Introduction Definition of seq C-M modules Main results Graded case Application References . Sequentially Cohen-Macaulay Rees modules . Naoki Taniguchi Meiji University Joint work with T. N. An, N. T. Dung and T. T. Phuong Mathematical Society of Japan at Meiji University March 21, 2015 . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 1 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 1 Introduction . 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-Macaulay property of R ( I ) where I is an m -primary ideal. . Question 1.1 . When is the Rees module R ( M ) sequentially Cohen-Macaulay? . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 2 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 1 Introduction . 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-Macaulay property of R ( I ) where I is an m -primary ideal. . Question 1.1 . When is the Rees module R ( M ) sequentially Cohen-Macaulay? . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 2 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 2 Definition of sequentially C-M modules Let R be a Noetherian ring, M ̸ = (0) a finitely generated R -module with d = dim R M < ∞ . 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 March 21, 2015 3 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References Let D i = M d i for 1 ≤ ∀ i ≤ ℓ . We then have a filtration 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 . Notice that dim R D i = dim R C i = d i for 1 ≤ ∀ i ≤ ℓ . . Definition 2.1 ([5, 6]) . (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 March 21, 2015 4 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References Let D i = M d i for 1 ≤ ∀ i ≤ ℓ . We then have a filtration 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 . Notice that dim R D i = dim R C i = d i for 1 ≤ ∀ i ≤ ℓ . . Definition 2.1 ([5, 6]) . (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 March 21, 2015 4 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 3 Main results In this section ( R, m ) a Noetherian local ring M ̸ = (0) a finitely generated R -module with d = dim R M F = { F n } n ∈ Z a filtration of ideals of R s.t. F 1 ̸ = R M = { M n } n ∈ Z an F -filtration of R -submodules of M R = R ( F ) a Noetherian ring R ( M ) a finitely generated R -module . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 5 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References Let 1 ≤ i ≤ ℓ . We set D i = { M n ∩ D i } n ∈ Z , C i = { [( M n ∩ D i ) + D i − 1 ] /D i − 1 } n ∈ Z . Then D i (resp. C i ) is an F -filtration of R -submodules of D i (resp. C i ). We have the exact sequence 0 → [ D i − 1 ] n → [ D i ] n → [ C i ] n → 0 of R -modules for ∀ n ∈ Z . Hence 0 → R ( D i − 1 ) → R ( D i ) → R ( C i ) → 0 0 → R ′ ( D i − 1 ) → R ′ ( D i ) → R ′ ( C i ) → 0 and 0 → G ( D i − 1 ) → G ( D i ) → G ( C i ) → 0 . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 6 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References . Theorem 3.1 . TFAE. (1) R ′ ( M ) is a sequentially C-M R ′ -module. (2) G ( M ) is a sequentially C-M G -module and {G ( D i ) } 0 ≤ i ≤ ℓ is the dimension filtration of G ( M ) . When this is the case, M is a sequentially C-M R -module. . . Theorem 3.2 . Suppose that M is a sequentially C-M R -module and F 1 ⊈ p for ∀ p ∈ Ass R M . Then TFAE. (1) R ( M ) is a sequentially C-M R -module. (2) G ( M ) is a sequentially C-M G -module, {G ( D i ) } 0 ≤ i ≤ ℓ is the dimension filtration of G ( M ) and a( G ( C i )) < 0 for 1 ≤ ∀ i ≤ ℓ . When this is the case, R ′ ( M ) is a sequentially C-M R ′ -module. . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 7 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 4 Graded case n ≥ 0 R n be a Z -graded ring. We put Let R = ∑ ∑ R k for ∀ n ∈ Z . F n = k ≥ n Then F n is a graded ideal of R , F = { F n } n ∈ Z is a filtration of ideals of R and F 1 := R + ̸ = R . Let E be a graded R -module with E n = (0) for ∀ n < 0 . Put ∑ E k for ∀ n ∈ Z . E ( n ) = k ≥ n Then E ( n ) is a graded R -submodule of E , E = { E ( n ) } n ∈ Z is an F -filtration of R -submodules of E . Then we have R = G ( F ) and E = G ( E ) . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 8 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References . Assumption 4.1 . R = ∑ n ≥ 0 R n a Noetherian Z -graded ring E ̸ = (0) a finitely generated graded R -module with d = dim R E < ∞ . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 9 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References . Proposition 4.2 . TFAE. (1) R ′ ( E ) is a sequentially C-M R ′ -module. (2) E is a sequentially C-M R -module. . . Theorem 4.3 . Suppose that R 0 is a local ring, E is a sequentially C-M R -module and F 1 ⊈ p for ∀ p ∈ Ass R E . Then TFAE. (1) R ( E ) is a sequentially C-M R -module. (2) a( C i ) < 0 for 1 ≤ ∀ i ≤ ℓ . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 10 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References § 5 Application –Stanley-Reisner algebras– . Notation 5.1 . V = { 1 , 2 , . . . , n } ( n > 0) a vertex set ∆ a simplicial complex on V s.t. ∆ ̸ = ∅ F (∆) a set of facets of ∆ m = ♯ F (∆) ( > 0) its cardinality S = k [ X 1 , X 2 , . . . , X n ] a polynomial ring over a field k I ∆ = ( X i 1 X i 2 · · · X i r | { i 1 < i 2 < · · · < i r } / ∈ ∆) R = k [∆] = S/I ∆ the Stanley-Reisner ring of ∆ . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 11 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References We consider the Z -graded ring R = k [∆] = ∑ n ≥ 0 R n and put R k = m n for ∀ n ∈ Z ∑ I n := k ≥ n where m := R + = ∑ n> 0 R n . Then F = { I n } n ∈ Z is an m -adic filtration of R and I 1 ̸ = R . . Proposition 5.2 . If ∆ is shellable, then R ′ ( m ) is a sequentially C-M ring. . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 12 / 18
Introduction Definition of seq C-M modules Main results Graded case Application References Notice that p ⊉ I 1 for ∀ p ∈ Ass R ⇐ ⇒ F ̸ = ∅ for ∀ F ∈ F (∆) ⇐ ⇒ ∆ ̸ = {∅} . . Theorem 5.3 . Suppose that ∆ is shellable with shelling order F 1 , F 2 , . . . , F m ∈ F (∆) s.t. dim F 1 ≥ dim F 2 ≥ · · · ≥ dim F m and ∆ ̸ = {∅} . Then TFAE. (1) R ( m ) is a sequentially C-M ring. (2) dim F i ≥ ♯ F ( ⟨ F 1 , F 2 , . . . , F i − 1 ⟩ ∩ ⟨ F i ⟩ ) for 2 ≤ ∀ i ≤ m . . . . . . . . Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 13 / 18
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