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 AGL rings arising as fiber products November 24, 2018 1 / 24
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 f g R − → T and T ← − S are homomorphisms of rings. Hence we have the exact sequence [ f ] ι − g 0 − → A − → R × S − → T of A -modules. S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 2 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References Question 1.1 When is R × T S an AGL ring? S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 3 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 ( S 1 ) 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 AGL rings arising as fiber products November 24, 2018 4 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 ∼ ( X , Y ) · ( Z , W ) + ( X a − Y b , Z c − W d ) [ ] = k [[ X , Y , Z , W ]] / is a CM local ring with r ( A ) = 3. How about the AGL property? S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 5 / 24
� � � � 1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 p 1 f A T p 2 g S where p 1 : A → R , ( x , y ) �→ x , p 2 : A → S , ( x , y ) �→ y . S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 6 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 φ ι 0 − → A − → B = R × S − → T − → 0 [ f ] where φ = . − g S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 7 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 AGL rings arising as fiber products November 24, 2018 8 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References Proof. J ℓ +1 = m ℓ +1 × n ℓ +1 ( ∀ ℓ ≥ 0), since J = m × n . (1) ℓ A ( J / J 2 ) = ℓ k ([ m / m 2 ] ⊕ [ n / n 2 ]) = ℓ R ( m / m 2 ) + ℓ S ( n / n 2 ). (2) ℓ A ( A / J ℓ +1 ) ℓ A ( A / J ) + ℓ A ( J / J ℓ +1 ) = ℓ R ( m / m ℓ +1 ) + ℓ S ( n / n ℓ +1 ) [ ] = 1 + [ ℓ R ( R / m ℓ +1 ) − 1] + [ ℓ S ( S / n ℓ +1 ) − 1] { } = 1 + [ ℓ R ( R / m ℓ +1 ) + ℓ S ( S / n ℓ +1 ) ] = − 1 ι φ (3) ( ⇒ ) By 0 → A → B → k → 0, ι → A → Ext 1 0 → A : B 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 AGL rings arising as fiber products November 24, 2018 9 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 3. AGL rings Suppose ( R , m ) a CM local ring, d = dim R , ♯ ( R / m ) = ∞ , ∃ K R . Definition 3.1 (Goto-Takahashi-T) We say that R is an almost Gorenstein local ring , if ∃ an exact sequence 0 → R → K R → C → 0 of R -modules such that µ R ( C ) = e 0 m ( C ). We have R is a Gorenstein ring ⇒ R is an AGL ring. µ R ( C ) = e 0 m ( C ) ⇔ m C = ( f 1 , f 2 , . . . , f d − 1 ) C , for some f 1 , f 2 , . . . , f d − 1 ∈ m S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 10 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References Suppose dim R = 1 and R ⊆ ∃ K ⊆ R s.t. K ∼ = K R . Then Remark 3.2 ( Goto-Matsuoka-Phuong, Goto-Takahashi-T, Kobayashi) R is an AGL ring ⇔ m K ⊆ R ⇔ m K = m ⇔ m K ∼ = m . Example 3.3 (1) k [[ t e , t e +1 , . . . , t 2 e − 3 , t 2 e − 1 ]] ( e ≥ 4) (2) k [[ X , Y , Z ]] / ( X , Y ) ∩ ( Y , Z ) ∩ ( Z , X ) (3) k [[ t 4 , t 5 , t 6 ]] ⋉ ( t 4 , t 5 , t 6 ) (4) 1-dimensional CM rings of finite CM-representation type (5) 2-dimensional rational singularity (6) k [[ X 1 , X 2 , . . . , X n , Y 1 , Y 2 , , . . . , Y n ]] / I 2 ( X 1 X 2 ··· X n Y 1 Y 2 ··· Y n ) ( n ≥ 2) S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 11 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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, ∃ K A , and ♯ k = ∞ . Hence, all the rings A , R , and S possess fractional canonical ideals. S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 12 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 AGL rings arising as fiber products November 24, 2018 13 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References Preliminaries for the proof of Theorem 4.2 We have R ⊆ K ⊆ R , K ∼ = K R , and S ⊆ L ⊆ S , L ∼ = K S . 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 · g 1 , L : n = L + S · g 2 for some g 1 ∈ ( R : m ) \ K and g 2 ∈ ( S : n ) \ L . We set X = ( K × L ) + A · g with g = ( g 1 , g 2 ) ∈ A . Then we have Lemma 4.3 A ⊆ X ⊆ A and X ∼ = K A . S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 14 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 ] = ( m K + m · g 1 ) × ( n L + n · g 2 ) = m ( K + R · g 1 ) × n ( L + S · g 2 ) = m · ( K : m ) × n · ( L : n ) = m K × n L . Therefore A is AGL ⇔ m K = m , n L = n ⇔ R , S are AGL. S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 15 / 24
1 Introduction 2 Basic facts 3 AGL rings 4 Results in dimension one 5 Higher dimensional cases References 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 ∼ = K A . Then K B = X : B ∼ = R × L . Therefore X : B = ξ · ( R × L ) for some ξ = ( ξ 1 , ξ 2 ) ∈ Q( A ). φ ι On the other hand, by 0 → A → B → 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 = ( ξ 1 R × ξ 2 L ) : J = ξ 1 · ( R : m ) × ξ 2 · ( L : n ) . S. Goto, R. Isobe, N. Taniguchi AGL rings arising as fiber products November 24, 2018 16 / 24
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