Hilbert Function and Betti Numbers of Algebras with Lefschetz - PowerPoint PPT Presentation

Main result Theorem 1. If A is an Artinian homogeneous K -algebra with m-times the WLP, then h A is a m-times weak Lefschetz O-sequence. 2. For every m-times weak Lefschetz O-sequence h, there exists an Artinian homogeneous K -algebra with h A = h. 1. is easy: ◮ The unimodality follows from the natural grading of the algebra: if × ℓ 1 : A j − → A j +1 is surjective, then × ℓ 1 : A d − → A d +1 is surjective ∀ d ≥ j . ◮ A / ( ℓ 1 ) in an algebra with ( m − 1)-times the WLP and h A / ( ℓ 1 ) : 1 , h 1 − h 0 , . . . , h k − h k − 1 . 38

2. is not so easy.... 39

Plan 40

Plan Fix h a m -times weak Lefschetz O-sequence. 41

Plan Fix h a m -times weak Lefschetz O-sequence. 1. We will construct inductively an ideal W m ( h ) of R 42

Plan Fix h a m -times weak Lefschetz O-sequence. 1. We will construct inductively an ideal W m ( h ) of R such that R / W m ( h ) will be the algebra we are looking for. 43

Plan Fix h a m -times weak Lefschetz O-sequence. 1. We will construct inductively an ideal W m ( h ) of R such that R / W m ( h ) will be the algebra we are looking for. 2. If R / I is an Artinian K -algebra with Hilbert function h and m -times the WLP then: 44

Plan Fix h a m -times weak Lefschetz O-sequence. 1. We will construct inductively an ideal W m ( h ) of R such that R / W m ( h ) will be the algebra we are looking for. 2. If R / I is an Artinian K -algebra with Hilbert function h and m -times the WLP then: β ij ( R / I ) ≤ β ij ( R / W m ( h )) , ∀ i , j ≥ 0. 45

Plan 3. Let I ⊂ R be an ideal such that R / I has Hilbert function h and m -times the weak Lefschetz property ( m ∈ N ). 46

Plan 3. Let I ⊂ R be an ideal such that R / I has Hilbert function h and m -times the weak Lefschetz property ( m ∈ N ). T.F.A.E.: 47

Plan 3. Let I ⊂ R be an ideal such that R / I has Hilbert function h and m -times the weak Lefschetz property ( m ∈ N ). T.F.A.E.: (a) R / I has maximal Betti numbers among K -algebras with the above properties. 48

Plan 3. Let I ⊂ R be an ideal such that R / I has Hilbert function h and m -times the weak Lefschetz property ( m ∈ N ). T.F.A.E.: (a) R / I has maximal Betti numbers among K -algebras with the above properties. (b) I is componentwise linear and the ideal ρ n − m (Gin( I )) is Gotzmann in K [ x 1 , . . . , x n − m ]. 49

Plan 3. Let I ⊂ R be an ideal such that R / I has Hilbert function h and m -times the weak Lefschetz property ( m ∈ N ). T.F.A.E.: (a) R / I has maximal Betti numbers among K -algebras with the above properties. (b) I is componentwise linear and the ideal ρ n − m (Gin( I )) is Gotzmann in K [ x 1 , . . . , x n − m ]. where: ρ i : K [ x 1 , . . . , x n ] − → K [ x 1 , . . . , x i ], with: � x j if j ≤ i ρ i ( x j ) = 0 if j > i . 50

Plan 4. Let R / I be as above. 51

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: 52

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) 53

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 54

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 5. Construct, starting from W m ( h ) and using a distraction matrix, another ideal I with : 55

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 5. Construct, starting from W m ( h ) and using a distraction matrix, another ideal I with : - the same Hilbert function 56

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 5. Construct, starting from W m ( h ) and using a distraction matrix, another ideal I with : - the same Hilbert function - the same Betti numbers 57

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 5. Construct, starting from W m ( h ) and using a distraction matrix, another ideal I with : - the same Hilbert function - the same Betti numbers - such that R / I still has m -times the WLP 58

Plan 4. Let R / I be as above. If ∃ q ∈ N such that: β q ( R / I ) = β q ( R / W m ( h )) then: β i ( R / I ) = β i ( R / W m ( h )) for all i ≥ q . 5. Construct, starting from W m ( h ) and using a distraction matrix, another ideal I with : - the same Hilbert function - the same Betti numbers - such that R / I still has m -times the WLP - I ≤ k 1 is the ideal of finite set of rational points in P n − 1 . K 59

The construction of W m ( h ) 60

The construction of W m ( h ) For h : 1 = h 0 < h 1 < . . . < h k ≥ h k +1 ≥ . . . ≥ h s a m -times weak Lefschetz O-sequence, denote: 61

