First Isomorphism Theorem for quivers
It can be easily checked that is an equivalence relation on .
Using standard techniques we can prove the following:
(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
is an isomorphism of quivers.
Proof. It easily follows from the definition of that is a well-defined morphism of quivers. Thus it is enough to show, that is both ,,onto” and ,,1-1” (in the sense that corresponding components of are).
We will show, that is onto, i.e. both are onto. Let and . By definition
for some , . It follows that
which completes this part.
is injective. Indeed, if
then . But then and thus . Analogously we prove the statement for .
This completes the proof.
|Title||First Isomorphism Theorem for quivers|
|Date of creation||2013-03-22 19:17:25|
|Last modified on||2013-03-22 19:17:25|
|Last modified by||joking (16130)|