the ratliff-rush operation on monomial idealsimns2010/porto2015/rr3dim.pdf · the ratli -rush...

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation on monomial ideals

Veronica Crispin Quinonez

Department of MathematicsUppsala University

June 10, 2015

1 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞

2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Definition and properties

R Noetherian ring and I ⊂ R regular ideal

(I `+1 : I `)`≥0 = ({r ∈ R | rI ` ⊆ I `+1})`≥0 increases with `

I =⋃∞`≥1(I `+1 : I `) studied by Ratliff and Rush (1978)

1 (I )` = I ` for `�∞2 I maximal with this property

3˜I = I

4 (I j) = I j for j �∞

I Ratliff-Rush ideal associated to I

I = I Ratliff-Rush ideal

r(I ) = min{` | I = I `+1 : I `} RR reduction number

2 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Why study RR?

(I `) = I ` for all ` ≥ 1⇔ depth of the associated graded ringGI (R) = ⊕n≥0I

n/I n+1 > 0 (Heinzer et al., 1992)

I regular m-primary ⊂ (R,m)⇒ I is the unique largest idealcontaining I with the same Hilbert polynomial as I (Heinzeret al., 1993)

I is coherent ⊂ (R,m)⇒ the tangent vector fields thatpreserve the Ratliff-Rush operation are liftable (Kallstrom,2009)

I m-primary ⊂ (R,m) Cohen-Macaulay

(I `+1) ⊆ I ` for all ` ≥ 1⇒ the 0-th Bockstein cohomology ofGI (R) vanishes (Puthenpurakal, 2012)

3 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Why study RR?

(I `) = I ` for all ` ≥ 1⇔ depth of the associated graded ringGI (R) = ⊕n≥0I

n/I n+1 > 0 (Heinzer et al., 1992)

I regular m-primary ⊂ (R,m)⇒ I is the unique largest idealcontaining I with the same Hilbert polynomial as I (Heinzeret al., 1993)

I is coherent ⊂ (R,m)⇒ the tangent vector fields thatpreserve the Ratliff-Rush operation are liftable (Kallstrom,2009)

I m-primary ⊂ (R,m) Cohen-Macaulay

(I `+1) ⊆ I ` for all ` ≥ 1⇒ the 0-th Bockstein cohomology ofGI (R) vanishes (Puthenpurakal, 2012)

3 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Why study RR?

(I `) = I ` for all ` ≥ 1⇔ depth of the associated graded ringGI (R) = ⊕n≥0I

n/I n+1 > 0 (Heinzer et al., 1992)

I regular m-primary ⊂ (R,m)⇒ I is the unique largest idealcontaining I with the same Hilbert polynomial as I (Heinzeret al., 1993)

I is coherent ⊂ (R,m)⇒ the tangent vector fields thatpreserve the Ratliff-Rush operation are liftable (Kallstrom,2009)

I m-primary ⊂ (R,m) Cohen-Macaulay

(I `+1) ⊆ I ` for all ` ≥ 1⇒ the 0-th Bockstein cohomology ofGI (R) vanishes (Puthenpurakal, 2012)

3 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Why study RR?

(I `) = I ` for all ` ≥ 1⇔ depth of the associated graded ringGI (R) = ⊕n≥0I

n/I n+1 > 0 (Heinzer et al., 1992)

I regular m-primary ⊂ (R,m)⇒ I is the unique largest idealcontaining I with the same Hilbert polynomial as I (Heinzeret al., 1993)

I is coherent ⊂ (R,m)⇒ the tangent vector fields thatpreserve the Ratliff-Rush operation are liftable (Kallstrom,2009)

I m-primary ⊂ (R,m) Cohen-Macaulay

(I `+1) ⊆ I ` for all ` ≥ 1⇒ the 0-th Bockstein cohomology ofGI (R) vanishes (Puthenpurakal, 2012)

3 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Other properties

I regular (!)R = k[x ]/(x2) and I = 〈x〉, then I =

⋃∞`≥1(I `+1 : I `) = R,

but (I )` = I ` only for I = I

Not a closure operation as inclusion is not preservedJ = 〈y4, xy3, x3y , x4〉 ⊂ 〈y3, x3〉 = IJ = J + 〈x2y2〉 (in fact, J` = (m4)` for all ` ≥ 2)but I = I and J * I

