Questions in commutative algebraImportance of monomial ideals

Answers to questions

Pictures of Commutative AlgebraCombinatorics of Monomial Ideals

Angela Kohlhaas

Department of MathematicsUniversity of Notre Dame

April 14, 2008

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Outline

1 Questions in commutative algebra

2 Importance of monomial idealsLatticeClosure propertiesIrreducible idealsGröbner basis

3 Answers to questionsIrreducible decompositionHilbert functionMinimal free resolution

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Basic (and not so basic) Questions

Given an ideal I in a commutative ring R, we would like to findthe following:

An irreducible decomposition for I

The minimal free resolution and Hilbert function of R/I

Simplifications of I includingIntegral closure IAdjoint or multiplier ideal adj(I)Core of I

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Basic (and not so basic) Questions

Given an ideal I in a commutative ring R, we would like to findthe following:

An irreducible decomposition for I

The minimal free resolution and Hilbert function of R/I

Simplifications of I includingIntegral closure IAdjoint or multiplier ideal adj(I)Core of I

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Basic (and not so basic) Questions

Given an ideal I in a commutative ring R, we would like to findthe following:

An irreducible decomposition for I

The minimal free resolution and Hilbert function of R/I

Simplifications of I includingIntegral closure IAdjoint or multiplier ideal adj(I)Core of I

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Basic (and not so basic) Questions

Given an ideal I in a commutative ring R, we would like to findthe following:

An irreducible decomposition for I

The minimal free resolution and Hilbert function of R/I

Simplifications of I includingIntegral closure IAdjoint or multiplier ideal adj(I)Core of I

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Monomial ideals

Setting: Let R = k[x1, . . . , xd] be a polynomial ring over a field k.

I is a monomial ideal if it is generated by monomials.

Example: R = k[x, y] and I = (x4, x2y2, y5).

Why consider monomial ideals?

Four reasons...

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Monomial ideals

Setting: Let R = k[x1, . . . , xd] be a polynomial ring over a field k.

I is a monomial ideal if it is generated by monomials.

Example: R = k[x, y] and I = (x4, x2y2, y5).

Why consider monomial ideals?

Four reasons...

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Monomial ideals

Setting: Let R = k[x1, . . . , xd] be a polynomial ring over a field k.

I is a monomial ideal if it is generated by monomials.

Example: R = k[x, y] and I = (x4, x2y2, y5).

Why consider monomial ideals?

Four reasons...

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Monomial ideals

Setting: Let R = k[x1, . . . , xd] be a polynomial ring over a field k.

I is a monomial ideal if it is generated by monomials.

Example: R = k[x, y] and I = (x4, x2y2, y5).

Why consider monomial ideals?

Four reasons...

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Pictures!

(1) Lattice diagrams:

I

y

xR = k[x, y]

I = (x4, x2y2, y5)R = k[x, y, z]

I = m3

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(2) Closed under intersections, sums, products, colons:

I, J monomial ideals =⇒ I ∩ J, I + J, IJ, J : I monomial ideals.

DefinitionGiven two ideals J and I of a ring R, define

J : I = {r ∈ R| rI ⊆ J}.

Example: Let R = k[x, y], J = (x2, y3), and I = (x2, xy, y3). Thenx, y2 ∈ J : I but y /∈ J : I. In fact, J : I = (x, y2).

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(2) Closed under intersections, sums, products, colons:

I, J monomial ideals =⇒ I ∩ J, I + J, IJ, J : I monomial ideals.

DefinitionGiven two ideals J and I of a ring R, define

J : I = {r ∈ R| rI ⊆ J}.

Example: Let R = k[x, y], J = (x2, y3), and I = (x2, xy, y3). Thenx, y2 ∈ J : I but y /∈ J : I. In fact, J : I = (x, y2).

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(2) Closed under intersections, sums, products, colons:

I, J monomial ideals =⇒ I ∩ J, I + J, IJ, J : I monomial ideals.

DefinitionGiven two ideals J and I of a ring R, define

J : I = {r ∈ R| rI ⊆ J}.

Example: Let R = k[x, y], J = (x2, y3), and I = (x2, xy, y3). Thenx, y2 ∈ J : I but y /∈ J : I. In fact, J : I = (x, y2).

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(2) Closed under intersections, sums, products, colons:

I, J monomial ideals =⇒ I ∩ J, I + J, IJ, J : I monomial ideals.

