Almost Gorenstein rings of higher dimension . Naoki Taniguchi - PowerPoint PPT Presentation

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Almost Gorenstein rings of higher dimension . Naoki Taniguchi Meiji University Joint work with Shiro Goto and Ryo Takahashi AMS Special Session on Algebra and Representation theory March 7, 2015 . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 1 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References § 1 History of almost Gorenstein rings In 1997, B. Barucci and R. Fr¨ oberg ([BF]) · · · one-dimensional analytically unramified local rings In 2013, S. Goto, N. Matsuoka and T. T. Phuong ([GMP]) · · · one-dimensional Cohen-Macaulay local rings which are not necessarily analytically unramified . . Question 1.1 . If it’s possible, what’s the definition of almost Gorenstein rings of higher . dimension? . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 2 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References § 1 History of almost Gorenstein rings In 1997, B. Barucci and R. Fr¨ oberg ([BF]) · · · one-dimensional analytically unramified local rings In 2013, S. Goto, N. Matsuoka and T. T. Phuong ([GMP]) · · · one-dimensional Cohen-Macaulay local rings which are not necessarily analytically unramified . . Question 1.1 . If it’s possible, what’s the definition of almost Gorenstein rings of higher . dimension? . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 2 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References § 1 History of almost Gorenstein rings In 1997, B. Barucci and R. Fr¨ oberg ([BF]) · · · one-dimensional analytically unramified local rings In 2013, S. Goto, N. Matsuoka and T. T. Phuong ([GMP]) · · · one-dimensional Cohen-Macaulay local rings which are not necessarily analytically unramified . . Question 1.1 . If it’s possible, what’s the definition of almost Gorenstein rings of higher . dimension? . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 2 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References Contents . . History of almost Gorenstein rings 1 . . Almost Gorenstein local rings 2 . . Semi-Gorenstein local rings 3 . . Almost Gorenstein graded rings 4 . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 3 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References § 2 Almost Gorenstein local rings . Setting 2.1 . ( R, m ) a Cohen-Macaulay local ring with d = dim R . ∃ the canonical module K R . | R/ m | = ∞ . . . Definition 2.2 . 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 ) . . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 4 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References Therefore in Definition 2.2, if C ̸ = (0) , then C is Cohen-Macaulay and dim R C = d − 1 . Moreover µ R ( C ) = e 0 m ( C ) ⇐ ⇒ m C = ( f 2 , f 3 , . . . , f d ) C for some f 2 , f 3 , . . . , f d ∈ m . Hence C is a maximally generated Cohen-Macaulay module in the sense of B. Ulrich (cf. [2]), which is called an Ulrich R -module . . Remark 2.3 . Suppose that d = 1 . Then TFAE. (1) R is almost Gorenstein in the sense of Definition 2.2. (2) R is almost Gorenstein in the sense of [GMP, Definition 3.1]. . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 5 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References Therefore in Definition 2.2, if C ̸ = (0) , then C is Cohen-Macaulay and dim R C = d − 1 . Moreover µ R ( C ) = e 0 m ( C ) ⇐ ⇒ m C = ( f 2 , f 3 , . . . , f d ) C for some f 2 , f 3 , . . . , f d ∈ m . Hence C is a maximally generated Cohen-Macaulay module in the sense of B. Ulrich (cf. [2]), which is called an Ulrich R -module . . Remark 2.3 . Suppose that d = 1 . Then TFAE. (1) R is almost Gorenstein in the sense of Definition 2.2. (2) R is almost Gorenstein in the sense of [GMP, Definition 3.1]. . