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

Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Order m

ALEXANDRU CONSTANTINESCU

Dipartimento di Matematica, Universit` a di Genova

8-9 May 2008 Barcellona-Genova Workshop on Commutative Algebra and Applications

1

Main Goal:

◮ Characterize the Hilbert function of algebras with the

Lefschetz property m times.

2

Main Goal:

◮ Characterize the Hilbert function of algebras with the

Lefschetz property m times.

◮ Give upper bounds for the Betti numbers of Artinian algebras

with a given Hilbert Function and with the Lefschetz property m times

3

Main Goal:

◮ Characterize the Hilbert function of algebras with the

Lefschetz property m times.

◮ Give upper bounds for the Betti numbers of Artinian algebras

with a given Hilbert Function and with the Lefschetz property m times and describe the cases in which these bounds are reached.

4

Some results in this direction

5

Some results in this direction

• T. Harima, J.C. Migliore, U. Nagel, J. Watanabe

6

Some results in this direction

• T. Harima, J.C. Migliore, U. Nagel, J. Watanabe

The Weak and Strong Lefschetz Properties for Artinian K-algebras, Journal of Algebra 262 (2003), 99–126

7

Some results in this direction

• T. Harima, J.C. Migliore, U. Nagel, J. Watanabe

The Weak and Strong Lefschetz Properties for Artinian K-algebras, Journal of Algebra 262 (2003), 99–126

• T. Harima, A. Wachi

8

Some results in this direction

• T. Harima, J.C. Migliore, U. Nagel, J. Watanabe

The Weak and Strong Lefschetz Properties for Artinian K-algebras, Journal of Algebra 262 (2003), 99–126

• T. Harima, A. Wachi

Generic initial ideals, graded Betti numbers and k-Lefschetz properties, arXiv:0707.2247 (2007)

9

Notations and Definitions

10

Notations and Definitions

• Let K be an infinite field, char(K) = 0.

11

Notations and Definitions

• Let K be an infinite field, char(K) = 0.
• A =

d≥0 Ad be a homogeneous, Artinian K-algebra,

12

Notations and Definitions

• Let K be an infinite field, char(K) = 0.
• A =

d≥0 Ad be a homogeneous, Artinian K-algebra, i.e.

A = R/I, where R = K[x1, . . . , xn] and I is a homogeneous ideal.

13

Notations and Definitions

• Let K be an infinite field, char(K) = 0.
• A =

d≥0 Ad be a homogeneous, Artinian K-algebra, i.e.

A = R/I, where R = K[x1, . . . , xn] and I is a homogeneous ideal.

• hA - the Hilbert function of A

hd(A) = dimK(Ad).

14

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP)

15

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t.

16

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t. ×ℓ : Ad− → Ad+1 has maximal rank ∀ d ≥ 1.

17

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t. ×ℓ : Ad− → Ad+1 has maximal rank ∀ d ≥ 1. ℓ is called a weak Lefschetz Element (WLE) for A.

18

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t. ×ℓ : Ad− → Ad+1 has maximal rank ∀ d ≥ 1. ℓ is called a weak Lefschetz Element (WLE) for A. A has m-times the weak Lefschetz Property (m ∈ N)

19

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t. ×ℓ : Ad− → Ad+1 has maximal rank ∀ d ≥ 1. ℓ is called a weak Lefschetz Element (WLE) for A. A has m-times the weak Lefschetz Property (m ∈ N) if ∃ ℓ1, . . . , ℓm ∈ A1 s. t.

20

Notations and Definitions

Definition (WLP)

A has the weak Lefschetz Property (WLP) if ∃ ℓ ∈ A1 s. t. ×ℓ : Ad− → Ad+1 has maximal rank ∀ d ≥ 1. ℓ is called a weak Lefschetz Element (WLE) for A. A has m-times the weak Lefschetz Property (m ∈ N) if ∃ ℓ1, . . . , ℓm ∈ A1 s. t. ℓi is a WLE for A/(ℓ1, . . . , ℓi−1), ∀ i ∈ 1, . . . , m.

21

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

22

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

23

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗

24

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗ i.e. h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs for

some k ∈ 0, . . . , s.