I ⊆ I ⊆ I ⊆√I

I monomial ⇒ I monomial

4 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Other properties

I regular (!)R = k[x ]/(x2) and I = 〈x〉, then I =

⋃∞`≥1(I `+1 : I `) = R,

but (I )` = I ` only for I = I

Not a closure operation as inclusion is not preservedJ = 〈y4, xy3, x3y , x4〉 ⊂ 〈y3, x3〉 = IJ = J + 〈x2y2〉 (in fact, J` = (m4)` for all ` ≥ 2)but I = I and J * I

I ⊆ I ⊆ I ⊆√I

I monomial ⇒ I monomial

4 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Other properties

I regular (!)R = k[x ]/(x2) and I = 〈x〉, then I =

⋃∞`≥1(I `+1 : I `) = R,

but (I )` = I ` only for I = I

Not a closure operation as inclusion is not preservedJ = 〈y4, xy3, x3y , x4〉 ⊂ 〈y3, x3〉 = IJ = J + 〈x2y2〉 (in fact, J` = (m4)` for all ` ≥ 2)but I = I and J * I

I ⊆ I ⊆ I ⊆√I

I monomial ⇒ I monomial

4 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Other properties

I regular (!)R = k[x ]/(x2) and I = 〈x〉, then I =

⋃∞`≥1(I `+1 : I `) = R,

but (I )` = I ` only for I = I

Not a closure operation as inclusion is not preservedJ = 〈y4, xy3, x3y , x4〉 ⊂ 〈y3, x3〉 = IJ = J + 〈x2y2〉 (in fact, J` = (m4)` for all ` ≥ 2)but I = I and J * I

I ⊆ I ⊆ I ⊆√I

I monomial ⇒ I monomial

4 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Calculation problems

What is r(I )?There are cases whenI `+1 : I ` = . . . = I `+n : I `+n−1 ⊂ I `+n+1 : I `+n

In some cases I ` = (I `) for 1 ≤ ` ≤ n − 1, but I n ⊂ (I n)

Heinzer et el., 1992Heinzer et al., 1993Rossi&Swanson, 2001Elias, 2004−, 2006 for certain classes of monomial ideals in k[x , y ] usingnumerical semigroups

5 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Calculation problems

What is r(I )?There are cases whenI `+1 : I ` = . . . = I `+n : I `+n−1 ⊂ I `+n+1 : I `+n

In some cases I ` = (I `) for 1 ≤ ` ≤ n − 1, but I n ⊂ (I n)

Heinzer et el., 1992Heinzer et al., 1993Rossi&Swanson, 2001Elias, 2004−, 2006 for certain classes of monomial ideals in k[x , y ] usingnumerical semigroups

5 / 15

The Ratliff-Rush operation on monomial ideals

The Ratliff-Rush operation

Calculation problems

What is r(I )?There are cases whenI `+1 : I ` = . . . = I `+n : I `+n−1 ⊂ I `+n+1 : I `+n

In some cases I ` = (I `) for 1 ≤ ` ≤ n − 1, but I n ⊂ (I n)

Heinzer et el., 1992Heinzer et al., 1993Rossi&Swanson, 2001Elias, 2004−, 2006 for certain classes of monomial ideals in k[x , y ] usingnumerical semigroups

5 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

R polynomial ring k[x , y , z ] or power series ring k[[x , y , z ]] over aninfinite field k, with m = 〈x , y , z〉.

Let I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j , and

let I be m-primary, that is, xd , yd and zd ∈ I .

Define three sets by deleting one of the coordinates:X = 〈(bj , cj)〉 = {

∑nj=0 λj(bj , cj) | λj ∈ Z≥0},

Y = 〈(aj , cj)〉 and Z = 〈(aj , bj)〉.

(0, 0) belongs to each of them ⇒ finitely generated subsemigroupsof Z2, affine semigroups.

6 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

R polynomial ring k[x , y , z ] or power series ring k[[x , y , z ]] over aninfinite field k, with m = 〈x , y , z〉.

