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# Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

Published:2014-06-12
Printed: Feb 2015
Department of Mathematics, University of Isfahan, Isfahan, Iran
• Rasool Hafezi,
Department of Mathematics, University of Isfahan, Isfahan, Iran
• Razieh Vahed,
Department of Mathematics, University of Isfahan, Isfahan, Iran
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## Abstract

We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.
 Keywords: derived category, Grothendieck duality, representation of quivers, reflection functor
 MSC Classifications: 18E30 - Derived categories, triangulated categories 16G20 - Representations of quivers and partially ordered sets 18E40 - Torsion theories, radicals [See also 13D30, 16S90] 16D90 - Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 18A40 - Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

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