Sequentially Cohen-Macaulay Rees modules . Naoki Taniguchi Meiji - - PowerPoint PPT Presentation

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Introduction Definition of seq C-M modules Main results Graded case Application References

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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

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Introduction Definition of seq C-M modules Main results Graded case Application References

§1 Introduction

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• N. T. Cuong, S. Goto and H. L. Truong

. . The equality I2 = qI 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

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Introduction Definition of seq C-M modules Main results Graded case Application References

§1 Introduction

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• N. T. Cuong, S. Goto and H. L. Truong

. . The equality I2 = qI 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

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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 = dimR M < ∞. Then ∀n ∈ Z, ∃Mn the largest R-submodule of M with dimR Mn ≤ n. Let S(M) = {dimR N | N is an R-submodule of M, N ̸= (0)} = {dim R/p | p ∈ AssR M} = {d1 < d2 < · · · < dℓ = d} where ℓ = ♯S(M).

Naoki Taniguchi (Meiji University) Sequentially Cohen-Macaulay Rees modules March 21, 2015 3 / 18

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Introduction Definition of seq C-M modules Main results Graded case Application References

Let Di = Mdi for 1 ≤ ∀i ≤ ℓ. We then have a filtration D0 := (0) ⊊ D1 ⊊ D2 ⊊ . . . ⊊ Dℓ = M which we call the dimension filtration of M. Put Ci = Di/Di−1 for 1 ≤ ∀i ≤ ℓ. Notice that dimR Di = dimR Ci = di for 1 ≤ ∀i ≤ ℓ. .

Definition 2.1 ([5, 6])

. . (1) M is a sequentially Cohen-Macaulay R-module

def

⇐ ⇒ Ci 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

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Introduction Definition of seq C-M modules Main results Graded case Application References

Let Di = Mdi for 1 ≤ ∀i ≤ ℓ. We then have a filtration D0 := (0) ⊊ D1 ⊊ D2 ⊊ . . . ⊊ Dℓ = M which we call the dimension filtration of M. Put Ci = Di/Di−1 for 1 ≤ ∀i ≤ ℓ. Notice that dimR Di = dimR Ci = di for 1 ≤ ∀i ≤ ℓ. .

Definition 2.1 ([5, 6])

. . (1) M is a sequentially Cohen-Macaulay R-module

def

⇐ ⇒ Ci 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

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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 = dimR M F = {Fn}n∈Z a filtration of ideals of R s.t. F1 ̸= R M = {Mn}n∈Z an F-filtration of R-submodules of M R = R(F) a Noetherian ring R(M) a finitely generated R-module

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Introduction Definition of seq C-M modules Main results Graded case Application References

Let 1 ≤ i ≤ ℓ. We set Di = {Mn ∩ Di}n∈Z, Ci = {[(Mn ∩ Di) + Di−1]/Di−1}n∈Z. Then Di (resp. Ci) is an F-filtration of R-submodules of Di (resp. Ci). We have the exact sequence 0 → [Di−1]n → [Di]n → [Ci]n → 0

• f R-modules for ∀n ∈ Z. Hence

0 → R(Di−1) → R(Di) → R(Ci) → 0 0 → R′(Di−1) → R′(Di) → R′(Ci) → 0 and 0 → G(Di−1) → G(Di) → G(Ci) → 0.

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Introduction Definition of seq C-M modules Main results Graded case Application References

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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(Di)}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 F1 ⊈ p for ∀p ∈ AssR M. Then TFAE. (1) R(M) is a sequentially C-M R-module. (2) G(M) is a sequentially C-M G-module, {G(Di)}0≤i≤ℓ is the dimension filtration of G(M) and a(G(Ci)) < 0 for 1 ≤ ∀i ≤ ℓ. When this is the case, R′(M) is a sequentially C-M R′-module.

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Introduction Definition of seq C-M modules Main results Graded case Application References

§4 Graded case

Let R = ∑

n≥0 Rn be a Z-graded ring. We put

Fn = ∑

k≥n

Rk for ∀n ∈ Z. Then Fn is a graded ideal of R, F = {Fn}n∈Z is a filtration of ideals of R and F1 := R+ ̸= R. Let E be a graded R-module with En = (0) for ∀n < 0. Put E(n) = ∑

k≥n

Ek for ∀n ∈ Z. 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).

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Assumption 4.1

. . R = ∑

n≥0 Rn a Noetherian Z-graded ring

E ̸= (0) a finitely generated graded R-module with d = dimR E < ∞

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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 R0 is a local ring, E is a sequentially C-M R-module and F1 ⊈ p for ∀p ∈ AssR E. Then TFAE. (1) R(E) is a sequentially C-M R-module. (2) a(Ci) < 0 for 1 ≤ ∀i ≤ ℓ.

