First Isomorphism Theorem for quivers

Let $Q=(Q_0,Q_1,s,t)$ and $Q^{\prime }=(Q_0^{\prime },Q_1^{\prime },s^{\prime },t^{\prime })$ be quivers. Assume, that $F:Q\to Q^{\prime }$ is a morphism of quivers. Define an equivalence relation $\sim$ on $Q$ as follows: for any $a,b\in Q_0$ and any $\alpha ,\beta \in Q_1$ we have

$$a{\sim }_{0}b\text{ if and only if }{F}_0(a)=F_0(b);$$

$$\alpha {\sim }_1\beta \text{if and only if }{F}_1(\alpha )=F_1(\beta ).$$

It can be easily checked that $\sim =({\sim }_0,{\sim }_1)$ is an equivalence relation on $Q$.

Using standard techniques we can prove the following:

First Isomorphism Theorem for quivers. The mapping

$$\overline{F}:(Q/\sim )\to \mathrm{Im}(F)$$

(where on the left side we have the quotient quiver (http://planetmath.org/QuotientQuiver) and on the right side the image of a quiver (http://planetmath.org/SubquiverAndImageOfAQuiver)) given by

$${\overline{F}}_0([a])=F_0(a),{\overline{F}}_1([\alpha ])=F_1(\alpha )$$

is an isomorphism of quivers.

Proof. It easily follows from the definition of $\sim$ that $\overline{F}$ is a well-defined morphism of quivers. Thus it is enough to show, that $\overline{F}$ is both ,,onto” and ,,1-1” (in the sense that corresponding components of $\overline{F}$ are).

1. We will show, that $\overline{F}$ is onto, i.e. both ${\overline{F}}_0,{\overline{F}}_1$ are onto. Let $b\in \mathrm{Im}{(F)}_0$ and $\beta \in \mathrm{Im}{(F)}_1$. By definition

   $$F_0(a)=b,F_1(\alpha )=\beta$$

   for some $a\in Q_0$, $\alpha \in Q_1$. It follows that

   $${\overline{F}}_0([a])=b,{\overline{F}}_1([\alpha ])=\beta .$$

   which completes this part.

2. $\overline{F}$ is injective. Indeed, if

   $${\overline{F}}_0([a])={\overline{F}}_0([b])$$

   then $F_0(a)=F_0(b)$. But then $a{\sim }_{0}b$ and thus $[a]=[b]$. Analogously we prove the statement for ${\overline{F}}_1$.

This completes the proof. $\mathrm{\square }$

Title	First Isomorphism Theorem for quivers
Canonical name	FirstIsomorphismTheoremForQuivers
Date of creation	2013-03-22 19:17:25
Last modified on	2013-03-22 19:17:25
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	5
Author	joking (16130)
Entry type	Definition
Classification	msc 14L24