Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References On Ratliff–Rush closure of modules Naoki Taniguchi Waseda University The 39th Japan Symposium on Commutative Algebra November 17, 2017 Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 1 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Introduction Throughout my talk A a 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 November 17, 2017 2 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Set Proj R ( I ) = { P ∈ Spec R ( I ) | P is a graded ideal , P ⊉ R ( I ) + } . Theorem 1.1 (Goto-Matsuoka, 2005) √ Let ( A , m ) be a two-dimensional RLR, I = m . Then TFAE. (1) � I = I. (2) � I n = I n for ∀ n > 0 . (3) I n = I n for ∃ n > 0 . (4) I n = I n for ∀ n ≫ 0 . (5) Proj R ( I ) is a normal scheme. (6) R ( I ) P is normal for ∀ P ∈ Spec R ( I ) \ { M } , where M = m R ( I ) + R ( I ) + . When this is the case, R ( I ) has FLC, H 1 M ( R ( I )) ∼ = R ( I ) / R ( I ) , and R ( I ) is CM ⇐ ⇒ I = I . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 3 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Question 1.2 Can we generalize Theorem 1.1 to the case of modules? Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 4 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results 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 November 17, 2017 5 / 37
� � � � Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References 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 November 17, 2017 6 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results 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 November 17, 2017 7 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results 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 November 17, 2017 8 / 37
� � � � Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Lemma 2.4 Suppose that A is a Noetherian domain and ℓ A ( F / M ) < ∞ . 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 November 17, 2017 9 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Proposition 2.5 Suppose that A is a normal domain and ℓ A ( F / M ) < ∞ . Let G be a finitely generated free A-module s.t. 0 → M → G is exact. Then S ∼ T R ( M ) = R ( M ) where T = Sym A ( G ) . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 10 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Ratliff–Rush closure of modules Setting 3.1 A a Noetherian ring M ̸ = (0) 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 November 17, 2017 11 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results 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 November 17, 2017 12 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results 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 A is a Noetherian domain. 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 November 17, 2017 13 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Proposition 3.7 Suppose that A is a normal domain and ℓ A ( F / M ) < ∞ . Let G be a finitely generated free A-module s.t. 0 → M → G is exact. Then S ∼ T � = � R ( M ) R ( M ) where T = Sym A ( G ) . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 14 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Definition 3.8 (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 3.9 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 November 17, 2017 15 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Example 3.10 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 3.11 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 November 17, 2017 16 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Proposition 3.12 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 3.13 If L is a reduction of M, then L ⊆ � � M . Naoki Taniguchi (Waseda University) On Ratliff–Rush closure of modules November 17, 2017 17 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References Remark 3.14 The implication L ⊆ � � L ⊆ M = ⇒ M does not hold in general. Example 3.15 (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 November 17, 2017 18 / 37
Introduction Preliminaries Ratliff–Rush closure of modules Main Results Application References The following is the key in our argument. Proposition 3.16 Suppose that A is a Noetherian domain. Then the following assertions hold. (1) � M n ⊆ M n for ∀ n ≫ 0 . (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 November 17, 2017 19 / 37
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