On the Hilbert function of one-dimensional semigroup rings Michela - PowerPoint PPT Presentation

On the Hilbert function of one-dimensional semigroup rings Michela Di Marca Joint work with Marco D’Anna and Vincenzo Micale Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 1 / 28

Introduction to the problem Hilbert function Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 2 / 28

Introduction to the problem Hilbert function Let ( R , m ) be a Noetherian local ring with | R \ m | = ∞ . gr ( R ) = ⊕ i ≥ 0 m i / m i + 1 is the associated graded ring of R . Definition The Hilbert function of R is H R ( i ) = dim k m i / m i + 1 , H R : N → N , where k = R / m . Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 3 / 28

Introduction to the problem Monomial curves Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 4 / 28

Introduction to the problem Monomial curves Definition C ⊆ A n k is an algebraic curve if ∃ I ( C ) ⊆ k [ x 1 , . . . , x n ] such that C = V ( I ( C )) ; k [ x 1 ,..., x n ] dim k = 1 . I ( C ) Suppose there are some numbers g 1 , . . . , g n ∈ N with gcd ( g 1 , . . . , g n ) = 1 , and an homomorphism ψ : k [ x 1 , . . . , x n ] → k [ t ] : x 1 �→ t g 1 . . . x n �→ t g n , such that I ( C ) = ker ψ , then C is called monomial curve, denoted by C = C ( g 1 , . . . , g n ) . Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 5 / 28

Introduction to the problem Monomial curves Let C = C ( g 1 , . . . , g n ) be a monomial curve determined by the homomorphism ψ . Then S = � g 1 , . . . , g n � is a numerical semigroup; 1 By extending ψ to ˆ ψ : k [[ x 1 , . . . , x n ]] → k [[ t ]] , we get 2 Im ( ˆ ψ ) = k [[ t S ]] , the semigroup ring associated to S ; k [[ t S ]] ∼ = k [[ x 1 ,..., x n ]] is the completion of the coordinate ring of C ; 3 I ( C ) e gr ( R ) ∼ = k [ x 1 ,..., x n ] is the coordinate ring of the tangent cone of C at 0. 4 I ( C ) ∗ Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 6 / 28

Introduction to the problem Monomial curves Example The cusp curve ψ : k [ x 1 , x 2 ] → k [ t ] x 1 �→ t 2 x 2 �→ t 3 S = � 2 , 3 � , I ( C ) = ( x 3 1 − x 2 2 ) gr ( R ) ∼ = k [ x 1 , x 2 ] ( x 2 2 ) Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 7 / 28

Introduction to the problem Questions Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 8 / 28

Introduction to the problem Questions Question (1) [Rossi’s conjecture] Is the Hilbert function of one-dimensional Gorenstein local rings non-decreasing? Answer: In general the problem is open. Question (2) Is the answer to the previous question affermative for rings associated to monomial curves? Partial answers: If gr ( R ) is Cohen-Macaulay, yes (A. Garcìa). Yes for some semigroups obtained by gluing (Arslan-Mete-Sahin, Jafari-Zarzuela Armengou). Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 9 / 28

Introduction to the problem Questions Question (3) Is the Hilbert function of rings associated to monomial curves non-decreasing for small embedding dimensions (e.g. edim = 3 , 4 , 5 )? Answer: edim = 3: Yes, more generally it is true for one-dimensional equicharacteristic rings (J. Elìas). edim = 4: Yes if the associated graded ring is Buchsbaum (Cortadellas Benitez-Jafari-Zarzuela Armengou). Open in general. edim = 5 , . . . , 9: The problem is totally open, the first counterexample is for edim = 10. Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 10 / 28

Some definitions and results Correspondences Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 11 / 28

Some definitions and results Correspondences v : k (( t )) → Z ∪ { 0 } � ∞ h = i r h t h , r h � = 0 �→ i Semigroup rings Semigroups R = k [[ t S ]] = k [[ t g 1 , . . . , t g n ]] → S = � g 1 , . . . , g n � m = ( t g 1 , . . . , t g n ) maximal ideal of R → M = S \ { 0 } maximal ideal of S m i → iM dim k m i / m i + 1 → | iM \ ( i + 1 ) M | R ′ = ∪ i ( m i : Q ( R ) m i ) blow-up of R S ′ = ∪ i ( iM − Z iM ) blow-up of S → R Gorenstein → S symmetric H R non-decreasing ⇔ | iM \ ( i + 1 ) M | ≤ | ( i + 1 ) M \ ( i + 2 ) M | , ∀ i Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 12 / 28

