ARS 讨论班
Chapter 3 Quiver
Let $k$ be a field , $Q$ be an acyclic quiver. Then
- $\mathfrak{r}_{kQ}=J$
Use the lemma:
Let $Λ$ be a left Artin ring, $\mathfrak{a}$ a nilpotent ideal of $Λ$, and $\Lambda/ \mathfrak{a}$ is a semisimple ring. Then $\mathfrak{a}= \mathrm{rad} Λ$.
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$P_i=kQe_i$’s are all the indecomposable projective $kQ$-modules.
- $kQ$ is a hereditary algebra.
Use the lemma:
$Λ$ be an Artin algebra. $Λ$ is hereditary ⇔ $\mathfrak{r}$ is projective as a left $Λ$-module.
It is easy to see that $$ \mathfrak{r}=\bigoplus_{i∈ Q_0} \mathfrak{r}e_i=\bigoplus_{i∈ Q_0} \bigoplus_{\substack{α∈ Q_1 \\ s(α)=i}} kQ\alpha. $$
Since $kQα≅kQe_{t(α)}=P_{t(α)}$, $\mathfrak{r}$ is a projective $kQ$-module.
(reptesentation of a quiver)
$\mathbf{Rep}_k(Q)$
Given $k$, $Q$, and $h: (V,f)→(V',f')$. $h$ is monomorphism (epimorphism, isomorphism) ⇔ $h_i$ is monomorphism (epimorphism, isomorphism) for all $i∈Q_0$.
$k$ is a field, $Q$ is a quiver. Then
$$ \mathbf{Rep}_k(Q) ≅ \mathrm{f.d.} (kQ) (≅ kQ\text1{-mod}, \text{if } Q \text{ is acyclic}) $$