Almost Gorenstien rings Naoyuki Matsuoka Meiji University - PowerPoint PPT Presentation

Preliminaries Definition m : m Idealization Recent Almost Gorenstien rings Naoyuki Matsuoka Meiji University September 9, 2014 Joint work with Shiro Goto and Tran Thi Phuong . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 1 / 29

Preliminaries Definition m : m Idealization Recent History 1997 The notion of almost Gorenstein rings was introduced by Valentina Barucci-Ralf Fr¨ oberg (analytically unramified case) with a result about the Gorenstein property of m : m = { α ∈ Q ( R ) | α m ⊆ m } . 2009 A counterexample for a result about m : m was given by Barucci. (But their result is true !) 2013 A new definition of almost Gorenstein rings of dimension one was given and repair the proof of the Gorenstein property of m : m . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 2 / 29

Preliminaries Definition m : m Idealization Recent History 1997 The notion of almost Gorenstein rings was introduced by Valentina Barucci-Ralf Fr¨ oberg (analytically unramified case) with a result about the Gorenstein property of m : m = { α ∈ Q ( R ) | α m ⊆ m } . 2009 A counterexample for a result about m : m was given by Barucci. (But their result is true !) 2013 A new definition of almost Gorenstein rings of dimension one was given and repair the proof of the Gorenstein property of m : m . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 2 / 29

Preliminaries Definition m : m Idealization Recent History 1997 The notion of almost Gorenstein rings was introduced by Valentina Barucci-Ralf Fr¨ oberg (analytically unramified case) with a result about the Gorenstein property of m : m = { α ∈ Q ( R ) | α m ⊆ m } . 2009 A counterexample for a result about m : m was given by Barucci. (But their result is true !) 2013 A new definition of almost Gorenstein rings of dimension one was given and repair the proof of the Gorenstein property of m : m . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 2 / 29

Preliminaries Definition m : m Idealization Recent Classes of local rings . regular ⇒ complete-intersection ⇒ Gorenstein ⇒ Cohen-Macaulay ⇒ Buchsbaum . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 3 / 29

Preliminaries Definition m : m Idealization Recent Classes of local rings . regular ⇒ complete-intersection ⇒ Gorenstein ⇒ almost Gorenstein ⇒ Cohen-Macaulay ⇒ Buchsbaum . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 4 / 29

Preliminaries Definition m : m Idealization Recent . . . Preliminaries (Hilbert coefficients, existence of canonical ideals) 1 . . Definition of almost Gorenstein rings 2 . . . The Gorenstein property of m : m 3 . . Almost Gorenstein rings obtained by idealization 4 . . . Recent researches 5 k [[ H ]] = k [[ t a 1 , t a 2 , . . . , t a ℓ ]] ⊆ k [[ t ]] for H = ⟨ a 1 , a 2 , . . . , a ℓ ⟩ . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 5 / 29

Preliminaries Definition m : m Idealization Recent § 1. Preliminaries . Let ( R , m ) a CM local ring, dim R = 1, I an m -primary ideal ⇒ ∃ e 0 ( I ) , e 1 ( I ) ∈ Z such that ( n + 1 ) ℓ R ( R / I n +1 ) = e 0 ( I ) − e 1 ( I ) ( ∀ n ≫ 0) . 1 We call e 0 ( I ) the multiplicity of R w.r.t. I and e 1 ( I ) the first Hilbert coefficient of R w.r.t. I . . . How to compute e 0 ( I ) and e 1 ( I ) ? . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 6 / 29

Preliminaries Definition m : m Idealization Recent Assume ∃ a ∈ I such that Q = ( a ) is a reduction of I (i.e., ∃ n ≥ 0, I n +1 = QI n ) For any n > 0, put I n a n = { x a n | x ∈ I n } ⊆ Q( R ) Let S = R [ I a ] ⊆ Q( R ). ⇒ S = ∪ a n = I r I n n > 0 a r where r = red Q ( I ) = min { n ≥ 0 | I n +1 = QI n } . Hence ℓ R ( R / I n +1 ) = ℓ R ( R / Q n +1 ) − ℓ R ( I n +1 / Q n +1 ) ( n + 1 ) = ℓ R ( R / Q ) − ℓ R ( S / R ) if n ≥ r − 1 1 ∥ ∥ e 0 ( I ) e 1 ( I ) . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 7 / 29

Preliminaries Definition m : m Idealization Recent . Theorem . e 0 ( I ) = ℓ R ( R / Q ) , e 1 ( I ) = ℓ R ( S / R ) . . . Corollary . µ R ( I / Q ) ≤ ℓ R ( I / Q ) ≤ e 1 ( I ) . . µ R ( I / Q ) = ℓ R ( I / Q ) ⇐ ⇒ m I ⊆ Q ( i.e. m I = m Q ) 1 ⇒ I 2 = QI ( i.e. red Q ( I ) ≤ 1) . . ℓ R ( I / Q ) = e 1 ( I ) ⇐ 2 . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 8 / 29

