whenever \(n\neq 0\). In the paper under review the author gives a criterion when given a triangulated structure on \(T\) there is a triangulated structure on \(T/F\) so that \(\pi\) is a functor of triangulated categories. In general there is no reason why such a structure might exist. The author gives quite striking examples.

The main result is the following. If \(T\) is equivalent as triangulated category to the derived category of bounded complexes of finitely generated modules over a finite-dimensional \(k\)-algebra \(A\), if under this equivalence \(F\) is given by the left derived tensor product with a bounded complex of \(A-A\)-bimodules, and suppose that the following three conditions are verified:

1) There is a hereditary abelian \(k\)-category \(H\) so that \(T\) is equivalent to the derived category of bounded complexes of finitely generated objects of \(H\). Call this equivalence \(G\).

2) For each indecomposable object \(U\) of \(T\) there are only finitely many objects of the form \(F^n(U)\) lying in \(G(H)\), where as usual \(H\) is the full subcategory of the derived category consisting of complexes with non zero-homogeneous component only in degree \(0\).

3) There is an integer \(N\geq 0\) such that the \(F\)-orbit of each indecomposable of \(T\) contains an object \(S^nU\) for some \(n\) with \(0\leq n\leq N\). Here \(S\) is the suspension functor of the triangulated category structure.

Then \(T/F\) admits a natural triangulated structure so that \(\pi\) is a functor of triangulated categories.

The proof of this result is done in the context of differential graded categories. The author constructs under the assumptions 1), 2), 3) a triangulated hull to the orbit category by an abstract argument. Then, by a more down-to earth construction the author uses the other hypotheses to construct an inverse to the triangulated hull functor.

The motivation to the paper comes from an application to the cluster algebras approach due to Buan, Marsh, Reineke, Reiten and Todorov for which this statement is crucial. Finally an application to Calabi-Yau categories is given.

The main result is the following. If \(T\) is equivalent as triangulated category to the derived category of bounded complexes of finitely generated modules over a finite-dimensional \(k\)-algebra \(A\), if under this equivalence \(F\) is given by the left derived tensor product with a bounded complex of \(A-A\)-bimodules, and suppose that the following three conditions are verified:

1) There is a hereditary abelian \(k\)-category \(H\) so that \(T\) is equivalent to the derived category of bounded complexes of finitely generated objects of \(H\). Call this equivalence \(G\).

2) For each indecomposable object \(U\) of \(T\) there are only finitely many objects of the form \(F^n(U)\) lying in \(G(H)\), where as usual \(H\) is the full subcategory of the derived category consisting of complexes with non zero-homogeneous component only in degree \(0\).

3) There is an integer \(N\geq 0\) such that the \(F\)-orbit of each indecomposable of \(T\) contains an object \(S^nU\) for some \(n\) with \(0\leq n\leq N\). Here \(S\) is the suspension functor of the triangulated category structure.

Then \(T/F\) admits a natural triangulated structure so that \(\pi\) is a functor of triangulated categories.

The proof of this result is done in the context of differential graded categories. The author constructs under the assumptions 1), 2), 3) a triangulated hull to the orbit category by an abstract argument. Then, by a more down-to earth construction the author uses the other hypotheses to construct an inverse to the triangulated hull functor.

The motivation to the paper comes from an application to the cluster algebras approach due to Buan, Marsh, Reineke, Reiten and Todorov for which this statement is crucial. Finally an application to Calabi-Yau categories is given.

Reviewer: Alexander Zimmermann (Amiens)

##### MSC:

18E30 | Derived categories, triangulated categories (MSC2010) |

16G20 | Representations of quivers and partially ordered sets |