§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References On Ratliff–Rush closure of modules Naoki Taniguchi Waseda University AMS Meeting Special Session on Homological Algebra March 17, 2018 Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 1 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References § 1 Introduction Throughout my talk A a commutative Noetherian ring I , J ideals of A ∪ [ I ℓ +1 : A I ℓ ] � I = the Ratliff–Rush closure of I ℓ ≥ 0 R ( I ) = A [ It ] ⊆ A [ t ] the Rees algebra of I Note that I ⊆ � I and � I · � J ⊆ � IJ � I ⊆ I , if grade A I > 0 If J ⊆ I and J is a reduction of I , then � J ⊆ � I . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 2 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References The projective scheme Proj R ( I ) = { P ∈ Spec R ( I ) | P is a graded ideal , P ⊉ R ( I ) + } of R ( I ) defines the blowup of Spec A along V ( I ). Theorem 1.1 (Goto-Matsuoka, 2005) √ Let ( A , m ) be a two-dimensional RLR, I = m . Then TFAE. (1) � I = I. (2) Proj R ( I ) is a normal scheme. M ( R ( I )) ∼ When this is the case, R ( I ) has FLC, H 1 = R ( I ) / R ( I ) , and R ( I ) is CM ⇐ ⇒ I = I . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 3 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References The notion of Rees algebra R ( I ) can be generalized to the module M , which is defined as R ( M ) = Sym A ( M ) / t (Sym A ( M )) . The Rees algebra of M includes the multi-Rees algebra, which corresponds to the case where M = I 1 ⊕ I 2 ⊕ · · · ⊕ I ℓ . The application to equisingularity theory needs this generalization ([2, 3]). Question 1.2 Can we generalize Theorem 1.1 to the case of modules? Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 4 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Contents Introduction 1 Preliminaries 2 Ratliff–Rush closure of modules 3 Main results 4 Application 5 Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 5 / 36
� � � � § 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References § 2 Preliminaries Setting 2.1 A a Noetherian ring M a finitely generated A -module F = A ⊕ r ( r > 0) s.t. M ⊆ F Look at the diagram ∃ 1 Sym( i ) Sym A ( M ) Sym A ( F ) = A [ t 1 , t 2 , . . . , t r ] =: S i i i M F Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 6 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References The Rees algebra R ( M ) of M is defined by R ( M ) = Im(Sym( i )) ⊆ S = A [ t 1 , t 2 , . . . , t r ] ⊕ M n . = n ≥ 0 Definition 2.2 For ∀ n ≥ 0, we define ( S ) M n = n ⊆ S n = F n R ( M ) and call it the integral closure of M n . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 7 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Proposition 2.3 For ∀ n ≥ 0 , we have ( ) M n = ( MS ) n n . ( ) In particular, M = 1 ⊆ F. MS More precisely, x ∈ M satisfies x n + c 1 x n − 1 + · · · + c n = 0 in S where n > 0, c i ∈ M i for 1 ≤ ∀ i ≤ n . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 8 / 36
� � � § 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Lemma 2.4 Suppose that rank A M = r. Then Q( R ( M )) = Q( S ) . Moreover, if A is a normal domain, then Q( R ( M )) = R ( M ) S R ( M ) Proof. Look at the diagram ∼ = Q( A ) ⊗ A Sym A ( M ) Q( A ) ⊗ A S Sym( i ) � Sym A ( M ) S We get 0 → t (Sym A ( M )) → Sym A ( M ) → R ( M ) → 0 which yields Q( A ) ⊗ A S ∼ = Q( A ) ⊗ A Sym A ( M ) ∼ = Q( A ) ⊗ A R ( M ) . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 9 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References § 3 Ratliff–Rush closure of modules Setting 3.1 A a Noetherian ring M a finitely generated A -module F = A ⊕ r ( r > 0) s.t. M ⊆ F R ( M ) = Im(Sym A ( M ) − → Sym A ( F )) ⊆ Sym A ( F ) We set a = R ( M ) + = ⊕ n > 0 M n , S = Sym A ( F ), and := ε − 1 ( ) S � H 0 R ( M ) a ( S / R ( M )) ⊆ S where ε : S → S / R ( M ). Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 10 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Definition 3.2 For ∀ n ≥ 0, we define ( S ) � � M n = ⊆ S n = F n R ( M ) n and call it the Ratliff–Rush closure of M n . Definition 3.3 (Liu, 1998) Suppose that A is a Noetherian domain. Then � M is defined to be the largest A -submodule N of F satisfying M ⊆ N ⊆ F , M n = N n for ∀ n ≫ 0. Remark 3.4 These definitions coincide, when A is a Noetherian domain. Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 11 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Proposition 3.5 For ∀ n ≥ 0 , we have ( ) ∪ [ ( M n ) ℓ +1 : F n ( M n ) ℓ ] � � M n = ( MS ) n = n . ℓ> 0 In particular ( ) ∪ [ M ℓ +1 : F M ℓ ] � � M = = 1 . MS ℓ> 0 Corollary 3.6 Suppose that M is a faithful A-module. Then M n ⊆ M n ⊆ F n � for ∀ n ≥ 0 . Hence S S ⊆ S . R ( M ) ⊆ � R ( M ) ⊆ R ( M ) Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 12 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Definition 3.7 (Buchsbaum-Rim, 1964, Hayasaka-Hyry, 2010) Suppose that ( A , m ) is a Noetherian local ring with d = dim A . Then M is called a parameter module in F , if ℓ A ( F / M ) < ∞ , M ⊆ m F , and µ A ( M ) = d + r − 1. Proposition 3.8 Suppose that ( A , m ) is a CM local ring with d = dim A > 0 . Let M be a parameter module in F. Then � M = M . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 13 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Example 3.9 Let A = k [[ X , Y ]]. Set ⟨( ) ( ) ( )⟩ X Y 0 M = , , ⊆ F = A ⊕ A . 0 X Y Then M is a parameter module in F and � M = M . Example 3.10 Let R = k [[ X , Y , Z , W ]]. Set A = R / ( X , Y ) ∩ ( Z , W ) , Q = ( X − Z , Y − W ) A . Then � Q = Q . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 14 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Proposition 3.11 Suppose that L = Ax 1 + Ax 2 + · · · + Ax ℓ ( ⊆ M ) is a reduction of M. Then ∪ [ ] M n +1 : F ( Ax 1 n + Ax 2 n + · · · + Ax ℓ � n ) M = . n > 0 Corollary 3.12 If L is a reduction of M, then L ⊆ � � M . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 15 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Remark 3.13 The implication L ⊆ � � L ⊆ M = ⇒ M does NOT hold in general. Example 3.14 (Heinzer-Johnston-Lantz-Shah, 1993) We consider A = k [[ t 3 , t 4 ]] ⊆ k [[ t ]] , I = ( t 8 ) , and J = ( t 11 , t 12 ) . Then J ⊆ I , but � J ⊈ � I . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 16 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References The following is the key in our argument. Proposition 3.15 Suppose that M is a faithful A-module. Then the following assertions hold. M ) n = M n for ∀ n ≫ 0 . (1) � M n = ( � (2) Let N be an A-submodule of F s.t. M ⊆ N. Then TFAE. ( i ) N ⊆ � M . ( ii ) M ℓ = N ℓ for ∃ ℓ > 0 . ( iii ) M n = N n for ∀ n ≫ 0 . ( iv ) � M = � N . (3) � � M = � M . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 17 / 36
§ 1 Introduction § 2 Preliminaries § 3 Ratliff–Rush closure of modules § 4 Main Results § 5 Application References Let us note the following. Lemma 3.16 Suppose that ( A , m ) is a Noetherian local ring. If M = F, then M = F. In particular, if M ̸ = F and M is faithful, then � M ̸ = F. Proof. Suppose M ̸ = F and choose a counterexample M so that r = rank A F > 0 is as small as possible. Then M ⊆ m F . Therefore F = M ⊆ m F = m F = m F so that F = m F , which is a contradiction. Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules March 17, 2018 18 / 36
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