Let I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j , and

let I be m-primary, that is, xd , yd and zd ∈ I .

Define three sets by deleting one of the coordinates:X = 〈(bj , cj)〉 = {

∑nj=0 λj(bj , cj) | λj ∈ Z≥0},

Y = 〈(aj , cj)〉 and Z = 〈(aj , bj)〉.

(0, 0) belongs to each of them ⇒ finitely generated subsemigroupsof Z2, affine semigroups.

6 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

R polynomial ring k[x , y , z ] or power series ring k[[x , y , z ]] over aninfinite field k, with m = 〈x , y , z〉.

Let I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j , and

let I be m-primary, that is, xd , yd and zd ∈ I .

Define three sets by deleting one of the coordinates:X = 〈(bj , cj)〉 = {

∑nj=0 λj(bj , cj) | λj ∈ Z≥0},

Y = 〈(aj , cj)〉 and Z = 〈(aj , bj)〉.

(0, 0) belongs to each of them ⇒ finitely generated subsemigroupsof Z2, affine semigroups.

6 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

R polynomial ring k[x , y , z ] or power series ring k[[x , y , z ]] over aninfinite field k, with m = 〈x , y , z〉.

Let I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j , and

let I be m-primary, that is, xd , yd and zd ∈ I .

Define three sets by deleting one of the coordinates:X = 〈(bj , cj)〉 = {

∑nj=0 λj(bj , cj) | λj ∈ Z≥0},

Y = 〈(aj , cj)〉 and Z = 〈(aj , bj)〉.

(0, 0) belongs to each of them ⇒ finitely generated subsemigroupsof Z2, affine semigroups.

6 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

Estimate on number of generators of an element in an affine semigroup

Lemma

Let Z = 〈(aj , bj)〉nj=0 with aj + bj ≤ d be an affine semigroup.

Assume (d , 0) and (0, d) belong to Z . Then for any two reals0 < α < 1 and 0 ≤ β, there is an L s.t. for all ` ≥ L we have:

(r , s) ∈ Z and r + s ≤ α · d`+ β⇓

(r , s) =∑n

j=0 λj(aj , bj) where∑n

j=0 λj ≤ `.

(r , s) may be a linear combination of the (aj , bj)’s in different ways,we choose the one where λj ≤ d − 1 for all 1 ≤ j ≤ n − 1.If λj ≥ d , then λj(aj , bj) = aj(d , 0) + (λj − d)(aj , bj) + bj(0, d).

L =β/d + (d − 1)(n − 1)

1− α

7 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

Estimate on number of generators of an element in an affine semigroup

Lemma

Let Z = 〈(aj , bj)〉nj=0 with aj + bj ≤ d be an affine semigroup.

Assume (d , 0) and (0, d) belong to Z . Then for any two reals0 < α < 1 and 0 ≤ β, there is an L s.t. for all ` ≥ L we have:

(r , s) ∈ Z and r + s ≤ α · d`+ β⇓

(r , s) =∑n

j=0 λj(aj , bj) where∑n

j=0 λj ≤ `.

(r , s) may be a linear combination of the (aj , bj)’s in different ways,we choose the one where λj ≤ d − 1 for all 1 ≤ j ≤ n − 1.If λj ≥ d , then λj(aj , bj) = aj(d , 0) + (λj − d)(aj , bj) + bj(0, d).

L =β/d + (d − 1)(n − 1)

1− α

7 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

Estimate on number of generators of an element in an affine semigroup

Lemma

Let Z = 〈(aj , bj)〉nj=0 with aj + bj ≤ d be an affine semigroup.

Assume (d , 0) and (0, d) belong to Z . Then for any two reals0 < α < 1 and 0 ≤ β, there is an L s.t. for all ` ≥ L we have:

(r , s) ∈ Z and r + s ≤ α · d`+ β⇓

(r , s) =∑n

j=0 λj(aj , bj) where∑n

j=0 λj ≤ `.

(r , s) may be a linear combination of the (aj , bj)’s in different ways,we choose the one where λj ≤ d − 1 for all 1 ≤ j ≤ n − 1.If λj ≥ d , then λj(aj , bj) = aj(d , 0) + (λj − d)(aj , bj) + bj(0, d).

