AGL rings arising as fiber products Shiro Goto (Meiji University) - - PowerPoint PPT Presentation

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AGL rings arising as fiber products Shiro Goto (Meiji University) - - PowerPoint PPT Presentation

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1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References AGL rings arising as fiber products Shiro Goto (Meiji University) Ryotaro Isobe (Chiba University) Naoki Taniguchi (Waseda University)

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1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References

AGL rings arising as fiber products

Shiro Goto (Meiji University) Ryotaro Isobe (Chiba University) Naoki Taniguchi (Waseda University)

The 40th Symposium on Commutative Algebra November 24, 2018

  • S. Goto, R. Isobe, N. Taniguchi

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1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References

1 Introduction

The fiber product A = R ×T S = {(a, b) ∈ R × S | f (a) = g(b)} is the subring of R × S, where R

f

− → T and T

g

← − S are homomorphisms of rings. Hence we have the exact sequence 0 − → A

ι

− → R × S

[ f −g ]

− → T

  • f A-modules.
  • S. Goto, R. Isobe, N. Taniguchi

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Question 1.1

When is R ×T S an AGL ring?

  • S. Goto, R. Isobe, N. Taniguchi

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Preceding results

Ogoma ([7]) the Gorensteinness of fiber product A = R ×T S, where R is a CM local ring, S is a equi-dimensional Noetherian local ring with (S1) D’Anna, Shapiro, Ananthnarayan-Avramov-Moore ([3, 8, 1]) the Gorensteinness of fiber product A = R ×R/I R, where R is a Noetherian local ring Nasseh-Sather-Wagstaff-Takahashi-VandeBogert ([4]) the CM fiber products of finite CM type

  • S. Goto, R. Isobe, N. Taniguchi

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Example 1.2 Let R = k[[X, Y ]]/(X a − Y b), S = k[[Z, W ]]/(Z c − W d) with a, b, c, d ≥ 2. Then

A = R ×k S ∼ = k[[X, Y , Z, W ]]/ [ (X, Y )·(Z, W ) + (X a − Y b, Z c − W d) ]

is a CM local ring with r(A) = 3.

How about the AGL property?

  • S. Goto, R. Isobe, N. Taniguchi

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2 Basic facts

For homomorphisms f : R → T, g : S → T, we consider A = R ×T S = {(a, b) ∈ R × S | f (a) = g(b)} ⊆ B = R × S. Then R

f

  • A

p2

  • p1
  • T

S

g

  • where p1 : A → R, (x, y) → x, p2 : A → S, (x, y) → y.
  • S. Goto, R. Isobe, N. Taniguchi

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Lemma 2.1 Suppose f and g are surjective. (1) A is a Noetherian ring ⇐ ⇒ R, S are Noetherian rings (2) (A, J) is a local ring ⇐ ⇒ (R, m), (S, n) are local rings When this is the case, J = (m × n) ∩ A. (3) (R, m), (S, n) are CM, dim R = dim S = d > 0, depth T ≥ d − 1 = ⇒ (A, J) is CM and dim A = d. Proof. Consider 0 − → A

ι

− → B = R × S

φ

− → T − → 0 where φ = [ f

−g

] .

  • S. Goto, R. Isobe, N. Taniguchi

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Let (R, m), (S, n) be Noetherian local rings, k = R/m = S/n, and f : R → k, g : S → k the canonical maps. Proposition 2.2 (1) v(A) = v(R) + v(S). (2) dim R = dim S > 0 = ⇒ e(A) = e(R) + e(S). (3) If R, S are CM and dim R = dim S = 1, A = R ×k S is Gorenstein ⇐ ⇒ R and S are DVRs.

  • S. Goto, R. Isobe, N. Taniguchi

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Proof.

Jℓ+1 = mℓ+1 × nℓ+1 (∀ℓ ≥ 0), since J = m × n. (1) ℓA(J/J2) = ℓk([m/m2] ⊕ [n/n2]) = ℓR(m/m2) + ℓS(n/n2). (2) ℓA(A/Jℓ+1) = ℓA(A/J) + ℓA(J/Jℓ+1) = 1 + [ ℓR(m/mℓ+1) + ℓS(n/nℓ+1) ] = 1 + { [ℓR(R/mℓ+1) − 1] + [ℓS(S/nℓ+1) − 1] } = [ ℓR(R/mℓ+1) + ℓS(S/nℓ+1) ] − 1 (3) (⇒) By 0 → A

ι

→ B

φ

→ k → 0, 0 → A : B

ι

→ A → Ext1

A(A/J, A) → 0.