DefinitionGiven two ideals J and I of a ring R, define

J : I = {r ∈ R| rI ⊆ J}.

Example: Let R = k[x, y], J = (x2, y3), and I = (x2, xy, y3). Thenx, y2 ∈ J : I but y /∈ J : I. In fact, J : I = (x, y2).

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(2) Closed under intersections, sums, products, colons:

I, J monomial ideals =⇒ I ∩ J, I + J, IJ, J : I monomial ideals.

DefinitionGiven two ideals J and I of a ring R, define

J : I = {r ∈ R| rI ⊆ J}.

Example: Let R = k[x, y], J = (x2, y3), and I = (x2, xy, y3). Thenx, y2 ∈ J : I but y /∈ J : I. In fact, J : I = (x, y2).

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Colon picture

Example:

J = (x2, y3),I = (x2, x1y1, y3),

J : I = (x, y2) = (x2−1, y3−1)

Remarks:1 A colon J : I, where J is a complete

intersection is called a link.2 Monomial complete intersection:

J = (xa11 , . . . , xan

n )

J

I(2, 3)

J : I

(1, 2)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Colon picture

Example:

J = (x2, y3),I = (x2, x1y1, y3),

J : I = (x, y2) = (x2−1, y3−1)

Remarks:1 A colon J : I, where J is a complete

intersection is called a link.2 Monomial complete intersection:

J = (xa11 , . . . , xan

n )

J

I(2, 3)

J : I

(1, 2)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Colon picture

Example:

J = (x2, y3),I = (x2, x1y1, y3),

J : I = (x, y2) = (x2−1, y3−1)

Remarks:1 A colon J : I, where J is a complete

intersection is called a link.2 Monomial complete intersection:

J = (xa11 , . . . , xan

n )

J

I(2, 3)

J : I

(1, 2)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Colon picture

Example:

J = (x2, y3),I = (x2, x1y1, y3),

J : I = (x, y2) = (x2−1, y3−1)

Remarks:1 A colon J : I, where J is a complete

intersection is called a link.2 Monomial complete intersection:

J = (xa11 , . . . , xan

n )

J

I(2, 3)

J : I

(1, 2)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Colon picture

Example:

J = (x2, y3),I = (x2, x1y1, y3),

J : I = (x, y2) = (x2−1, y3−1)

Remarks:1 A colon J : I, where J is a complete

intersection is called a link.2 Monomial complete intersection:

J = (xa11 , . . . , xan

n )

J

I(2, 3)

J : I

(1, 2)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(3) Simple description of irreducible ideals:

DefinitionAn ideal I is irreducible if I 6= J ∩ K for any ideals J and Kproperly containing I.

Remarks:Let R = k[x1, . . . , xd]. Irreduciblemonomial ideals are of the form(xa1

i1 , . . . , xadid ) for some subset

{i1, . . . , in} of the variables.In R = k[x, y], (x2, y3) is irreducible.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(3) Simple description of irreducible ideals:

DefinitionAn ideal I is irreducible if I 6= J ∩ K for any ideals J and Kproperly containing I.

Remarks:Let R = k[x1, . . . , xd]. Irreduciblemonomial ideals are of the form(xa1

i1 , . . . , xadid ) for some subset

{i1, . . . , in} of the variables.In R = k[x, y], (x2, y3) is irreducible.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(3) Simple description of irreducible ideals:

DefinitionAn ideal I is irreducible if I 6= J ∩ K for any ideals J and Kproperly containing I.

Remarks:Let R = k[x1, . . . , xd]. Irreduciblemonomial ideals are of the form(xa1

i1 , . . . , xadid ) for some subset

{i1, . . . , in} of the variables.In R = k[x, y], (x2, y3) is irreducible.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(3) Simple description of irreducible ideals:

DefinitionAn ideal I is irreducible if I 6= J ∩ K for any ideals J and Kproperly containing I.

Remarks:Let R = k[x1, . . . , xd]. Irreduciblemonomial ideals are of the form(xa1

i1 , . . . , xadid ) for some subset

{i1, . . . , in} of the variables.In R = k[x, y], (x2, y3) is irreducible.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Properties of monomial ideals

(4) Connection to other ideals via Gröbner bases:

Definition

A monomial order on R = k[x1, . . . , xd] is a total order > on themonomials of R such that if m1, m2, n are monomials of R with n 6= 1,then m1 > m2 =⇒ nm1 > nm2 > m2.

