Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Sequentially Cohen-Macaulay Rees modules Naoki Taniguchi Meiji University Joint work with T. N. An, N. T. Dung and T. T. Phuong at Purdue University October 29, 2014 . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 1 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Introduction . [CGT] . N. T. Cuong, S. Goto and H. L. Truong, The equality I 2 = q I in sequentially Cohen-Macaulay rings , J. Algebra, (379) (2013), 50-79. . In [CGT], Characterized the sequentially Cohen-Macaulayness of R ( I ) where I is an m -primary ideal which contains a good parameter ideal as a reduction. ([Theorem 5.3]). . Question 1.1 . When is the Rees module R ( M ) sequentially Cohen-Macaulay ? . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 2 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Contents . . Introduction 1 . . Filtration 2 . . Survey on sequentially Cohen-Macaulay modules 3 . . Main results 4 . . Sequentially Cohen-Macaulay property in E ♮ 5 . . Application –Stanley-Reisner algebras– 6 . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 3 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Filtration Let R be a commutative ring. . Definition 2.1 . F = { F n } n ∈ Z is a filtration of ideals of R def ⇐ ⇒ . . F n is an ideal of R , 1 . . F n ⊇ F n +1 for ∀ n ∈ Z , 2 . . F m F n ⊆ F m + n for ∀ m, n ∈ Z and 3 . . F 0 = R . . 4 Then we put F n t n ⊆ R [ t ] , R ′ = R ′ ( F ) = F n t n ⊆ R [ t, t − 1 ] . ∑ ∑ R = R ( F ) = n ≥ 0 n ∈ Z . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 4 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Let M be an R -module. . Definition 2.2 . M = { M n } n ∈ Z is an F -filtration of R -submodules of M def ⇐ ⇒ . . M n is an R -submodule of M , 1 . . M n ⊇ M n +1 for ∀ n ∈ Z , 2 . . F m M n ⊆ M m + n for ∀ m, n ∈ Z and 3 . . M 0 = M . . 4 We set t n ⊗ M n ⊆ R [ t ] ⊗ R M, ∑ R ( M ) = n ≥ 0 t n ⊗ M n ⊆ R [ t, t − 1 ] ⊗ R M. ∑ R ′ ( M ) = n ∈ Z . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 5 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Here t n ⊗ M n = { t n ⊗ x | x ∈ M n } ⊆ R [ t, t − 1 ] ⊗ R M for ∀ n ∈ Z . If F 1 ̸ = R , then we put G = G ( F ) = R ′ /u R ′ , G ( M ) = R ′ ( M ) /u R ′ ( M ) where u = t − 1 . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 6 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References For the rest of this section, we assume F 1 ̸ = R . . Lemma 2.3 . Suppose R is Noetherian and M is finitely generated. Then TFAE. (1) R ( M ) is a finitely generated graded R -module. (2) R ′ ( M ) is a finitely generated graded R ′ -module. (3) ∃ n 1 , n 2 , . . . , n ℓ ≥ 0 ( ℓ > 0) s.t. M n = ∑ ℓ i =1 F n − n i M n i for ∀ n ≥ max { n 1 , n 2 , . . . , n ℓ } . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 7 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . The composite map i ε → R ′ ( M ) ψ : R ( M ) − − → G ( M ) is surjective and Ker ψ = u R ′ ( M ) ∩ R ( M ) = u [ R ( M )] + , n> 0 t n ⊗ M n . where [ R ( M )] + = ∑ . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 8 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . Assumption 2.4 . R ( F ) a Noetherian ring R ( M ) a finitely generated R -module . Then R is Noetherian and M is finitely generated. . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 9 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . Proposition 2.5 . The following assertions hold true. (1) Let P ∈ Ass R R ( M ) . Then p ∈ Ass R M , P = p R [ t ] ∩ R and { dim R/ p + 1 if dim R/ p < ∞ , F 1 ⊈ p , dim R /P = dim R/ p otherwise , where p = P ∩ R . (2) Suppose M ̸ = (0) , d = dim R M < ∞ and ∃ p ∈ Ass R M s.t. dim R/ p = d , F 1 ⊈ p . Then dim R R ( M ) = d + 1 . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 10 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . Proof. . (1) Let P ∈ Ass R R ( M ) . Then P ∈ Ass R R [ t ] ⊗ R M , so that P = Q ∩ R for some ∪ Q ∈ Ass R [ t ] R [ t ] ⊗ R M = Ass R [ t ] R [ t ] / p R [ t ] . p ∈ Ass R M Thus p = Q ∩ R and Q = p R [ t ] for ∃ p ∈ Ass R M . Therefore P = p R [ t ] ∩ R , p = P ∩ R. Let R = R/ p . Then F = { F n R } n ∈ Z is a filtration of ideals of R and R /P ∼ = R ( F ) as graded R -algebras. . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 11 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . Corollary 2.6 . Suppose R is local, M ̸ = (0) . Then { d + 1 if ∃ p ∈ Ass R M s.t. dim R/ p = d, F 1 ⊈ p , dim R R ( M ) = d otherwise , where d = dim R M . . . Proposition 2.7 . The following assertions hold true. (1) Let P ∈ Ass R ′ R ′ ( M ) . Then p ∈ Ass R M , P = p R [ t, t − 1 ] ∩ R ′ and dim R ′ /P = dim R/ p + 1 , where p = P ∩ R . (2) Suppose M ̸ = (0) . Then dim R ′ R ′ ( M ) = dim R M + 1 . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 12 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References . Lemma 2.8 . Suppose that R is a local ring, M ̸ = (0) . Then G ( M ) ̸ = (0) and dim G G ( M ) = dim R M . . . Proof. . Let N be a unique graded maximal ideal of an H -local ring R ′ . Then R ′ ( M ) N ̸ = (0) and u ∈ N . Therefore G ( M ) N ̸ = (0) , so that G ( M ) ̸ = (0) . Hence dim G G ( M ) = dim R M . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 13 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Survey on sequentially C-M modules Let R be a Noetherian ring and M ̸ = (0) a finitely generated R -module with d = dim R M < ∞ . We put Assh R M = { p ∈ Supp R M | dim R/ p = d } . Then ∀ n ∈ Z , ∃ M n the largest R -submodule of M with dim R M n ≤ n . Let S ( M ) = { dim R N | N is an R -submodule of M, N ̸ = (0) } { dim R/ p | p ∈ Ass R M } = = { d 1 < d 2 < · · · < d ℓ = d } where ℓ = ♯ S ( M ) . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 14 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Let D i = M d i for 1 ≤ ∀ i ≤ ℓ . We then have a filtration D : D 0 := (0) ⊊ D 1 ⊊ D 2 ⊊ . . . ⊊ D ℓ = M which we call the dimension filtration of M . Put C i = D i /D i − 1 for 1 ≤ ∀ i ≤ ℓ . . Definition 3.1 ([Sch, St]) . (1) M is a sequentially Cohen-Macaulay R -module def ⇐ ⇒ C i is a C-M R -module for 1 ≤ ∀ i ≤ ℓ . (2) R is a sequentially Cohen-Macaulay ring def ⇐ ⇒ dim R < ∞ and R is a sequentially C-M module over itself. . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 15 / 49
Seq C-M property in E ♮ Intro Filtration Survey on seq C-M modules Main results Application References Let ∩ (0) = M ( p ) p ∈ Ass R M be a primary decomposition of (0) in M , where Ass R M/M ( p ) = { p } for ∀ p ∈ Ass R M . . Fact 3.2 ([Sch]) . The following assertions hold true. (1) D i = ∩ dim R/ p ≥ d i +1 M ( p ) for 0 ≤ ∀ i < ℓ . (2) Ass R C i = { p ∈ Ass R M | dim R/ p = d i } and Ass R D i = { p ∈ Ass R M | dim R/ p ≤ d i } for 1 ≤ ∀ i ≤ ℓ . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules October 29, 2014 16 / 49
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