L =β/d + (d − 1)(n − 1)

1− α7 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

Estimate on number of generators of an element in an affine semigroup

The bound provided by the proof is very rough.

Example

Consider Z = 〈(3, 0), (1, 0), (0, 3)〉 = 〈(aj , bj)〉. Let α = 23 , β = 0.

Then for any ` ≥ 2 and (r , s) ∈ Z s. t. r + s ≤ 2` we have(r , s) =

∑λj(aj , bj) with

∑λj ≤ `.

The bound provided in the proof is ` ≥ 6.

8 / 15

The Ratliff-Rush operation on monomial ideals

Affine semigroups defined by monomial ideals

Estimate on number of generators of an element in an affine semigroup

The bound provided by the proof is very rough.

Example

Consider Z = 〈(3, 0), (1, 0), (0, 3)〉 = 〈(aj , bj)〉. Let α = 23 , β = 0.

Then for any ` ≥ 2 and (r , s) ∈ Z s. t. r + s ≤ 2` we have(r , s) =

∑λj(aj , bj) with

∑λj ≤ `.

The bound provided in the proof is ` ≥ 6.

8 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Use of affine semigroups

Proposition (I)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j is m-primary.

X = 〈(bj , cj)〉,Y = 〈(aj , cj)〉,Z = 〈(aj , bj)〉 are affine semigroups.

Then there is an integer L such that for any ` ≥ L we have

I ` = 〈x ry sz t |(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X , and r +s + t = d`〉.

⊆ is true by definition.

⊇ may seem trivial but needs a proof, since r + s + t = d` doesnot necessarily imply (r , s) =

∑nj=0 λj(aj , bj) for some

non-negative λj ’s with∑n

j=0 λj = `.

9 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Use of affine semigroups

Proposition (I)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j is m-primary.

X = 〈(bj , cj)〉,Y = 〈(aj , cj)〉,Z = 〈(aj , bj)〉 are affine semigroups.

Then there is an integer L such that for any ` ≥ L we have

I ` = 〈x ry sz t |(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X , and r +s + t = d`〉.

⊆ is true by definition.

⊇ may seem trivial but needs a proof, since r + s + t = d` doesnot necessarily imply (r , s) =

∑nj=0 λj(aj , bj) for some

non-negative λj ’s with∑n

j=0 λj = `.

9 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Use of affine semigroups

Proposition (I)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j is m-primary.

X = 〈(bj , cj)〉,Y = 〈(aj , cj)〉,Z = 〈(aj , bj)〉 are affine semigroups.

Then there is an integer L such that for any ` ≥ L we have

I ` = 〈x ry sz t |(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X , and r +s + t = d`〉.

⊆ is true by definition.

⊇ may seem trivial but needs a proof, since r + s + t = d` doesnot necessarily imply (r , s) =

∑nj=0 λj(aj , bj) for some

non-negative λj ’s with∑n

j=0 λj = `.

9 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Use of affine semigroups

Proposition (I)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R with aj + bj + cj = d for all j is m-primary.

X = 〈(bj , cj)〉,Y = 〈(aj , cj)〉,Z = 〈(aj , bj)〉 are affine semigroups.

Then there is an integer L such that for any ` ≥ L we have

I ` = 〈x ry sz t |(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X , and r +s + t = d`〉.

⊆ is true by definition.

⊇ may seem trivial but needs a proof, since r + s + t = d` doesnot necessarily imply (r , s) =

∑nj=0 λj(aj , bj) for some

non-negative λj ’s with∑n

j=0 λj = `.

9 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Definition

We define the following ideals associated to I :

IZ = 〈x ry szd−r−s | (r , s) ∈ Z and r + s ≤ d〉IY = 〈x ryd−r−tz t | (r , t) ∈ Y and r + t ≤ d〉IX = 〈xd−s−ty sz t | (s, t) ∈ X and s + t ≤ d〉.

10 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉

Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉

X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

11 / 15

The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

The corresponding affine semigroups are:

Z = 〈(7, 0), (2, 5), (5, 0), (0, 7), (0, 2), (0, 0)〉Y = 〈(7, 0), (2, 0), (5, 2), (0, 0), (0, 5), (0, 7)〉X = 〈(0, 0), (5, 0), (0, 2), (7, 0), (2, 5), (0, 7)〉.