25

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗ i.e. h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs for

some k ∈ 0, . . . , s.

• the sequence 1, h1 − h0, . . . , hk − hk−1 is again an O-sequence.

26

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗ i.e. h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs for

some k ∈ 0, . . . , s.

• the sequence 1, h1 − h0, . . . , hk − hk−1 is again an O-sequence.

h is a m-times weak Lefschetz O − sequence if:

27

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗ i.e. h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs for

some k ∈ 0, . . . , s.

• the sequence 1, h1 − h0, . . . , hk − hk−1 is again an O-sequence.

h is a m-times weak Lefschetz O − sequence if:

• h is unimodal∗ .

28

Notations and Definitions

Let h : 1 = h0, h1, . . . , hs be a finite O-sequence.

Definition (WL O-sequence)

h is a weak Lefschetz O − sequence if :

• h is unimodal∗ i.e. h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs for

some k ∈ 0, . . . , s.

• the sequence 1, h1 − h0, . . . , hk − hk−1 is again an O-sequence.

h is a m-times weak Lefschetz O − sequence if:

• h is unimodal∗ .
• the sequence 1, h1 − h0, . . . , hk − hk−1 is a (m − 1)-times

weak Lefschetz O-sequence.

29

Main result

30

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA is a m-times weak Lefschetz O-sequence.

31

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

32

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

33

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

◮ The unimodality follows from the natural grading of the

algebra:

34

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

◮ The unimodality follows from the natural grading of the

algebra: if ×ℓ1 : Aj − → Aj+1 is surjective,

35

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

◮ The unimodality follows from the natural grading of the

algebra: if ×ℓ1 : Aj − → Aj+1 is surjective, then ×ℓ1 : Ad − → Ad+1 is surjective ∀ d ≥ j.

36

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

◮ The unimodality follows from the natural grading of the

algebra: if ×ℓ1 : Aj − → Aj+1 is surjective, then ×ℓ1 : Ad − → Ad+1 is surjective ∀ d ≥ j.

◮ A/(ℓ1) in an algebra with (m − 1)-times the WLP and

37

Main result

Theorem

• 1. If A is an Artinian homogeneous K-algebra with m-times the

WLP, then hA 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 hA = h.

• 1. is easy:

◮ The unimodality follows from the natural grading of the

algebra: if ×ℓ1 : Aj − → Aj+1 is surjective, then ×ℓ1 : Ad − → Ad+1 is surjective ∀ d ≥ j.

◮ A/(ℓ1) in an algebra with (m − 1)-times the WLP and

hA/(ℓ1) : 1, h1 − h0, . . . , hk − hk−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 Wm(h) of R

42

Plan

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

• 1. We will construct inductively an ideal Wm(h) of R such that

R/Wm(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 Wm(h) of R such that

R/Wm(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 Wm(h) of R such that

R/Wm(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: βij(R/I) ≤ βij(R/Wm(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 ρn−m(Gin(I)) is Gotzmann in K[x1, . . . , xn−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 ρn−m(Gin(I)) is Gotzmann in K[x1, . . . , xn−m].

where: ρi : K[x1, . . . , xn] − → K[x1, . . . , xi], with: ρi(xj) = xj if j ≤ i 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:

βq(R/I) = βq(R/Wm(h))

53

Plan

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

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

54

Plan

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

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

• 5. Construct, starting from Wm(h) and using a distraction

matrix, another ideal I with :

55

Plan

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

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

• 5. Construct, starting from Wm(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:

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

• 5. Construct, starting from Wm(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:

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

• 5. Construct, starting from Wm(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:

βq(R/I) = βq(R/Wm(h)) then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

• 5. Construct, starting from Wm(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≤k1 is the ideal of finite set of rational points in Pn−1

K

.

59

The construction of Wm(h)

60

The construction of Wm(h)

For h : 1 = h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs a m-times weak Lefschetz O-sequence, denote:

61

The construction of Wm(h)

For h : 1 = h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs a m-times weak Lefschetz O-sequence, denote: ∆h := 1, h1 − h0, . . . , hk − hk−1.

