Gentle algebras are Gorenstein

The aim of this note is to show that that gentle algebras are Gorenstein. These are both interesting classes of (finite dimensional) algebras. The first class was introduced in [2] as appropriate context for the investigation of algebras derived equivalent to hereditary algebras of type Ãn. The gentle algebras which are trees are precisely the algebras derived equivalent to hereditary algebras of type An, see [1]. It is also known from [12] that the algebras with derived discrete category which are not piecewise hereditary are gentle algebras with one cycle violating the clock condition, see [4] for their derived classification. It is interesting to notice that the class of gentle algebras is closed under derived equivalence [11]. See also [10] for further interesting properties. On the other hand the concept of a Gorenstein algebra Λ, where by definition Λ has finite injective dimension both as a left and a right Λ-module, is inspired from commutative ring theory. This class of algebras contains both the selfinjective algebras and the algebras of finite global dimension, and were investigated in [3],[7]. In the bounded derived category D(Λ) consider the subcategories K (P) of bounded complexes of projectives and K (I) of bounded complexes of injectives. Then Λ is Gorenstein exactly when the last two coincide, and it is easy to see that the property of being Gorenstein is preserved under derived equivalence, see [7]. Moreover the AR-translation τ : K (P) −→ K(I) shows that we have AR-triangles in K (P) in this case, see [6, 1.4]. The property of beeing Gorenstein is also preserved under the skew group ring construction with a finite group whose order is invertible in Λ, see [9],[3]. Thus we may conclude that also the skewed-gentle algebras considered in [5] are Gorenstein, at least if the field is not of characteristic 2. Skewed gentle algebras form a class of derived tame algebras which is not closed under derived equivalence.