The ideals defined by these affine semigroups are:

IZ = 〈x7, x2y5, x5z2, y7, y2z5, x5y2, y4z3, y6z , z7〉

IY = 〈x7, x2y5, x2z5, x4y3, x6y , x5z2, y7, y2z5, z7〉

IX = 〈x7, x2y5, x5z2, y5z2, x3z4, xz6, y7, y2z5, z7〉.

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The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Proposition (II)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R, with aj + bj + cj = d for all j , is m-primaryideal. Then for all sufficiently large ` we have

I ` = zd`−d IZ + yd`−d IY + xd`−d IX + 〈x3y3, y3z3, x3z3〉Im,` (1)

for some ideal Im,` generated by monomials of degree d(`− 2).

Example

Let I = 〈x7, x2y5, y7, z7〉. Then IY = 〈x7, x2y5, x4y3, x6y , y7, z7〉and IX = IZ = I .Then for all ` ≥ 3 the ideal I ` is on the form (1).It doesn’t work for ` = 2 because of the monomial x2y5, whichgives rise to x4y3 and x6y in IY . We have (x2y5)3 = y14(x6y), butcannot rewrite any monomial generator of I 2 into a product of x6y .

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The Ratliff-Rush operation on monomial ideals

Powers of monomial ideals

Ideals defined by affine semigroups

Proposition (II)

I = 〈xaj ybj zcj 〉nj=0 ⊂ R, with aj + bj + cj = d for all j , is m-primaryideal. Then for all sufficiently large ` we have

I ` = zd`−d IZ + yd`−d IY + xd`−d IX + 〈x3y3, y3z3, x3z3〉Im,` (1)

for some ideal Im,` generated by monomials of degree d(`− 2).

Example

Let I = 〈x7, x2y5, y7, z7〉. Then IY = 〈x7, x2y5, x4y3, x6y , y7, z7〉and IX = IZ = I .Then for all ` ≥ 3 the ideal I ` is on the form (1).It doesn’t work for ` = 2 because of the monomial x2y5, whichgives rise to x4y3 and x6y in IY . We have (x2y5)3 = y14(x6y), butcannot rewrite any monomial generator of I 2 into a product of x6y .

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The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

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The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

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The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

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The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

13 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

13 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

13 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Theorem

Let I = 〈xaj ybj zcj 〉nj=0 with aj + bj + cj = d for all j be anm-primary ideal in R and IZ , IY , IX be defined as previously. Thenthe Ratliff-Rush ideal I = IZ ∩ IY ∩ IX .

Steps of the proof:

Show that I `+1 : I ` = IZ ∩ IY ∩ IX for all ` big enough.

It suffices to consider monomials as I is monomial.

For ` big enough I ` is generated by x ry sz t where(r , s) ∈ Z , (r , t) ∈ Y , (s, t) ∈ X and r + s + t = d`.

⊇ Consider p ∈ IZ ∩ IY ∩ IX . Then p · x ry sz t = . . .= p1

∏(xaj ybj zcj )λj ∈ I `+1 for some p1 by the lemma on the

estimate of the generators of an affine semigroup element.

⊆ If a monomial p /∈ IZ , then zdlp /∈ zdl IZ and zdlp /∈ I l+1 byProposition (II).

13 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.

By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) =

I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Example

Let I = 〈x7, x2y5, x5z2, y7, y2z5, z7〉.By the previous theorem and example we have

I = (I + 〈x5y2, y4z3, y6z〉) ∩ (I + 〈x2z5, x4y3, x6y〉) ∩∩(I + 〈y5z2, x3z4, xz6〉) = I + 〈x4y4z4〉.

Some bounds/estimates:

the ideal I ` is on the form from Proposition (II) for ` ≥ 3

(I )` = I ` for all ` ≥ 2

I `+1 : I ` = I for all ` ≥ 1.

14 / 15

The Ratliff-Rush operation on monomial ideals

RR of monomial ideals in three variables

Thank you for your attention!

15 / 15