62

The construction of Wm(h)

For h : 1 = h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs a m-times weak Lefschetz O-sequence, denote: ∆h := 1, h1 − h0, . . . , hk − hk−1. Inductively, ∆1h = ∆h,

63

The construction of Wm(h)

For h : 1 = h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs a m-times weak Lefschetz O-sequence, denote: ∆h := 1, h1 − h0, . . . , hk − hk−1. Inductively, ∆1h = ∆h, ∆ih := ∆(∆i−1h) for i = 2, . . . , m

64

The construction of Wm(h)

For h : 1 = h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs a m-times weak Lefschetz O-sequence, denote: ∆h := 1, h1 − h0, . . . , hk − hk−1. Inductively, ∆1h = ∆h, ∆ih := ∆(∆i−1h) for i = 2, . . . , m ∆ih is a (m − i)-times weak Lefschetz O-sequence.

65

The construction of Wm(h)

66

The construction of Wm(h)

The case m = 1

67

The construction of Wm(h)

The case m = 1 Set n = h1

68

The construction of Wm(h)

The case m = 1 Set n = h1 and define: I0 :=Lex(∆h) ⊂ R′ = K[x1, . . . , xn−1] .

69

The construction of Wm(h)

The case m = 1 Set n = h1 and define: I0 :=Lex(∆h) ⊂ R′ = K[x1, . . . , xn−1] . I1 := I0 · R ⊂ R.

70

The construction of Wm(h)

The case m = 1 Set n = h1 and define: I0 :=Lex(∆h) ⊂ R′ = K[x1, . . . , xn−1] . I1 := I0 · R ⊂ R. It is easy to see that:

71

The construction of Wm(h)

The case m = 1 Set n = h1 and define: I0 :=Lex(∆h) ⊂ R′ = K[x1, . . . , xn−1] . I1 := I0 · R ⊂ R. It is easy to see that:

• the Hilbert function of R/I1 is:

1 = h0, h1, . . . , hk−1, hk, hk, . . . , hk, . . .

72

The construction of Wm(h)

The case m = 1 Set n = h1 and define: I0 :=Lex(∆h) ⊂ R′ = K[x1, . . . , xn−1] . I1 := I0 · R ⊂ R. It is easy to see that:

• the Hilbert function of R/I1 is:

1 = h0, h1, . . . , hk−1, hk, hk, . . . , hk, . . .

• (x1, . . . , xn−1)k+1 ⊆ I1

73

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0.

74

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0. Let r0 := hk − hd0.

75

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0. Let r0 := hk − hd0. Take M1 , . . . , Mr0 ∈ R, the largest (in rev-lex order) r0 monomials of degree d0 NOT in I1.

76

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0. Let r0 := hk − hd0. Take M1 , . . . , Mr0 ∈ R, the largest (in rev-lex order) r0 monomials of degree d0 NOT in I1. We define: I2 := I1 + (M1 , . . . , Mr0).

77

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0. Let r0 := hk − hd0. Take M1 , . . . , Mr0 ∈ R, the largest (in rev-lex order) r0 monomials of degree d0 NOT in I1. We define: I2 := I1 + (M1 , . . . , Mr0). The Hilbert function of R/I2 will be: 1 = h0, h1, . . . , hd0−1, hd0, hd0, . . . , hd0, . . .

78

The construction of Wm(h)

Let d0 > k be the smallest degree for which hk > hd0. Let r0 := hk − hd0. Take M1 , . . . , Mr0 ∈ R, the largest (in rev-lex order) r0 monomials of degree d0 NOT in I1. We define: I2 := I1 + (M1 , . . . , Mr0). The Hilbert function of R/I2 will be: 1 = h0, h1, . . . , hd0−1, hd0, hd0, . . . , hd0, . . . Technical proof...

