2024/02/19 by Valentine Soto, Soto, Valentine
Mathematics · #16E35 #16G #18G65 #18G80 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2402.12312
openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The bounded derived category of a finite dimensional algebra of finite global dimension is equivalent the stable category of ℤ-graded modules over its trivial extension \citeHappel. In particular, given two derived equivalent finite dimensional algebras Λ1 and Λ2 of finite global dimension, their trivial extensions are stable equivalent. The converse is not true in general. The goal of this paper is to study cases where derived equivalences between Λ1 and Λ2 arise from an equivalence of categories involving their trivial extension. Thanks to a graded version of Happel's theorem, we show that one can construct a ℤ-grading on the Λi so that such an equivalence involving their trivial extension yield a derived equivalence between the category of ℤ-graded module over Λi. We describe explicitly the tilting object associated to this derived equivalence (in the non-graded and in the graded case) for triangular matrix algebras. Finally, we apply these results to the particular case of gentle algebras. In this context, we study how one can obtain a derived equivalence between Λ1 and Λ2 (in the non-graded and the graded case) from graded generalized Kauer moves.