Betti tables of ideals in a polynomial ring

Matteo Varbaro

Dipartimento di Matematica, Universit` a di Genova

From a joint work J¨ urgen Herzog and Leila Sharifan

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

F r e e r e s o l u t i o n s

Let S = K [x ₀ , . . . , x _n ] be a polynomial ring over a field K and I ⊂ S a graded ideal. The minimal graded free resolution of I is:

0 → M

j ∈N

S (−j )

→ . . . → M

j ∈N

S (−j )

→ M

j ∈N

S (−j )

→ I → 0.

The invariants β _{i ,j} (I ) = β _{i ,j} are the graded Betti numbers of I .

If X ⊂ P ⁿ is a projective scheme, we will refer to its free resolution

(and related concepts) as the one of the saturated ideal defining it.

B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i t a b l e s

The Betti table of I is the matrix (β i ,i +d (I )) i ,d . It can be thought as a (n + 1) × reg(I ) matrix: For example, if

I = (x 0 x 1 , x 1 x 2 , x 2 x 3 , x ₃ ² ) ⊂ S = K [x 0 , . . . , x 3 ], the resolution of I is:

0 → S (−4) → S (−3) ³ ⊕ S(−4) → S(−2) ⁴ → I → 0 Therefore its Betti table is:





0 0 0 0

4 3 0 0

0 1 1 0





I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

I d e a l s w i t h l i n e a r r e s o l u t i o n

The ideal I is said to have d -linear resolution if all its minimal generators are of degree d and reg(I ) = d . Equivalently, if the Betti tables of I has only one nonzero row, the d th.

For example, the rational normal curve in P ⁴ has Betti table:

 0 0 0 0 0

6 8 3 0 0



thus it has 2-linear resolution. Indeed it is known that all varieties of minimal degree have a 2-linear resolution. More generally:

(Bruns, Conca, -). If I defines a variety of minimal degree,

then I ^k has 2k-linear resolution for each k.

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

Ex t r e m a l B e t t i n u m b e r s

A graded Betti number β i ,i +d of I is said extremal if β i ,i +d (I ) 6= 0 and β _h,h+k (I ) = 0 for all (h, k) 6= (i , d ) with h ≥ i and k ≥ d . For example, let us look at the Betti table of

I = (x ₀ ⁴ , x ₀ ³ x ₁ , x ₀ ³ x ₂ , x ₀ ³ x ₃ , x ₀ ² x ₁ ³ , x ₀ x ₁ ⁴ , x ₀ ² x ₁ ² x ₂ , x ₀ x ₁ ³ x ₂ , x ₁ ⁶ )

















0 0 0 0

0 0 0 0

0 0 0 0

4 6 4 1

4 6 2 0

1 1 0 0

















The extremal Betti numbers of I are those marked in red.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I (s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I

(s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I (s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I

(s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I (s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I

(s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

C o h o m o l o g i c a l i n t e r p r e t a t i o n

Let X ⊂ P ⁿ be a projective scheme and I _X its ideal sheaf. Then β _{i ,i +d} is an extremal Betti number of L

m∈Z Γ(X , I _X (m)) ⊂ S iff:

(i) i < n;

(ii) dim

(H

(X , I

(q − p))) = β

6= 0 for p = n − i and q = d − 1;

(iii) H

(X , I (s − r )) = 0 for all (r , s) 6= (p, q), 1 ≤ r ≤ p and s ≥ q.

G o a l s o f t h e t a l k

We will explain how to give a numerical characterization of:

(i) The Betti tables of ideals I ⊂ S with d -linear resolution.

(ii) The extremal Betti numbers of any graded ideal I ⊂ S .

For simplicity, we will assume that the characteristic of K is 0.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

G o a l s o f t h e t a l k

We will explain how to give a numerical characterization of:

(i) The Betti tables of ideals I ⊂ S with d -linear resolution.

(ii) The extremal Betti numbers of any graded ideal I ⊂ S .