The construction of W m ( h ) For h : 1 = h 0 < h 1 < . . . < h k ≥ h k +1 ≥ . . . ≥ h s a m -times weak Lefschetz O-sequence, denote: ∆ h := 1 , h 1 − h 0 , . . . , h k − h k − 1 . 62

The construction of W m ( h ) For h : 1 = h 0 < h 1 < . . . < h k ≥ h k +1 ≥ . . . ≥ h s a m -times weak Lefschetz O-sequence, denote: ∆ h := 1 , h 1 − h 0 , . . . , h k − h k − 1 . Inductively, ∆ 1 h = ∆ h , 63

The construction of W m ( h ) For h : 1 = h 0 < h 1 < . . . < h k ≥ h k +1 ≥ . . . ≥ h s a m -times weak Lefschetz O-sequence, denote: ∆ h := 1 , h 1 − h 0 , . . . , h k − h k − 1 . Inductively, ∆ 1 h = ∆ h , ∆ i h := ∆(∆ i − 1 h ) for i = 2 , . . . , m 64

The construction of W m ( h ) For h : 1 = h 0 < h 1 < . . . < h k ≥ h k +1 ≥ . . . ≥ h s a m -times weak Lefschetz O-sequence, denote: ∆ h := 1 , h 1 − h 0 , . . . , h k − h k − 1 . Inductively, ∆ 1 h = ∆ h , ∆ i h := ∆(∆ i − 1 h ) for i = 2 , . . . , m ∆ i h is a ( m − i )-times weak Lefschetz O-sequence. 65

The construction of W m ( h ) 66

The construction of W m ( h ) The case m = 1 67

The construction of W m ( h ) The case m = 1 Set n = h 1 68

The construction of W m ( h ) The case m = 1 Set n = h 1 and define: I 0 :=Lex(∆ h ) ⊂ R ′ = K [ x 1 , . . . , x n − 1 ] . 69

The construction of W m ( h ) The case m = 1 Set n = h 1 and define: I 0 :=Lex(∆ h ) ⊂ R ′ = K [ x 1 , . . . , x n − 1 ] . I 1 := I 0 · R ⊂ R . 70

The construction of W m ( h ) The case m = 1 Set n = h 1 and define: I 0 :=Lex(∆ h ) ⊂ R ′ = K [ x 1 , . . . , x n − 1 ] . I 1 := I 0 · R ⊂ R . It is easy to see that: 71

The construction of W m ( h ) The case m = 1 Set n = h 1 and define: I 0 :=Lex(∆ h ) ⊂ R ′ = K [ x 1 , . . . , x n − 1 ] . I 1 := I 0 · R ⊂ R . It is easy to see that: - the Hilbert function of R / I 1 is: 1 = h 0 , h 1 , . . . , h k − 1 , h k , h k , . . . , h k , . . . 72

The construction of W m ( h ) The case m = 1 Set n = h 1 and define: I 0 :=Lex(∆ h ) ⊂ R ′ = K [ x 1 , . . . , x n − 1 ] . I 1 := I 0 · R ⊂ R . It is easy to see that: - the Hilbert function of R / I 1 is: 1 = h 0 , h 1 , . . . , h k − 1 , h k , h k , . . . , h k , . . . - ( x 1 , . . . , x n − 1 ) k +1 ⊆ I 1 73

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . 74

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . Let r 0 := h k − h d 0 . 75

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . Let r 0 := h k − h d 0 . Take M 1 , . . . , M r 0 ∈ R , the largest (in rev-lex order) r 0 monomials of degree d 0 NOT in I 1 . 76

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . Let r 0 := h k − h d 0 . Take M 1 , . . . , M r 0 ∈ R , the largest (in rev-lex order) r 0 monomials of degree d 0 NOT in I 1 . We define: I 2 := I 1 + ( M 1 , . . . , M r 0 ). 77

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . Let r 0 := h k − h d 0 . Take M 1 , . . . , M r 0 ∈ R , the largest (in rev-lex order) r 0 monomials of degree d 0 NOT in I 1 . We define: I 2 := I 1 + ( M 1 , . . . , M r 0 ). The Hilbert function of R / I 2 will be: 1 = h 0 , h 1 , . . . , h d 0 − 1 , h d 0 , h d 0 , . . . , h d 0 , . . . 78

The construction of W m ( h ) Let d 0 > k be the smallest degree for which h k > h d 0 . Let r 0 := h k − h d 0 . Take M 1 , . . . , M r 0 ∈ R , the largest (in rev-lex order) r 0 monomials of degree d 0 NOT in I 1 . We define: I 2 := I 1 + ( M 1 , . . . , M r 0 ). The Hilbert function of R / I 2 will be: 1 = h 0 , h 1 , . . . , h d 0 − 1 , h d 0 , h d 0 , . . . , h d 0 , . . . Technical proof... 79

