§3. R-MODULES VS. R-MODULES 7 xm ≡ xn (mod mΛ), we see that z ≡ 0 (mod mΛ), so 1 − z is a unit of Λ. Since z(1 − z) ∈ mnΛ, it follows that z ∈ mnΛ. Thus we have xm + xn ≡ 2xmxn, xm ≡ x2 m , xn ≡ x2 n (mod mnΛ) . Multiplying the first congruence, in turn, by xm and by xn, we learn that xm ≡ xmxn ≡ xn (mod mnΛ). If, now, n and m n, we see that x ≡ xm (mod mnΛ). This verifies the claim. Since Λ is mΛ-adically complete (cf. Exercise 1.24), we let x be the limit of the sequence (xn) and check that x is an idempotent lifting e. 1.10. Corollary. KRS holds for finitely generated modules over complete local rings. Henselian local rings are almost characterized as those having the Krull-Remak-Schmidt property. Indeed, a theorem due to Evans states that a local ring R is Henselian if and only if for every module-finite commutative local R-algebra A the finitely generated A-modules satisfy KRS [Eva73]. §3. R-modules vs. R-modules A major theme in this book is the study of direct-sum decomposi- tions over local rings that are not necessarily complete. Here we record a few results that will allow us to use KRS over the completion R to understand R-modules. We begin with a result due to Guralnick [Gur85, Theorem A] on lifting homomorphisms modulo high powers of the maximal ideal of a local ring. Given finitely generated modules M and N over a local ring (R, m), we define a lifting number for the pair (M, N) to be a non- negative integer e satisfying the following property: For each positive integer f and each R-homomorphism ξ : M/me+fM −→ N/me+fN, there exists σ ∈ HomR(M, N) such that σ and ξ induce the same homomorphism M/mfM −→ N/mfN. (Thus the outer and bottom squares in the diagram below both commute, though the top square may not.) M σ N M/me+fM ξ N/me+fN M/mfM ξ=σ N/mfN For example, 0 is a lifting number for the pair (M, N) if M is free.

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