For simplicity, we will assume that the characteristic of K is 0.

G o a l s o f t h e t a l k

We will explain how to give a numerical characterization of:

(i) The Betti tables of ideals I ⊂ S with d -linear resolution.

(ii) The extremal Betti numbers of any graded ideal I ⊂ S .

For simplicity, we will assume that the characteristic of K is 0.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

G o a l s o f t h e t a l k

We will explain how to give a numerical characterization of:

(i) The Betti tables of ideals I ⊂ S with d -linear resolution.

(ii) The extremal Betti numbers of any graded ideal I ⊂ S .

For simplicity, we will assume that the characteristic of K is 0.

G o a l s o f t h e t a l k

We will explain how to give a numerical characterization of:

(i) The Betti tables of ideals I ⊂ S with d -linear resolution.

(ii) The extremal Betti numbers of any graded ideal I ⊂ S .

For simplicity, we will assume that the characteristic of K is 0.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

R e d u c t i o n t o B o r e l - f i x e d i d e a l s

If I has d -linear resolution, then it has the same Betti table of its generic initial ideal Gin(I ) (Aramova, Herzog, Hibi).

Gin(I ) is strongly stable ( Gin(I ) : x i ⊂ Gin(I ) : x _j ∀ j < i ).

Viceversa, any strongly stable ideal generated in degree d has d -linear resolution (Elihaou, Kervaire).

So we are allowed to focus on the Betti tables of strongly stable

monomials ideals J ⊂ S generated in degree d .

R e d u c t i o n t o B o r e l - f i x e d i d e a l s

If I has d -linear resolution, then it has the same Betti table of its generic initial ideal Gin(I ) (Aramova, Herzog, Hibi).

Gin(I ) is strongly stable ( Gin(I ) : x i ⊂ Gin(I ) : x _j ∀ j < i ).

Viceversa, any strongly stable ideal generated in degree d has d -linear resolution (Elihaou, Kervaire).

So we are allowed to focus on the Betti tables of strongly stable

monomials ideals J ⊂ S generated in degree d .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

R e d u c t i o n t o B o r e l - f i x e d i d e a l s

If I has d -linear resolution, then it has the same Betti table of its generic initial ideal Gin(I ) (Aramova, Herzog, Hibi).

Gin(I ) is strongly stable ( Gin(I ) : x i ⊂ Gin(I ) : x _j ∀ j < i ).

Viceversa, any strongly stable ideal generated in degree d has d -linear resolution (Elihaou, Kervaire).

So we are allowed to focus on the Betti tables of strongly stable

monomials ideals J ⊂ S generated in degree d .

R e d u c t i o n t o B o r e l - f i x e d i d e a l s

If I has d -linear resolution, then it has the same Betti table of its generic initial ideal Gin(I ) (Aramova, Herzog, Hibi).

Gin(I ) is strongly stable ( Gin(I ) : x i ⊂ Gin(I ) : x _j ∀ j < i ).

Viceversa, any strongly stable ideal generated in degree d has d -linear resolution (Elihaou, Kervaire).

So we are allowed to focus on the Betti tables of strongly stable

monomials ideals J ⊂ S generated in degree d .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

R e d u c t i o n t o B o r e l - f i x e d i d e a l s

If I has d -linear resolution, then it has the same Betti table of its generic initial ideal Gin(I ) (Aramova, Herzog, Hibi).

Gin(I ) is strongly stable ( Gin(I ) : x i ⊂ Gin(I ) : x _j ∀ j < i ).

Viceversa, any strongly stable ideal generated in degree d has d -linear resolution (Elihaou, Kervaire).

So we are allowed to focus on the Betti tables of strongly stable

monomials ideals J ⊂ S generated in degree d .