79

The construction of Wm(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 Wm(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 W1(h) such that: hR/W1(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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ 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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · 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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1:

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1: d = 4 : z3t, y2t2, yzt2, z2t2, xt3, yt3, zt3, t4

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1: d = 4 : z3t, y2t2, yzt2, z2t2, xt3, yt3, zt3, t4 d = 5 : z3t2, y2t3, yzt3, z2t3, xt4, yt4, zt4, t5

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1: d = 4 : z3t, y2t2, yzt2, z2t2, xt3, yt3, zt3, t4 d = 5 : z3t2, y2t3, yzt3, z2t3, xt4, yt4, zt4, t5 d = 6 : z3t3, y2t4, yzt4, z2t4, xt5, yt5, zt5, t6

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1: d = 4 : z3t, y2t2, yzt2, z2t2, xt3, yt3, zt3, t4 d = 5 : z3t2, y2t3, yzt3, z2t3, xt4, yt4, zt4, t5 d = 6 : z3t3, y2t4, yzt4, z2t4, xt5, yt5, zt5, t6 d = 7 : z3t4, y2t5, yzt5, z2t5, xt6, yt6, zt6, t7

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]. I0 := Lex(∆h) = (x2, xy, xz, y3, y2z, yz2, z4) ⊂ K[x, y, z]. I1 = I0 · R For d ≥ 4, the monomials in R \ I1 are: z3td−3, y2td−2, yztd−2, z2td−2, xtd−1, ytd−1, ztd−1, td Add to I1: d = 4 : z3t, y2t2, yzt2, z2t2, xt3, yt3, zt3, t4 d = 5 : z3t2, y2t3, yzt3, z2t3, xt4, yt4, zt4, t5 d = 6 : z3t3, y2t4, yzt4, z2t4, xt5, yt5, zt5, t6 d = 7 : z3t4, y2t5, yzt5, z2t5, xt6, yt6, zt6, t7 W1(h) = I1 + (z3t, y2t2, yzt3, z2t3, xt4, yt5, zt5, t7).

92

The construction of Wm(h)

In order to apply induction we also need to show that W1(h) is strongly stable.

93

The construction of Wm(h)

In order to apply induction we also need to show that W1(h) is strongly stable. This is not difficult:

94

The construction of Wm(h)

In order to apply induction we also need to show that W1(h) is strongly stable. This is not difficult:

• For the monomial generators in the first n − 1 variables

95

The construction of Wm(h)

In order to apply induction we also need to show that W1(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 Wm(h)

In order to apply induction we also need to show that W1(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 xn

97

The construction of Wm(h)

In order to apply induction we also need to show that W1(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 xn it follows from

the fact that we chose the largest monomilas in rev-lex order as generators.

98

The construction of Wm(h)

The general case

99

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2.

100

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

101

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,

102

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].

103

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

104

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih.

105

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih. To construct Wm(h)

106

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih. To construct Wm(h) take I1 := Wm−1(∆h) · R and

107

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih. To construct Wm(h) take I1 := Wm−1(∆h) · R and follow the same steps as in the case m = 1.

108

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih. To construct Wm(h) take I1 := Wm−1(∆h) · R and follow the same steps as in the case m = 1. Everything works because we only used the fact that Lex(∆h) is strongly stable.

109

The construction of Wm(h)

The general case Let m ∈ N, m≥ 2. Assume we can construct an algebra R′/Wm−1(∆h), with:

• Hilbert function ∆h,
• Wm−1(∆h) is a strongly stable ideal of R′ = K[x1, . . . , xn−1].
• (x1, . . . , xn−i)ki+1 ⊆ Wm−1(∆h) for all i = 2, . . . , m − 1,

where ki is the length of ∆ih. To construct Wm(h) take I1 := Wm−1(∆h) · R and follow the same steps as in the case m = 1. Everything works because we only used the fact that Lex(∆h) is strongly stable. The choice of Lex(∆h) as a starting point is useful for obtaining maximal Betti numbers.

110

R/Wm(h) has the WLP m times

111

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

112

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.

113

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

114

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs,

115

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs, (b) (x1, . . . , xn−1)k+1 ⊆ I,

116

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs, (b) (x1, . . . , xn−1)k+1 ⊆ I, (c) If M ∈ Gens(I) is divisible by xn, then deg(M) ≥ k + 1.

117

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs, (b) (x1, . . . , xn−1)k+1 ⊆ I, (c) If M ∈ Gens(I) is divisible by xn, then deg(M) ≥ k + 1.

