6 1. THE KRULL-REMAK-SCHMIDT THEOREM therefore is a left (and right) Artinian ring. Now apply Corollary 1.5 and Theorem 1.3. §2. KRS over Henselian rings We now proceed to prove KRS for finitely generated modules over complete and, more generally, Henselian local rings. Here we define a local ring (R, m,k) to be Henselian provided, for every module-finite R- algebra Λ (not necessarily commutative), each idempotent of Λ J (Λ) lifts to an idempotent of Λ. For the classical definition of “Henselian” in terms of factorization of polynomials, and for other equivalent con- ditions, see Theorem A.30. 1.7. Lemma. Let R be a commutative ring and Λ a module-finite R-algebra (not necessarily commutative). Then ΛJ (R) J (Λ). Proof. Let f ΛJ (R). We want to show that Λ(1 λf) = Λ for every λ Λ. Clearly Λ(1 λf) + ΛJ (R) = Λ, and now NAK completes the proof. 1.8. Theorem. Let (R, m,k) be a Henselian local ring, and let M be an indecomposable finitely generated R-module. Then EndR(M) is nc-local. In particular, KRS holds for the category of finitely generated modules over a Henselian local ring. Proof. Let E = EndR(M) and J = J (E). Since E is a module- finite R-algebra (cf. Exercise 1.22), Lemma 1.7 implies that mE J and hence that E/J is a finite-dimensional k-algebra. It follows that E/J is semisimple Artinian. Moreover, since E has no non- trivial idempotents, neither does E/J. By the Wedderburn-Artin The- orem [Lam91, (3.5)], E/J is a division ring. 1.9. Corollary (Hensel’s Lemma). Let (R, m,k) be a complete local ring. Then R is Henselian. Proof. Let Λ be a module-finite R-algebra, and put J = J (Λ). Again, J, and J/mΛ is a nilpotent ideal of Λ/mΛ (since Λ/mΛ is Artinian). By Proposition 1.4 we can lift each idempotent of Λ/J to an idempotent of Λ/mΛ. Therefore it will suffice to show that every idem- potent e of Λ/mΛ lifts to an idempotent of Λ. Using Proposition 1.4, we can choose, for each positive integer n, an element xn Λ such that xn + = e and xn x2 n (mod mnΛ). (Of course mnΛ = (mΛ)n.) We claim that (xn) is a Cauchy sequence for the mΛ-adic topology on Λ. To see this, let n be an arbitrary positive integer. Given any m n, put z = xm + xn 2xmxn. Then z z2 (mod mnΛ). Also, since
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