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m _k (J) = |{u ∈ G(I ) : m(u) = k}|

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m (J) = |{u ∈ G(I ) : m(u) = k}|

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m _k (J) = |{u ∈ G(I ) : m(u) = k}|

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m (J) = |{u ∈ G(I ) : m(u) = k}|

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m _k (J) = |{u ∈ G(I ) : m(u) = k}|

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m (J) = |{u ∈ G(I ) : m(u) = k}|

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m _k (J) = |{u ∈ G(I ) : m(u) = k}|

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m (J) = |{u ∈ G(I ) : m(u) = k}|

T h e E l i h a o u - K e r v a i r e f o r m u l a

Given an ideal I ⊂ S = K [x ₀ , . . . , x _n ], we define:

n

X

k=0

m k (I )t ^k =

n

X

i =0

β i (I )(t − 1) ⁱ .

Obviously to characterize the possible Betti tables of ideals with linear resolution we can characterize the possible sequences (m 0 (I ), m 1 (I ), . . . , m n (I )).

For a monomial u ∈ S , let us set m(u) = max{i : x _i |u}. Elihaou and Kervaire showed that, if J ⊂ S is strongly stable, then:

m _k (J) = |{u ∈ G(I ) : m(u) = k}|

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x x x x ∗ x x x x = x x x x .

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x 0 x 1 x 1 x 3 ∗ x ₂ x 2 x 3 x 3 = x 2 x 3 x 4 x 6 .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x x x x ∗ x x x x = x x x x .

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x 0 x 1 x 1 x 3 ∗ x ₂ x 2 x 3 x 3 = x 2 x 3 x 4 x 6 .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x x x x ∗ x x x x = x x x x .

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x 0 x 1 x 1 x 3 ∗ x ₂ x 2 x 3 x 3 = x 2 x 3 x 4 x 6 .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x x x x ∗ x x x x = x x x x .

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x 0 x 1 x 1 x 3 ∗ x ₂ x 2 x 3 x 3 = x 2 x 3 x 4 x 6 .

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

A m u l t i p l i c a t i v e s t r u c t u r e o n S

Given two monomials u, v ⊂ S d , write them as u = x i

· · · x _i

and x _j

· · · x _j

with i ₁ ≤ . . . ≤ i _d and j ₁ ≤ . . . ≤ j _d , and define:

u ∗ v =

( x i

+j

x i

+j

· · · x _i

_+j

if i d + j d ≤ n,

0 otherwise

We extend the operation to the whole S _d by K -linearity, and denote by S _d the gotten K -algebra. For example, if d = 4, n ≥ 6, u = x 0 x ₁ ² x 3 and v = x ₂ ² x ₃ ² , then:

u ∗ v = x x x x ∗ x x x x = x x x x .

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m 0 (J), m 1 (J), . . . , m n (J)) satisfies Macaulay’s conditions.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m (J), m (J), . . . , m (J)) satisfies Macaulay’s conditions.

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m 0 (J), m 1 (J), . . . , m n (J)) satisfies Macaulay’s conditions.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m (J), m (J), . . . , m (J)) satisfies Macaulay’s conditions.

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m 0 (J), m 1 (J), . . . , m n (J)) satisfies Macaulay’s conditions.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m (J), m (J), . . . , m (J)) satisfies Macaulay’s conditions.

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m 0 (J), m 1 (J), . . . , m n (J)) satisfies Macaulay’s conditions.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m (J), m (J), . . . , m (J)) satisfies Macaulay’s conditions.

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m 0 (J), m 1 (J), . . . , m n (J)) satisfies Macaulay’s conditions.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

T h e r i n g S

Notice that S _d has a natural N-grading, namely S d = L n

k=0 (S _d ) _k , (S _d ) _k = hu ∈ S _d : m(u) = ki.

One can show that there is a graded isomorphism of K -algebras:

S _d ∼ = K [y ₁ , . . . , y _d ] (y ₁ , . . . , y _d ) ⁿ⁺¹ .

We showed that, if J ⊂ S is a strongly stable monomial ideal, then G(J) = (G(J), ∗) is a quotient of S _d . The Hilbert function of G(J) is:

dim _K (G(J) _k ) = m _k (J),

so (m (J), m (J), . . . , m (J)) satisfies Macaulay’s conditions.

