Folding derived categories with Frobenius functors |
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Authors: | Bangming Deng |
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Institution: | a Department of Mathematics, Beijing Normal University, Beijing 100875, China b School of Mathematics, University of New South Wales, Sydney 2052, Australia |
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Abstract: | Following the work B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591-3622], we show that a Frobenius morphism F on an algebra A induces naturally a functor F on the (bounded) derived category Db(A) of , and we further prove that the derived category Db(AF) of for the F-fixed point algebra AF is naturally embedded as the triangulated subcategory Db(A)F of F-stable objects in Db(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in Db(AF) and those in Db(A). Thus, the AR-quiver of Db(AF) can be obtained by folding the AR-quiver of Db(A). Finally, we further extend this relation to the root categories ?(AF) of AF and ?(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ?(AF) and ?(A) results in the same relation on the associated root systems as induced from the graph folding relation. |
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Keywords: | 16G20 18E30 16G70 |
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