Hence, J = A : B. Thus, because A is Gorenstein and A : J = J : J, R × S = B = A : (A : B) = A : J = J : J = (m : m) × (n : n). Therefore, R = m : m and S = n : n, whence R, S are DVRs.

  • S. Goto, R. Isobe, N. Taniguchi

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  • 3. AGL rings

Suppose (R, m) a CM local ring, d = dim R, ♯(R/m) = ∞, ∃ KR. Definition 3.1 (Goto-Takahashi-T) We say that R is an almost Gorenstein local ring, if ∃ an exact sequence 0 → R → KR → C → 0

  • f R-modules such that µR(C) = e0

m(C).

We have R is a Gorenstein ring ⇒ R is an AGL ring. µR(C) = e0

m(C) ⇔ mC = (f1, f2, . . . , fd−1)C, for some

f1, f2, . . . , fd−1 ∈ m

  • S. Goto, R. Isobe, N. Taniguchi

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Suppose dim R = 1 and R ⊆ ∃K ⊆ R s.t. K ∼ = KR. Then Remark 3.2 (Goto-Matsuoka-Phuong, Goto-Takahashi-T, Kobayashi) R is an AGL ring ⇔ mK ⊆ R ⇔ mK = m ⇔ mK ∼ = m. Example 3.3 (1) k[[te, te+1, . . . , t2e−3, t2e−1]] (e ≥ 4) (2) k[[X, Y , Z]]/(X, Y ) ∩ (Y , Z) ∩ (Z, X) (3) k[[t4, t5, t6]] ⋉ (t4, t5, t6) (4) 1-dimensional CM rings of finite CM-representation type (5) 2-dimensional rational singularity (6) k[[X1, X2, . . . , Xn, Y1, Y2, , . . . , Yn]]/I2( X1 X2 ··· Xn

Y1 Y2 ··· Yn ) (n ≥ 2)

  • S. Goto, R. Isobe, N. Taniguchi

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4 Results in dimension one

Setting 4.1 (R, m), (S, n) CM local rings, dim R = dim S = 1 k = R/m = S/n, f : R → k, g : S → k canonical maps A = R ×k S ⊆ B = R × S, J = m × n (the maximal ideal of A) Then Q(A) = Q(B) = Q(R) × Q(S) A = B = R × S We assume that Q(A) = Q(R) × Q(S) is a Gorenstein ring, ∃ KA, and ♯k = ∞. Hence, all the rings A, R, and S possess fractional canonical ideals.

  • S. Goto, R. Isobe, N. Taniguchi

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Theorem 4.2 TFAE. (1) A = R ×k S is an AGL ring. (2) A = R ×k S is a GGL ring. (3) R and S are AGL rings.

  • S. Goto, R. Isobe, N. Taniguchi

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Preliminaries for the proof of Theorem 4.2

We have R ⊆ K ⊆ R, K ∼ = KR, and S ⊆ L ⊆ S, L ∼ = KS. Firstly, suppose R and S are not DVRs. Then K : m ⊆ R, L : n ⊆ S. Hence, because R : m ̸⊆ K and S : n ̸⊆ L, we have K : m = K + R·g1, L : n = L + S·g2 for some g1 ∈ (R : m) \ K and g2 ∈ (S : n) \ L. We set X = (K × L) + A·g with g = (g1, g2) ∈ A. Then we have Lemma 4.3 A ⊆ X ⊆ A and X ∼ = KA.

  • S. Goto, R. Isobe, N. Taniguchi

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Theorem 4.4 Suppose R and S are not DVRs. TFAE. (1) A = R ×k S is an AGL ring. (2) R and S are AGL rings. Proof. Note A is AGL ⇔ JX = J (= m × n), while JX = (m × n) · [(K × L) + A·g] = (mK + m·g1) × (nL + n·g2) = m(K + R·g1) × n(L + S·g2) = m·(K : m) × n·(L : n) = mK × nL. Therefore A is AGL ⇔ mK = m, nL = n ⇔ R, S are AGL.

  • S. Goto, R. Isobe, N. Taniguchi

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Proof of (1) ⇔ (3) in Theorem 4.2

Assume R is a DVR but S is not. Choose X so that A ⊆ X ⊆ A and X ∼ = KA. Then KB = X : B ∼ = R × L. Therefore X : B = ξ · (R × L) for some ξ = (ξ1, ξ2) ∈ Q(A). On the other hand, by 0 → A

ι

→ B

φ

→ k = A/J → 0, we get 0 − → X : B − → X − → A/J − → 0. Hence JX ⊆ X : B ⊆ X. Thus Lemma 4.5 X : B ⊆ X ⊆ (X : B) : J = (ξ1R × ξ2L) : J = ξ1·(R : m) × ξ2·(L : n).

