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