Krull-Schmidt Theorem for Quivers

The Krull-Schmidt theorem guarantees unique decomposition of quiver representations into indecomposables, providing a fundamental structure theorem.

TheoremKrull-Schmidt Theorem

Let $Q$ be a quiver and $\mathbb{F}$ a field. Every finite-dimensional representation $V \in \text{Rep}(Q)$ decomposes as a direct sum of indecomposable representations: $V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_k$

This decomposition is unique up to isomorphism and reordering of summands. That is, if: $V \cong V_1 \oplus \cdots \oplus V_k \cong W_1 \oplus \cdots \oplus W_m$ where all $V_i, W_j$ are indecomposable, then $k = m$ and (after reordering) $V_i \cong W_i$ .

Proof strategy: The key is that the endomorphism ring $\text{End}(V_i)$ of an indecomposable representation is a local ring (has a unique maximal ideal). This property, combined with finite-dimensionality, implies unique decomposition via a general theorem in module theory.

DefinitionLocal Ring

A ring $R$ (with unit) is local if it has a unique maximal (left) ideal $\mathfrak{m}$ . Equivalently, the non-units form an ideal.

For indecomposable representations, non-isomorphisms form the unique maximal ideal in $\text{End}(V)$ .

ExampleDecomposition for $A_2$

For $Q = 1 \to 2$ , consider the representation: $V = (\mathbb{F}^2, \mathbb{F}^3, \phi)$ where $\phi = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$ .

This decomposes as: $V \cong I \oplus S_2$ where:

$I = (\mathbb{F}, \mathbb{F}, [1])$ : the indecomposable $\mathbb{F} \xrightarrow{1} \mathbb{F}$
$S_2 = (0, \mathbb{F}, \text{N/A})$ : the simple at vertex 2

This decomposition is unique (up to reordering).

TheoremConsequences

Classification reduces to indecomposables: To understand all representations of $Q$ , it suffices to classify indecomposable representations.

Grothendieck group: The Grothendieck group $K_0(\text{Rep}(Q))$ is the free abelian group generated by isomorphism classes of indecomposables, providing a complete invariant up to direct sum.

Dimension vectors: For finite type quivers, dimension vectors of indecomposables are precisely the positive roots of the associated root system (Gabriel's theorem).

Remark

Krull-Schmidt holds more generally for:

Artinian rings: Rings satisfying descending chain condition on ideals
Noetherian modules: Modules satisfying ascending chain condition
Length-finite categories: Abelian categories where objects have finite composition series

The theorem fails for:

Infinite-dimensional representations: May have non-unique decompositions
Non-algebraically closed fields: Indecomposables over $\mathbb{R}$ may decompose over $\mathbb{C}$
Certain non-commutative rings: Without suitable finiteness conditions

Krull-Schmidt is fundamental for classification problems: it reduces the infinite problem (all representations) to a finite or manageable problem (indecomposables), which for finite type quivers is completely solved by Gabriel's theorem.