Category equivalences involving graded modules over path algebras of quivers

Abstract

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and kQ its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(kQ)≔Gr(kQ)/Fdim(kQ) of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(kQ)≡ModS(Q)≡GrL(Q∘)≡ModL(Q∘)0≡QGr(kQ(n)). Here S(Q)=lim⟶EndkI(kQ1⊗n) is a direct limit of finite dimensional semisimple algebras; Q∘ is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from Q; L(Q∘) is the Leavitt path algebra of Q∘; L(Q∘)0 is its degree zero component; and Q(n) is the quiver whose incidence matrix is the nth power of that for Q. It is also shown that all short exact sequences in qgr(kQ), the full subcategory of finitely presented objects in QGr(kQ), split. Consequently qgr(kQ) can be given the structure of a triangulated category with suspension functor the Serre degree twist (−1); this triangulated category is equivalent to the “singularity category” Db(Λ)/Dperf(Λ) where Λ is the radical square zero algebra kQ/kQ≥2, and Db(Λ) is the bounded derived category of finite dimensional left Λ-modules.