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

B e t t i n u m b e r s o f i d e a l s w i t h l i n e a r r e s o l u t i o n

Indeed we can show that, for a sequence (m ₀ , . . . , m _n ), TFAE:

(i) There exists an ideal I ⊂ S with d -linear resolution such that m _k (I ) = m _k for all k = 0, . . . , n.

(ii) There exists a strongly stable monomial ideal J ⊂ S generated in degree d such that m k (J) = m k for all k = 0, . . . , n.

(iii) There exists a standard graded K -algebra A with dim _K A 1 ≤ d and dim _K A _k = m _k for all k = 0, . . . , n.

(iv) m 0 = 1, m 1 ≤ d , m _{i +1} ≤ m _i ^{hi i} for all i = 1, . . . , n − 1.

The same result has been shown, with a different proof, by Murai.

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

E x a m p l e

We will never find an ideal I ⊂ S with the below minimal free resolution:

0 → S (−6) ⁶ → S(−5) ²² → S(−4) ²⁹ → S(−3) ¹⁴ → I → 0.

Indeed we would have m

(I ) = 1, m

(I ) = 3, m

(I ) = 4 and m

(I ) = 6,

but m ₂ (I ) ^h2i = 4 ^h2i = 5 < 6 = m ₃ (I ).

E x a m p l e

We will never find an ideal I ⊂ S with the below minimal free resolution:

0 → S (−6) ⁶ → S(−5) ²² → S(−4) ²⁹ → S(−3) ¹⁴ → I → 0.

Indeed we would have m

(I ) = 1, m

(I ) = 3, m

(I ) = 4 and m

(I ) = 6,

but m ₂ (I ) ^h2i = 4 ^h2i = 5 < 6 = m ₃ (I ).

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

E x a m p l e

We will never find an ideal I ⊂ S with the below minimal free resolution:

0 → S (−6) ⁶ → S(−5) ²² → S(−4) ²⁹ → S(−3) ¹⁴ → I → 0.

Indeed we would have m

(I ) = 1, m

(I ) = 3, m

(I ) = 4 and m

(I ) = 6,

but m ₂ (I ) ^h2i = 4 ^h2i = 5 < 6 = m ₃ (I ).

E x a m p l e

We will never find an ideal I ⊂ S with the below minimal free resolution:

0 → S (−6) ⁶ → S(−5) ²² → S(−4) ²⁹ → S(−3) ¹⁴ → I → 0.

Indeed we would have m

(I ) = 1, m

(I ) = 3, m

(I ) = 4 and m

(I ) = 6,

but m ₂ (I ) ^h2i = 4 ^h2i = 5 < 6 = m ₃ (I ).

BETTI TABLES OF IDEALS IN A POLYNOMIAL RING

E x a m p l e

We will never find an ideal I ⊂ S with the below minimal free resolution:

0 → S (−6) ⁶ → S(−5) ²² → S(−4) ²⁹ → S(−3) ¹⁴ → I → 0.

Indeed we would have m

(I ) = 1, m

(I ) = 3, m

(I ) = 4 and m

(I ) = 6,

but m ₂ (I ) ^h2i = 4 ^h2i = 5 < 6 = m ₃ (I ).

C o m p o n e n t w i s e l i n e a r i d e a l s

Our initial dream was to characterize the possible Betti tables of componentwise linear ideals I ⊂ S, that is such that I hmi has m-linear resolution for all m, where I _hmi = (f ∈ I : deg(f ) = m).

The interest in this comes from the fact that the generic initial ideal of every homogeneous ideal is componentwise linear.

Our characterization of the Betti tables of ideals with linear

resolution gives some necessary conditions that a Betti table of a

componentwise linear ideal must satisfy. We show that these

conditions are also sufficient up to three variables. Unfortunately,

this is no longer true when the number of variables is bigger.