A module endomorphism is a module homomorphism from a module to itself. These maps form the endomorphism ring under pointwise addition and composition. An invertible module endomorphism is a module automorphism.
A module automorphism is an invertible module endomorphism. Equivalently, it is a bijective module homomorphism from a module to itself; its inverse is automatically a module homomorphism. They form a group under composition.
A module retraction onto a submodule is a module homomorphism with . It gives : write . Thus a nonzero proper submodule admitting a retraction contradicts indecomposability. Extending a projection from a larger submodule can produce such a retraction through the long exact sequence of Ext groups.
The endomorphism ring consists of the -module homomorphisms , with pointwise addition and composition as multiplication.
An endomorphism ring is local when its nonunits form a proper two-sided ideal; commutativity is not required. The Fitting lemma gives this property for an indecomposable finite-length module. If two nonunits had an invertible sum, multiplication by its inverse would give with noninvertible . Fitting makes nilpotent, so would be invertible, a contradiction. Products with a nonunit are nonunits by finite-length injectivity/surjectivity. The quotient by the nonunit ideal is a division ring.
For a finite-dimensional module over an algebraically closed field, write a Krull-Schmidt decomposition with distinct indecomposable types. Each endomorphism ring of is a local endomorphism ring with residue division algebra , by the finite-dimensional division algebra over an algebraically closed field result. Modulo the Jacobson radical of , the blocks of one type become and all maps between different types vanish. A composite through a different indecomposable type cannot be invertible, since that would make one type a direct summand of the other. This yields the displayed product and the Levi decomposition of a quiver automorphism group.
A nonzero module is a brick when every endomorphism is either zero or an isomorphism. Equivalently, its endomorphism ring is a division ring.
An indecomposable finite-dimensional quiver representation that is not a brick module contains a brick with nonzero self-extensions. The Ringel form then gives , impossible for a positive definite Tits form of a quiver.
The Fitting lemma supplies a nonzero nilpotent endomorphism of a non-brick indecomposable . Minimize its nonzero rank; nilpotence and minimality give . Set . Choose a nonzero component . The square-zero endomorphism has rank at least , so is injective.
If , the projection extends to by the long exact sequence of Ext groups, giving a module retraction and contradicting indecomposability. Thus this extension group is nonzero. Since path algebras are hereditary, induces a surjection . Hence is a proper indecomposable submodule with self-extensions. Iterate until a brick module is reached; dimensions strictly decrease.
This minimal-rank argument is given in section 2 of William Crawley-Boevey's quiver lectures.
