Gorenstein flat representations of left rooted quivers

2020/06/30 by Zhenxing Di, Di, Zhenxing, Sergio Estrada +5
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2006.16468

openalex publication_date 2020/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Gorenstein flat objects in the category \sf Rep(Q,R) of representations of a left rooted quiver Q with values in \sf Mod(R), the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in \sf Rep(Q,R) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φ_i^X: ⊕_{a:j→ i}X(j)→ X(i) is injective, and the left R-modules X(i) and \rm Cokerφ_i^X are Gorenstein flat. As an application of this result, we show that there is a hereditary abelian model structure on \sf Rep(Q,R) whose cofibrant objects are precisely the Gorenstein flat representations, fibrant objects are precisely the cotorsion representations, and trivial objects are precisely the representations with values in the right orthogonal category of all projectively coresolved Gorenstein flat left R-modules.

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