The construction of W m ( h ) This ensures that we can proceed in the same way, that is by adding in each degree where it is needed the largest in rev-lex order monomials. 80

The construction of W m ( h ) This ensures that we can proceed in the same way, that is by adding in each degree where it is needed the largest in rev-lex order monomials. After at most s − k steps we will obtain an ideal W 1 ( h ) such that: h R / W 1 ( h ) = h . 81

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. 82

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. 83

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. 84

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R 85

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d 86

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : 87

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : z 3 t , y 2 t 2 , yzt 2 , z 2 t 2 , xt 3 , yt 3 , zt 3 , t 4 d = 4 : 88

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : z 3 t , y 2 t 2 , yzt 2 , z 2 t 2 , xt 3 , yt 3 , zt 3 , t 4 d = 4 : z 3 t 2 , y 2 t 3 , yzt 3 , z 2 t 3 , xt 4 , yt 4 , zt 4 , t 5 d = 5 : 89

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : z 3 t , y 2 t 2 , yzt 2 , z 2 t 2 , xt 3 , yt 3 , zt 3 , t 4 d = 4 : z 3 t 2 , y 2 t 3 , yzt 3 , z 2 t 3 , xt 4 , yt 4 , zt 4 , t 5 d = 5 : z 3 t 3 , y 2 t 4 , yzt 4 , z 2 t 4 , xt 5 , yt 5 , zt 5 , t 6 d = 6 : 90

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : z 3 t , y 2 t 2 , yzt 2 , z 2 t 2 , xt 3 , yt 3 , zt 3 , t 4 d = 4 : z 3 t 2 , y 2 t 3 , yzt 3 , z 2 t 3 , xt 4 , yt 4 , zt 4 , t 5 d = 5 : z 3 t 3 , y 2 t 4 , yzt 4 , z 2 t 4 , xt 5 , yt 5 , zt 5 , t 6 d = 6 : z 3 t 4 , y 2 t 5 , yzt 5 , z 2 t 5 , xt 6 , yt 6 , zt 6 , t 7 d = 7 : 91

Example Let h : 1 , 4 , 7 , 8 , 6 , 3 , 1 . Then we have ∆ h : 1 , 3 , 3 , 1. R = K [ x , y , z , t ]. I 0 := Lex(∆ h ) = ( x 2 , xy , xz , y 3 , y 2 z , yz 2 , z 4 ) ⊂ K [ x , y , z ]. I 1 = I 0 · R For d ≥ 4, the monomials in R \ I 1 are: z 3 t d − 3 , y 2 t d − 2 , yzt d − 2 , z 2 t d − 2 , xt d − 1 , yt d − 1 , zt d − 1 , t d Add to I 1 : z 3 t , y 2 t 2 , yzt 2 , z 2 t 2 , xt 3 , yt 3 , zt 3 , t 4 d = 4 : z 3 t 2 , y 2 t 3 , yzt 3 , z 2 t 3 , xt 4 , yt 4 , zt 4 , t 5 d = 5 : z 3 t 3 , y 2 t 4 , yzt 4 , z 2 t 4 , xt 5 , yt 5 , zt 5 , t 6 d = 6 : z 3 t 4 , y 2 t 5 , yzt 5 , z 2 t 5 , xt 6 , yt 6 , zt 6 , t 7 d = 7 : W 1 ( h ) = I 1 + ( z 3 t , y 2 t 2 , yzt 3 , z 2 t 3 , xt 4 , yt 5 , zt 5 , t 7 ). 92

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. 93

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. This is not difficult: 94

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. This is not difficult: - For the monomial generators in the first n − 1 variables 95

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. This is not difficult: - For the monomial generators in the first n − 1 variables it follows from the fact that Lex(∆ h ) is strongly stable. 96

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. This is not difficult: - For the monomial generators in the first n − 1 variables it follows from the fact that Lex(∆ h ) is strongly stable. - For the monomial generators divisible by x n 97

The construction of W m ( h ) In order to apply induction we also need to show that W 1 ( h ) is strongly stable. This is not difficult: - For the monomial generators in the first n − 1 variables it follows from the fact that Lex(∆ h ) is strongly stable. - For the monomial generators divisible by x n it follows from the fact that we chose the largest monomilas in rev-lex order as generators. 98

The construction of W m ( h ) The general case 99

The construction of W m ( h ) The general case Let m ∈ N , m ≥ 2. 100