On Blow-Ups and Injectivity of Quivers

TL;DR

This paper explores the relationship between blow-ups and injectivity in quivers, establishing conditions under which quivers maintain injectivity through blow-ups and characterizing certain injective quivers.

Contribution

It introduces a novel connection between blow-ups and injectivity in quivers, providing a characterization of quivers injective with respect to specific embeddings.

Findings

Injectivity of quivers is preserved under certain blow-ups.

Characterization of quivers injective with respect to the embedding of $P_{n}$ into $C_{n}$.

Establishment of a criterion linking blow-ups and injectivity for a class of monic maps.

Abstract

This work connects the idea of a "blow-up" of a quiver with that of injectivity, showing that for a class of monic maps $\Phi$, a quiver is $\Phi$-injective if and only if all blow-ups of it are as well. This relationship is then used to characterize all quivers that are injective with respect to the natural embedding of $P_{n}$ into $C_{n}$.