|
    Emmanuel Letellier\
    [ *Université Paris Diderot, IMJ-PRG, CNRS*]{}\
    [ letellier@math.univ-paris-diderot.fr]{}
- |
    Fernando Rodriguez Villegas\
    [*ICTP Trieste*]{}\
    [villegas@ictp.it]{}
title: Locally free representations of quivers over commutative Frobenius algebras
---

Introduction {#intro}
============

Let $\R$ be a commutative Frobenius $\k$-algebra over a field $\k$ (which will be either finite or algebraically closed). This means that $\R$ is finite dimensional as a $\k$-vector space and equipped with a Frobenius $1$-form $\lambda:\R\to \k$, i.e. one which is non-zero on any nontrivial ideal of $\R$. Examples to keep in mind are the truncated polynomial rings $\k_d:=\k[t]/(t^d)$, in particular $\k_2=\k[\epsilon]/(\epsilon^2)$ which we usually abbreviate to $\k[\epsilon]$ the algebra of dual numbers. More generally if $\R$ is a Frobenius $\k$-algebra so is $\R[\epsilon]:=\R[\epsilon]/(\epsilon^2)$ (see §\[Frobsection\] for more details).

We fix $Q=(I,E)$ a finite quiver, with set of vertices $I$ and set of arrows $E$. A representation of $Q$ over the ring $\R$ is given by an $\R$-module $M_i$ at each vertex $i\in I$ together with a $\R$-module homomorphism $M_a:M_i\rightarrow M_j$ for any arrow $a:i\rightarrow j\in E$. The category of representations of $Q$ over $\R$ is equivalent to the category $\R Q$-Mod where $\R Q$ denotes the path algebra of $Q$. According to [@GLS Definition 1.1], 