Our construction satisfies (a), (b) and (c), so it has the WLP.

118

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs, (b) (x1, . . . , xn−1)k+1 ⊆ I, (c) If M ∈ Gens(I) is divisible by xn, then deg(M) ≥ k + 1.

Our construction satisfies (a), (b) and (c), so it has the WLP. It is also easy to check that xn is a WLE for R/Wm(h).

119

R/Wm(h) has the WLP m times

Lemma

If I is a strongly stable ideal of R = K[x1, . . . , xn], then T.F.A.E:

• 1. R/I has the WLP.
• 2. The following three conditions hold:

(a) hR/I is unimodal∗ : h0 < h1 < . . . < hk ≥ hk+1 ≥ . . . ≥ hs, (b) (x1, . . . , xn−1)k+1 ⊆ I, (c) If M ∈ Gens(I) is divisible by xn, then deg(M) ≥ k + 1.

Our construction satisfies (a), (b) and (c), so it has the WLP. It is also easy to check that xn is a WLE for R/Wm(h). As R/Wm(h) + (xn) = R′/Wm−1(∆h), which by induction has the WLP (m −1) times, we have that R/Wm(h) has m-times the WLP.

120

R/Wm(h) has the WLP m times

The proof of the Lemma is based on tahe following result:

121

R/Wm(h) has the WLP m times

The proof of the Lemma is based on tahe following result:

Lemma (A. Wiebe)

If I is a strongly stable ideal of R = K[x1, . . . , xn] then:

122

R/Wm(h) has the WLP m times

The proof of the Lemma is based on tahe following result:

Lemma (A. Wiebe)

If I is a strongly stable ideal of R = K[x1, . . . , xn] then: R/I has the WLP ⇐ ⇒ xn is a WLE for R/I.

123

R/Wm(h) has maximal Betti numbers

124

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then:

125

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0.

126

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts:

127

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts: 1. ρn−m(Wm(h)) = Lex(∆m(h)).

128

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts: 1. ρn−m(Wm(h)) = Lex(∆m(h)).

• 2. Because
• βij(R/I) ≤ βij(R/Gin(I)),

∀i, j ≥ 0,

129

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts: 1. ρn−m(Wm(h)) = Lex(∆m(h)).

• 2. Because
• βij(R/I) ≤ βij(R/Gin(I)),

∀i, j ≥ 0,

• Gin(I) is a strongly stable ideal in characteristic 0,

130

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts: 1. ρn−m(Wm(h)) = Lex(∆m(h)).

• 2. Because
• βij(R/I) ≤ βij(R/Gin(I)),

∀i, j ≥ 0,

• Gin(I) is a strongly stable ideal in characteristic 0,
• R/I has the m-WLP ⇔ R/Gin(I) has the m-WLP,

131

R/Wm(h) has maximal Betti numbers

Proposition

If R/I is an Artinian K-algebra with Hilbert function h and m-times the WLP then: βij(R/I) ≤ βij(R/Wm(h)) , ∀i, j ≥ 0. To prove this we mainly use three facts: 1. ρn−m(Wm(h)) = Lex(∆m(h)).

• 2. Because
• βij(R/I) ≤ βij(R/Gin(I)),

∀i, j ≥ 0,

• Gin(I) is a strongly stable ideal in characteristic 0,
• R/I has the m-WLP ⇔ R/Gin(I) has the m-WLP,

we can restrict to the case when I is strongly stable.

132

R/Wm(h) has maximal Betti numbers

• 3. Proposition (A.M. Bigatti)

Let I, J be strongly stable ideals with the same Hilbert

• function. Assume that m≤i(Ij) ≤ m≤i(Jj),

∀i, j ≥ 0. Then

• ne has:

133

R/Wm(h) has maximal Betti numbers

• 3. Proposition (A.M. Bigatti)

Let I, J be strongly stable ideals with the same Hilbert

• function. Assume that m≤i(Ij) ≤ m≤i(Jj),

∀i, j ≥ 0. Then

• ne has:

(a) mi(J) ≤ mi(I), ∀ i > 0.

