Abstract
According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.
Highlights
Supported by Università di Padova (Progetto di ricerca di Ateneo CPDA105885/10 “Differential graded categories” and Progetto ex 60% “Anelli e categorie di moduli”)
According to the “Classical Krull–Schmidt Theorem”, if the module MR∈Ob(C) I (MR) is of finite composition length, MR is a direct sum of indecomposable modules in an essentially unique way
U1,1 ⊕ U2,2 ⊕ · · · ⊕ Un,n ∼= Uσ (1),τ (1) ⊕ Uσ (2),τ (2) ⊕ · · · ⊕ Uσ (n),τ (n) for every pair of permutations σ, τ of {1, 2, . . . , n}; that is, the module U1,1 ⊕ U2,2 ⊕ · · · ⊕ Un,n is a direct sum of n uniserial modules with n! essentially different direct-sum decompositions into non-zero uniserial submodules, which is the maximum allowed by Theorem 5.2. (By Theorem 15.2, this is equivalent to saying that the module has n! essentially different direct-sum decompositions into indecomposable submodules, because every indecomposable direct summand of a finite direct sum of uniserial modules is a uniserial module.)
Summary
A first generalization to finite non-commutative groups is due to the Scottish mathematician Maclagan-Wedderburn [66] Wedderburn’s result can be stated by saying that if a finite group G has two directproduct decompositions G = G1 × G2 × · · · × Gt = H1 × H2 × · · · × Hs, t = s and there is an automorphism φ of G such that φ(Gi ) = Hσ (i) for all i’s. The present form of the Krull–Schmidt Theorem in Group Theory is the following. Theorem 1.1 If a group G satisfies both the ascending chain condition and the descending chain condition on normal subgroups, G is a direct product G1 × G2 × · · · × Gt of finitely many indecomposable groups in an essentially unique way in the following sense.
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