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**1 - 6**of**6**### THE ALMOST SPLIT TRIANGLES FOR PERFECT COMPLEXES OVER GENTLE ALGEBRAS

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"... Throughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N+, we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i, j ∈ Z, [i, j]: = {l ∈ Z | i ≤ l ≤ j} (in particular, ..."

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Throughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N+, we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i, j ∈ Z, [i, j]: = {l ∈ Z | i ≤ l ≤ j} (in particular, [i, j] = ∅ if i> j).

### ON THE EXISTENCE OF A DERIVED EQUIVALENCE BETWEEN A KOSZUL ALGEBRA AND ITS YONEDA ALGEBRA

"... Abstract. In this paper we focus on the relations between the derived cat-egories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to t ..."

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Abstract. In this paper we focus on the relations between the derived cat-egories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We consider derived discrete Koszul algebras, and we give necessary and sufficient conditions for these Koszul algebras to be derived equivalent to their Yoneda algebras. Finally, we look at the Koszul algebras such that they are derived equivalent to a hereditary algebra. In the case that the hereditary algebra is tame, we characterize when these algebras are derived equivalent to their Yoneda algebras We dedicate this work to the memory of Dieter Happel. In our context, algebras will always be finite dimensional, and of the form A = kQ/I, where Q is a quiver and I is an homogeneous ideal with generators in degrees bigger equal to two and k is an algebraically closed field. Most of the cases, the homogeneous ideal I is generated in degree two. The ideal of kQ generated by the