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Introduction Definition of seq C-M modules Main results Graded case Application References

§5 Application –Stanley-Reisner algebras–

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Notation 5.1

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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[X1, X2, . . . , Xn] a polynomial ring over a field k I∆ = (Xi1Xi2 · · · Xir | {i1 < i2 < · · · < ir} / ∈ ∆) R = k[∆] = S/I∆ the Stanley-Reisner ring of ∆

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Introduction Definition of seq C-M modules Main results Graded case Application References

We consider the Z-graded ring R = k[∆] = ∑

n≥0 Rn and put

In := ∑

k≥n

Rk = mn for ∀n ∈ Z where m := R+ = ∑

n>0 Rn. Then F = {In}n∈Z is an m-adic filtration of

R and I1 ̸= R. .

Proposition 5.2

. . If ∆ is shellable, then R′(m) is a sequentially C-M ring.

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Introduction Definition of seq C-M modules Main results Graded case Application References

Notice that p ⊉ I1 for ∀p ∈ Ass R ⇐ ⇒ F ̸= ∅ for ∀F ∈ F(∆) ⇐ ⇒ ∆ ̸= {∅}. .

Theorem 5.3

. . Suppose that ∆ is shellable with shelling order F1, F2, . . . , Fm ∈ F(∆) s.t. dim F1 ≥ dim F2 ≥ · · · ≥ dim Fm and ∆ ̸= {∅}. Then TFAE. (1) R(m) is a sequentially C-M ring. (2) dim Fi ≥ ♯F(⟨F1, F2, . . . , Fi−1⟩ ∩ ⟨Fi⟩) for 2 ≤ ∀i ≤ m.

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Introduction Definition of seq C-M modules Main results Graded case Application References

By Theorem 5.3 we get the following. .

Corollary 5.4

. . Suppose that dim Fm ≥ 1. If ⟨F1, F2, . . . , Fi−1⟩ ∩ ⟨Fi⟩ is a simplex for 2 ≤ ∀i ≤ m, then R(m) is a sequentially C-M ring.

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Example 5.5

. . Let ∆ = ⟨F1, F2, F3⟩, where F1 = {1, 2, 3}, F2 = {2, 3, 4} and F3 = {4.5}. Then ∆ is shellable with numbering F(∆) = {F1, F2, F3} and ⟨F1⟩ ∩ ⟨F2⟩ , ⟨F1, F2⟩ ∩ ⟨F3⟩ are simplexes, so that R(m) is a sequentially C-M ring. ∆ = 1 4 5 2 3

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Example 5.6

. . Let ∆ = ⟨F1, F2, F3, F4⟩, where F1 = {1, 2, 5}, F2 = {2, 3}, F3 = {3, 4} and F4 = {4, 5}. Then ∆ is shellable with numbering F(∆) = {F1, F2, F3, F4}. We put ∆1 = ⟨F1, F2, F3⟩, ∆2 = ⟨F4⟩. Then ♯F(∆1 ∩ ∆2) = 2 = dim F4 + 1 > dim F4, so that R(m) is NOT a sequentially C-M ring by Theorem 5.3. 1 2 3 4 5 ∆ =

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Thank you so much for your attention.

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Introduction Definition of seq C-M modules Main results Graded case Application References

References

[1]

• N. T. Cuong, S. Goto and H. L. Truong, The equality I2 = qI in sequentially

Cohen-Macaulay rings, J. Algebra, (379) (2013), 50-79. [2]

• G. Faltings, ¨

Uber die Annulatoren lokaler Kohomologiegruppen, Archiv der Math., 30 (1978), 473–476. [3]

• S. Goto, Y. Horiuchi and H. Sakurai, Sequentially Cohen-Macaulayness versus parametric

decomposition of powers of parameter ideals, J. Comm. Algebra, 2 (2010), 37–54. [4]

• S. Goto and K. Watanabe, On graded rings, I, J. Math. Soc. Japan, 30 (1978), 179–213.

[5]

• P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, in: Proc.
• f the Ferrara Meeting in honour of Mario Fiorentini, University of Antwerp, Wilrijk,

Belgium, (1998), 245–264. [6]

• R. P. Stanley, Combinatorics and commutative algebra, Second Edition, Birkh¨

auser, Boston, 1996. [7]

• N. Taniguchi, T.T. Phuong, N. T. Dung and T. N. An, Sequentially Cohen-Macaulay Rees

modules, preprint 2014. [8]

• D. Q. Viet, A note on the Cohen-Macaulayness of Rees Algebra of filtrations, Comm.

Algebra 21 (1993), 221-229.

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