§4. EXERCISES 9 f f , and it follows that σ and ξ induce the same homomorphism M/mf M −→ N/mf N. We denote e(M, N) the smallest lifting number for the pair (M, N). 1.13. Theorem (Guralnick). Let (R, m) be a local ring, and let M and N be finitely generated R-modules. If M/mr+1M | N/mr+1N for some r max {e(M, N), e(N, M)}, then M | N. Proof. Choose reciprocal homomorphisms ξ : M/mr+1M −→ N/mr+1N and η : N/mr+1N −→ M/mr+1M such that ηξ = 1M/mr+1M. Since r is a lifting number (Lemma 1.12), there exist R-homomorphisms σ : M −→ N and τ : N −→ M such that σ agrees with ξ and τ agrees with η modulo m. By NAK, τσ : M −→ M is surjective and therefore, by Exercise 1.27, an automorphism. It follows that M | N. 1.14. Corollary. Let (R, m) be a local ring and M, N finitely generated R-modules. If M/mnM N/mnN for all n 0, then M N. Proof. By Theorem 1.13, M | N and N | M. In particular, we have surjections N α −→ M and M β −→ N. Then βα is a surjective en- domorphism of N and therefore is an automorphism (cf. Exercise 1.27). It follows that α is one-to-one and therefore an isomorphism. 1.15. Corollary. Let (R, m) be a local ring and (R, m) its m-adic completion. Let M and N be finitely generated R-modules. (i) If R ⊗R M | R ⊗R N, then M | N. (ii) If R ⊗R M R ⊗R N , then M N. 1.16. Corollary. Let M, N and P be finitely generated modules over a local ring (R, m). If P M P N, then M N. Proof. We have (R ⊗R P ) (R ⊗R M) (R ⊗R P ) (R ⊗R N). Using KRS for complete rings (Corollary 1.9) we see that R ⊗R M R ⊗R N. Now apply Corollary 1.15. §4. Exercises 1.17. Exercise. Prove Proposition 1.1: For a non-zero object M in an additive category A, and E = EndA(M), if 0 and 1 are the only idempotents of E, then M is indecomposable. Conversely, suppose e = e2 E, with e = 0, 1. If both e and 1 e split, then M is decomposable.
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