  • S. Goto, R. Isobe, N. Taniguchi

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Proof of (1) ⇔ (3) in Theorem 4.2

Corollary 4.6 J(X : B) ⊆ JX ⊆ J · [ξ1(R : m) × ξ2(L : n)]. (1) ⇒ (3) We have JX = J. Hence n·ξ2L ⊆ n ⊆ n·ξ2(L : n) = ξ2·nL because n(L : n) = nL. Thus n = ξ2·nL ∼ = nL, so that S is AGL. (3) ⇒ (1) We have JX = [JX ∩ A · (1, 0) · ξ] + Jξ, and JX ∩ A · (1, 0) · ξ ⊆ Jξ ⇒ JX = Jξ ∼ = J JX ∩A·(1, 0)·ξ ̸⊆ Jξ ⇒ JX = ξ(R ×n) ∼ = ξ(m×n) = ξJ ∼ = J This will prove that A is AGL.

  • S. Goto, R. Isobe, N. Taniguchi

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Theorem 4.2 R ×k S is an AGL ring ⇔ R and S are AGL rings. Letting S = R, we have Corollary 4.7 R ×R/m R is AGL ⇔ R is AGL ⇔ R ⋉ m is AGL.

  • S. Goto, R. Isobe, N. Taniguchi

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Comment to the case of 2-AGL rings

Let c = R : R[K]. R is a Gorenstein ring ⇔ c = R R is a non-Gorenstein AGL ring ⇔ c = m We also have Theorem 4.8 R ×R/c R is 2-AGL ⇔ R is 2-AGL ⇔ R ⋉ c is 2-AGL

  • S. Goto, R. Isobe, N. Taniguchi

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Theorem 4.9 R ×k S is 2-AGL ⇔ R is AGL, S is 2-AGL, or R is 2-AGL, S is AGL Example 4.10 (1) k[[t3, t7, t8]] ×k k[[t]] (2) k[[t3, t7, t8]] ×k k[[t3, t4, t5]]

  • S. Goto, R. Isobe, N. Taniguchi

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  • 5. Higher dimensional cases

(R, m), (S, n) CM local ring with d = dim R = dim S > 0 (T, mT) a RLR with dim T = d − 1, ♯(T/mT) = ∞. f : R → T, g : S → T surjective A = R ×T S, J = (m × n) ∩ A. Then A is a CM local ring with dim A = d. Proposition 5.1 A = R ×T S is Gorenstein ⇔ R and S are RLRs.

  • S. Goto, R. Isobe, N. Taniguchi

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Theorem 5.2 Assume that ∃ KA and that Q(A) is a Gorenstein ring. Then TFAE. (1) A = R ×T S is an AGL ring. (2) R and S are AGL rings.

  • S. Goto, R. Isobe, N. Taniguchi

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Thank you for your attention.

  • S. Goto, R. Isobe, N. Taniguchi

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References

[1]

  • H. Ananthnarayan, L. L. Avramov, and W. F. Moore, Connected sums of

Gorenstein local rings, J. Reine Angew. Math., 667 (2012), 149–176. [2]

  • V. Barucci and R. Fr¨
  • berg, One-dimensional almost Gorenstein rings, J. Algebra, 188

(1997), no. 2, 418–442. [3]

  • M. D’Anna, A construction of Gorenstein rings, J. Algebra, 306 (2006), 507–519.

[4]

  • S. Nasseh, S. Sather-Wagstaff, R. Takahashi, and K. VandeBogert,

Applications and homological properties of local rings with decomposable maximal ideals,

  • J. Pure and Appl. Algebra, 223 (2019), no.3, 1272–1287.

[5]

  • S. Goto, N. Matsuoka, and T. T. Phuong, Almost Gorenstein rings, J. Algebra, 379

(2013), 355–381. [6]

  • S. Goto, R. Takahashi, and N. Taniguchi, Almost Gorenstein rings - towards a

theory of higher dimension, J. Pure Appl. Algebra, 219 (2015), 2666–2712. [7]

  • T. Ogoma, Fiber products of Noetherian rings, Commutative algebra and combinatorics

(Kyoto, 1985), 173–182, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987. [8]

  • J. Shapiro, On a construction of Gorenstein rings proposed by M. D’Anna, J. Algebra,

323 (2010), 1155–1158.

  • S. Goto, R. Isobe, N. Taniguchi

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