Category equivalences involving graded modules over path algebras of quivers

Details Category equivalences involving graded modules over path algebras of quivers Category equivalences involving graded modules over path algebras of quivers

2011-07-18

arxiv.org/abs/1107.3511v3

Mathematics

Rings and Algebras

Several changes made as a result of the referee's report. Added Lemma 3.5 and Prop. 3.6 showing that O is a generator

Scientific paper

Let kQ be the path algebra of a quiver Q with its standard grading. We show that the category of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, QGr(kQ), is equivalent to several other categories: the graded modules over a suitable Leavitt path algebra, the modules over a certain direct limit of finite dimensional multi-matrix algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th} power of that for Q, and others. A relation with a suitable Cuntz-Krieger algebra is established. All short exact sequences in the full subcategory of finitely presented objects in QGr(kQ), split so that subcategory can be given the structure of a triangulated category with suspension functor the Serre degree twist (-1); it is shown that this triangulated category is equivalent to the "singularity category" for the radical square zero algebra kQ/kQ_{\ge 2}.

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