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 5 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Example 2.4 . (1) k [[ t 3 , t 4 , t 5 ]] . (2) k [[ t a , t a +1 , . . . , t 2 a − 3 , t 2 a − 1 ]] ( a ≥ 4) . (3) k [[ X, Y, Z ]] / ( X, Y ) ∩ ( Y, Z ) ∩ ( Z, X ) . (4) Suppose that R is not Gorenstein. If R is an almost Gorenstein local ring, then R is G-regular. (5) 1-dimensional finite CM-representation type. (6) 2-dimensional rational singularity. . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 6 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Theorem 2.5 (NZD characterization) . (1) If R is an almost Gorenstein local ring of dimension d > 1 , then so is R/ ( f ) for genaral NZD f ∈ m . (2) Let f ∈ m be a NZD on R . If R/ ( f ) is an almost Gorenstein local ∈ m 2 , if R is not ring, then so is R . When this is the case, f / . Gorenstein. . Corollary 2.6 . Suppose that d > 0 . If R/ ( f ) is an almost Gorenstein local ring for every NZD f ∈ m , then R is Gorenstein. . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 7 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Theorem 2.5 (NZD characterization) . (1) If R is an almost Gorenstein local ring of dimension d > 1 , then so is R/ ( f ) for genaral NZD f ∈ m . (2) Let f ∈ m be a NZD on R . If R/ ( f ) is an almost Gorenstein local ∈ m 2 , if R is not ring, then so is R . When this is the case, f / . Gorenstein. . Corollary 2.6 . Suppose that d > 0 . If R/ ( f ) is an almost Gorenstein local ring for every NZD f ∈ m , then R is Gorenstein. . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 7 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Example 2.7 . Let U = k [[ X 1 , X 2 , . . . , X n , Y 1 , Y 2 , , . . . , Y n ]] ( n ≥ 2) be the formal power series ring over an infinite field k and put R = U/ I 2 ( M ) , where I 2 ( M ) denotes the ideal of U generated by 2 × 2 minors of the matrix ( ) X 1 X 2 ··· X n M = . Y 1 Y 2 ··· Y n Then R is almost Gorenstein with dim R = n + 1 and r( R ) = n − 1 . . . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 8 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Proof of Example 2.7. . Notice that { X i − Y i − 1 } 1 ≤ i ≤ n (here Y 0 = Y n ) forms a regular sequence on R R/ ( X i − Y i − 1 | 1 ≤ i ≤ n ) R ∼ = k [[ X 1 , X 2 , . . . , X n ]] / I 2 ( N ) = S , ( ) X 1 X 2 ··· X n − 1 X n where N = . X 2 X 3 ··· X n X 1 Then S is Cohen-Macaulay with dim S = 1 , n 2 = x 1 n and K S ∼ = ( x 1 , x 2 , . . . , x n − 1 ) , where n is the maximal ideal of S , x i is the image of X i in S . Hence S is an almost Gorenstein local ring, since n ( x 1 , x 2 , . . . , x n − 1 ) ⊆ ( x 1 ) . Thus R . is almost Gorenstein. . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 9 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Theorem 2.8 . Let ( S, n ) be a Noetherian local ring, φ : R → S a flat local homomorphism. Suppose that S/ m S is a RLR. Then TFAE. (1) R is an almost Gorenstein local ring. (2) S is an almost Gorenstein local ring. . Therefore R is almost Gorenstein ⇐ ⇒ R [[ X 1 , X 2 , . . . , X n ]] ( n ≥ 1) is almost Gorenstein. ⇒ � R is almost Gorenstein ⇐ R is almost Gorenstein. . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 10 / 22

History Almost Gorenstein local rings Semi-Gorenstein local rings Almost Gorenstein graded rings References . Theorem 2.8 . Let ( S, n ) be a Noetherian local ring, φ : R → S a flat local homomorphism. Suppose that S/ m S is a RLR. Then TFAE. (1) R is an almost Gorenstein local ring. (2) S is an almost Gorenstein local ring. . Therefore R is almost Gorenstein ⇐ ⇒ R [[ X 1 , X 2 , . . . , X n ]] ( n ≥ 1) is almost Gorenstein. ⇒ � R is almost Gorenstein ⇐ R is almost Gorenstein. . . . . . . Naoki Taniguchi (Meiji University) Almost Gorenstein rings of higher dimension March 7, 2015 10 / 22