Preliminaries Definition m : m Idealization Recent The case H = ⟨ 3 , 4 , 5 ⟩ . Example . Let H = ⟨ 3 , 4 , 5 ⟩ and R = k [[ H ]] = k [[ t 3 , t 4 , t 5 ]] ( k a field). Take I = ( t 3 , t 4 ) and Q = ( t 3 ), then Q is a reduction of I . In fact, I 3 = QI 2 . Hence S = I 2 t 6 . For e 0 ( I ): ⇒ e 0 ( I ) = ℓ R ( R / Q ) = 3 . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 9 / 29

Preliminaries Definition m : m Idealization Recent The case H = ⟨ 3 , 4 , 5 ⟩ . Example . Let H = ⟨ 3 , 4 , 5 ⟩ and R = k [[ H ]] = k [[ t 3 , t 4 , t 5 ]] ( k a field). Take I = ( t 3 , t 4 ) and Q = ( t 3 ), then Q is a reduction of I . In fact, I 3 = QI 2 . Hence S = I 2 t 6 . For e 1 ( I ): ⇒ e 1 ( I ) = ℓ R ( S / R ) = 2 . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 10 / 29

Preliminaries Definition m : m Idealization Recent Existence of canonical ideals Let K R denote the canonical module of R . ∃ K R ⇔ R ∼ = a Gorenstein ring / ∼ . . Definition . We say that I ⊊ R is a canonical ideal of R if I ∼ = K R . . . When ∃ I ⊊ R a canonical ideal? . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 11 / 29

Preliminaries Definition m : m Idealization Recent . Theorem (Herzog-Kunz) . TFAE . . ∃ I ⊊ R a canonical ideal of R. 1 . . Q( � R ) is a Gorenstein ring. 2 Hence if R is analytically unramified then ∃ I a canonical ideal of R. . . Corollary . Suppose that Q( � R ) is Gorenstein. If | R / m | = ∞ , then R ⊆ ∃ K ⊆ R such that K ∼ = K R where R is the integral closure of R. . . Proof. . ∃ a ∈ I such that Q = ( a ) is a reduction of I . Put K = I . a . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 12 / 29

Preliminaries Definition m : m Idealization Recent . Example . Let R = k [[ X , Y , Z ]] / ( X , Y ) ∩ ( Y , Z ) ∩ ( Z , X ). Then I = ( x + y , y + z ) ∼ = K R . If k = Z / (2), then ∀ a ∈ I , ( a ) is not a reduction of I . . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 13 / 29

Preliminaries Definition m : m Idealization Recent . Proposition . Let k = R / m and � k / k an extension of fields. then ∃ φ : ( R , m ) → ( � R , � m ) a flat homomorphism of local rings such that . . m = m � � R 1 . . � m ∼ = � R / � k as k-algebras. 2 Moreover we have the following (a) Q ( � R ) is Gorenstein ⇔ Q ( � � R ) is Gorenstein. In this case, ∀ I a canonical ideal of R, I � R is a canonical ideal of � R and e 1 ( I � R ) = e 1 ( I ) . (b) m : m is Gorenstein ⇔ � m : � m is Gorenstein. . . Problem . ⇒ � ? R is analytically unramified ⇐ R is analytically unramified . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 14 / 29

Preliminaries Definition m : m Idealization Recent § 2. Definition of almost Gorenstein rings . Definition . We say that R is an almost Gorenstein ring, if . . Q( � R ) is Gorenstein. Hence ∃ I ⊊ R a canonical ideal of R . 1 . . e 1 ( I ) ≤ r( R ) (the Cohen-Macaulay type of R ) = µ R ( I ). 2 . . Remark . Let I , J ⊊ R canonical ideals, then e 1 ( I ) = e 1 ( J ). . . R is Gorenstein ⇒ r( R ) = 1 and I is a parameter ideal. Hence e 1 ( I ) = 0 ≤ r( R ). Thus R is almost Gorenstein . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 15 / 29

Preliminaries Definition m : m Idealization Recent Examples of almost Gorenstein rings . Example . . R = k [[ t 3 , t 4 , t 5 ]] ⊆ k [[ t ]] (r( R ) = 2; an integral domain) 1 . . R = k [[ X , Y , Z ]] / ( X , Y ) ∩ ( Y , Z ) ∩ ( Z , X ) (r( R ) = 2; a 2 reduced ring) . . R = k [[ X , Y , Z , W ]] / ( Y 2 , Z 2 , W 2 , YW , ZW , XW − YZ ) 3 (r( R ) = 3; not a reduced ring) . . Let 3 ≤ a ∈ Z and R = k [[ t a , t a +1 , t a 2 − a − 1 ]]. 4 Let I be a canonical ideal of R ⇒ e 1 ( I ) = a ( a − 1) − 1, r( R ) = 2. 2 Hence R is an almost Gorenstein ring ⇔ a = 3. On the other ( x ) y a − 2 z hand, R ∼ = k [[ x , y , z ] / I 2 . Hence, by x a − 1 y z Nari-Numata-Watanabe, R is almost Gorenstein ⇔ a − 2 = 1. . . . . . . . Naoyuki Matsuoka (Meiji University) Almost Gorenstein rings September 9, 2014 16 / 29