134

R/Wm(h) has maximal Betti numbers

• 3. Proposition (A.M. Bigatti)

Let I, J be strongly stable ideals with the same Hilbert

• function. Assume that m≤i(Ij) ≤ m≤i(Jj),

∀i, j ≥ 0. Then

• ne has:

(a) mi(J) ≤ mi(I), ∀ i > 0. (b) βij(R/J) ≤ βij(R/I), ∀ i, j ≥ 0.

135

R/Wm(h) has maximal Betti numbers

• 3. Proposition (A.M. Bigatti)

Let I, J be strongly stable ideals with the same Hilbert

• function. Assume that m≤i(Ij) ≤ m≤i(Jj),

∀i, j ≥ 0. Then

• ne has:

(a) mi(J) ≤ mi(I), ∀ i > 0. (b) βij(R/J) ≤ βij(R/I), ∀ i, j ≥ 0.

For a monomial M = xa1

1 . . . xan n in K[x1, . . . , xn] we define:

max(M) = max{i : ai > 0}.

136

R/Wm(h) has maximal Betti numbers

• 3. Proposition (A.M. Bigatti)

Let I, J be strongly stable ideals with the same Hilbert

• function. Assume that m≤i(Ij) ≤ m≤i(Jj),

∀i, j ≥ 0. Then

• ne has:

(a) mi(J) ≤ mi(I), ∀ i > 0. (b) βij(R/J) ≤ βij(R/I), ∀ i, j ≥ 0.

For a monomial M = xa1

1 . . . xan n in K[x1, . . . , xn] we define:

max(M) = max{i : ai > 0}. For a set of monomials A ⊂ K[x1, . . . , xn] and for i = 1, . . . , n we write: mi(A) = ♯{M ∈ A : max(M) = i}, m≤i(A) = ♯{M ∈ A : max(M) ≤ i}.

137

All ideals with maximal Betti numbers

138

All ideals with maximal Betti numbers

Proposition

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 ideals with the above properties. (b) I is componentwise linear and the ideal ρn−m(Gin(I)) is Gotzmann in K[x1, . . . , xn−m].

139

All ideals with maximal Betti numbers

Proposition

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 ideals with the above properties. (b) I is componentwise linear and the ideal ρn−m(Gin(I)) is Gotzmann in K[x1, . . . , xn−m]. I is Componentwise linear ⇔ βij(R/I) = βij(R/Gin(I)), ∀i, j ≥ 0.

140

All ideals with maximal Betti numbers

Proposition

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 ideals with the above properties. (b) I is componentwise linear and the ideal ρn−m(Gin(I)) is Gotzmann in K[x1, . . . , xn−m]. I is Componentwise linear ⇔ βij(R/I) = βij(R/Gin(I)), ∀i, j ≥ 0. I is Gotzmann ⇔ βij(R/I) = βij(R/ Lex(I)), ∀i, j ≥ 0.

141

Rigidity

Proposition

Let R/I be as above. If ∃ q ∈ N such that: βq(R/I) = βq(R/Wm(h)) , then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q.

142

Rigidity

Proposition

Let R/I be as above. If ∃ q ∈ N such that: βq(R/I) = βq(R/Wm(h)) , then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q. In a general context, this true if we replace Wm(h) by Gin(I) or by Lex(I).

143

Rigidity

Proposition

Let R/I be as above. If ∃ q ∈ N such that: βq(R/I) = βq(R/Wm(h)) , then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q. In a general context, this true if we replace Wm(h) by Gin(I) or by Lex(I). The proposition follows from the Eliahou-Kervaire formula: βi(R/I) = n

s=i ms(I)

s−1

i−1

• 144

Rigidity

Proposition

Let R/I be as above. If ∃ q ∈ N such that: βq(R/I) = βq(R/Wm(h)) , then: βi(R/I) = βi(R/Wm(h)) for all i ≥ q. In a general context, this true if we replace Wm(h) by Gin(I) or by Lex(I). The proposition follows from the Eliahou-Kervaire formula: βi(R/I) = n

s=i ms(I)

s−1

i−1

• and from: mj(I) ≤ mj(Wm(h)), ∀ j > 0 .

145