Some definitions and results Apéry-sets and numerical invariants of S Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 13 / 28

Some definitions and results Apéry-sets and numerical invariants of S Let S = � g 1 , . . . , g n � , where g 1 < . . . < g n are the generators of the minimal system of generators. Definition The Apéry-set of S is the set Ap ( S ) = { ω 0 , ω 1 , . . . , ω g 1 − 1 } , where ω i = min { s ∈ S | s ≡ i ( mod g 1 ) } . Similarly, one can define the Apéry-set for S ′ = � g 1 , g 2 − g 1 , . . . , g n − g 1 � Ap ( S ′ ) = { ω ′ 0 , ω ′ 1 , . . . , ω ′ g 1 − 1 } , i = min { s ′ ∈ S ′ | s ′ ≡ i ( mod where ω ′ g 1 ) } . Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 14 / 28

Some definitions and results Apéry-sets and numerical invariants of S Definition a i = the positive number such that ω i = ω ′ i + a i g 1 , i = 0 , 1 , . . . , g 1 − 1 b i = max { l | ω i ∈ lM } , i = 0 , 1 , . . . , g 1 − 1 In general a i ≥ b i for every i . Example R = Q [[ t 8 , t 9 , t 12 , t 13 , t 19 ]] S = � 8 , 9 , 12 , 13 , 19 � = { 0 , 8 , 9 , 12 , 13 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 24 , →} M \ 2 M = { 8 , 9 , 12 , 13 , 19 } 2 M \ 3 M = { 16 , 17 , 18 , 20 , 21 , 22 } 3 M \ 4 M = { 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 } 4 M \ 5 M = { 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 } reduction number = 4 Ap ( S ) = { 0 , 9 , 18 , 19 , 12 , 13 , 22 , 31 } Ap ( S ′ ) = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } a 0 = 0 , a 1 = 1 , a 2 = 2 , a 3 = 2 , a 4 = 1 , a 5 = 1 , a 6 = 2 , a 7 = 3 b 0 = 0 , b 1 = 1 , b 2 = 2 , b 3 = 1 , b 4 = 1 , b 5 = 1 , b 6 = 2 , b 7 = 3 Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 15 / 28

Some definitions and results Apéry-sets and numerical invariants of S Let R = k [[ t S ]] , where S = � g 1 , . . . , g n � , g 1 < g 2 < . . . < g n . Proposition (A. Garcìa) gr ( R ) is Cohen-Macaulay if and only if t g 1 is a regular element. Proposition (Barucci-Fröberg) gr ( R ) is Cohen-Macaulay if and only if a i = b i , for every i. Definition We call order of an element s ∈ S the integer i such that s ∈ iM \ ( i + 1 ) M, denoted by ord ( s ) ; we also say that s is on the i-th level. An element s skips the level when adding g 1 if ord ( s + g 1 ) > ord ( s ) + 1 . t g 1 is a zerodivisor in R ⇔ ∃ s ∈ S that skips the level when adding g 1 . Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 16 / 28

Some definitions and results Apéry-sets and numerical invariants of S Example S = � 8 , 9 , 12 , 13 , 19 � M \ 2 M = { 8 , 9 , 12 , 13 , 19 } 2 M \ 3 M = { 16 , 17 , 18 , 20 , 21 , 22 } 3 M \ 4 M = { 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 } 4 M \ 5 M = { 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 } reduction number = 4 19 skips the order when adding 8 ; 18 , 22 , 27 , 31 do not come from the previous level. Definition D i = { s ∈ ( i − 1 ) M \ iM : s + g 1 ∈ ( i + 1 ) M } , i ≥ 2 . C i = { s ∈ iM \ ( i + 1 ) M : s − g 1 / ∈ ( i − 1 ) M \ iM } , i ≥ 1 . H R is non-decreasing ⇔ | D i | ≤ | C i | , ∀ i ∈ { 2 , . . . , r } Example 19 ∈ D 2 ; 18 , 22 ∈ C 2 , 27 , 31 ∈ C 3 . Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 17 / 28

Our results Characterization of the skipping elements Indice Introduction to the problem 1 Hilbert function Monomial curves Questions Some definitions and results 2 Correspondences Apéry-sets and numerical invariants of S Our results 3 Characterization of the skipping elements The main theorem Applications Future goals Michela Di Marca On the Hilbert function of one-dimensional semigroup rings 18 / 28