CHAPTER 1

The Krull-Remak-Schmidt Theorem

In this chapter we will prove the Krull-Remak-Schmidt uniqueness

theorem for direct-sum decompositions of finitely generated modules

over complete local rings. The first such theorem, in the context of

finite groups, was stated by Wedderburn [Wed09]: Let G be a finite

group with two direct-product decompositions G = H1 × · · · × Hm and

G = K1 × · · · × Kn, where each Hi and each Kj is indecomposable.

Then m = n, and, after renumbering, Hi

∼

= Ki for each i. In 1911

Remak [Rem11] gave a complete proof, and actually proved more:

Hi and Ki are centrally isomorphic, that is, there are isomorphisms

fi : Hi −→ Ki such that

x−1f(x)

is in the center of G for each x ∈ Hi,

i = 1,...,m. These results were extended to groups with operators sat-

isfying the ascending and descending chain conditions by Krull [Kru25]

and Schmidt [Sch29]. In 1950 Azumaya [Azu50] proved an analogous

result for possibly infinite direct sums of modules, with the assump-

tion that the endomorphism ring of each factor is local in the non-

commutative sense.

§1. KRS in an additive category

Looking ahead to an application in Chapter 3, we will clutter things

up slightly by working in an additive category, rather than a category

of modules. An additive category is a category A with 0-object such

that (i) HomA(M1,M2) is an abelian group for each pair M1, M2 of

objects, (ii) composition is bilinear, and (iii) every finite set of objects

has a biproduct. A biproduct of M1,... , Mm consists of an object M

together with maps ui : Mi −→ M and pi : M −→ Mi, i = 1,...,m,

such that piuj = δij and u1p1 + · · · + umpm = 1M . We denote the

biproduct by M1 ⊕ · · · ⊕ Mm.

We will need an additional condition on our additive category, that

idempotents split (cf. [Bas68, Chapter I, §3, p. 19]). Given an object

M and an idempotent e ∈ EndA(M), we say that e splits provided there

is a factorization M

p

− − → K

u

−→ − M such that e = up and pu = 1K.

The reader is probably familiar with the notion of an abelian cat-

egory, that is, an additive category in which every map has a kernel

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http://dx.doi.org/10.1090/surv/181/01