Common term orders: Set m = xa11 · · · x

add , m′ = xb1

1 · · · xbdd .

Homog. lexicographic order: m >hlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai > bi for the first index i with ai 6= bi.

x4y2z3 >hlex x2y5z2

Reverse lexicographic order: m >rlex m′ ⇐⇒ deg(m) > deg(m′)or deg(m) = deg(m′) and ai < bi for the last index i with ai 6= bi.

x4y2z3 <rlex x2y5z2

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Initial ideal

DefinitionLet f ∈ R. The initial term of f , in(f ), is the greatest term of fwith respect to a given monomial order. If I is an ideal of R,in(I) = (in(f )| f ∈ I) is the initial ideal of I.

Example: inhlex(x4 + x3y2 + xy4) = x3y2.

A Gröbner basis for an ideal I is a set of elementsg1, . . . , gr ∈ I such that in(I) = (in(g1), . . . , in(gr)).

Theorem (Macaulay)

Let I be an ideal of R. Then the set of monomials not in in(I)forms a basis for R/I as a k-vector space.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Initial ideal

DefinitionLet f ∈ R. The initial term of f , in(f ), is the greatest term of fwith respect to a given monomial order. If I is an ideal of R,in(I) = (in(f )| f ∈ I) is the initial ideal of I.

Example: inhlex(x4 + x3y2 + xy4) = x3y2.

A Gröbner basis for an ideal I is a set of elementsg1, . . . , gr ∈ I such that in(I) = (in(g1), . . . , in(gr)).

Theorem (Macaulay)

Let I be an ideal of R. Then the set of monomials not in in(I)forms a basis for R/I as a k-vector space.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Initial ideal

DefinitionLet f ∈ R. The initial term of f , in(f ), is the greatest term of fwith respect to a given monomial order. If I is an ideal of R,in(I) = (in(f )| f ∈ I) is the initial ideal of I.

Example: inhlex(x4 + x3y2 + xy4) = x3y2.

A Gröbner basis for an ideal I is a set of elementsg1, . . . , gr ∈ I such that in(I) = (in(g1), . . . , in(gr)).

Theorem (Macaulay)

Let I be an ideal of R. Then the set of monomials not in in(I)forms a basis for R/I as a k-vector space.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Initial ideal

DefinitionLet f ∈ R. The initial term of f , in(f ), is the greatest term of fwith respect to a given monomial order. If I is an ideal of R,in(I) = (in(f )| f ∈ I) is the initial ideal of I.

Example: inhlex(x4 + x3y2 + xy4) = x3y2.

A Gröbner basis for an ideal I is a set of elementsg1, . . . , gr ∈ I such that in(I) = (in(g1), . . . , in(gr)).

Theorem (Macaulay)

Let I be an ideal of R. Then the set of monomials not in in(I)forms a basis for R/I as a k-vector space.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

LatticeClosure propertiesIrreducible idealsGröbner basis

Initial ideal

DefinitionLet f ∈ R. The initial term of f , in(f ), is the greatest term of fwith respect to a given monomial order. If I is an ideal of R,in(I) = (in(f )| f ∈ I) is the initial ideal of I.

Example: inhlex(x4 + x3y2 + xy4) = x3y2.

A Gröbner basis for an ideal I is a set of elementsg1, . . . , gr ∈ I such that in(I) = (in(g1), . . . , in(gr)).

Theorem (Macaulay)

Let I be an ideal of R. Then the set of monomials not in in(I)forms a basis for R/I as a k-vector space.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Irreducible decomposition

Fact: Every ideal I of R can be written as the irredundantintersection of finitely many irreducible ideals, called anirreducible decomposition of I.

Remark: If I is monomial, this decomposition is unique.

Proposition

Let I = (xa1yb1 , . . . , xanybn) in k[x, y], with a1 > . . . > an ≥ 0 and0 ≤ b1 < . . . < bn. Then

I = (yb1) ∩ (xa1 , yb2) ∩ . . . ∩ (xan),

with the first or last components deleted if b1 = 0 or an = 0.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Irreducible decomposition

Fact: Every ideal I of R can be written as the irredundantintersection of finitely many irreducible ideals, called anirreducible decomposition of I.

Remark: If I is monomial, this decomposition is unique.

Proposition

Let I = (xa1yb1 , . . . , xanybn) in k[x, y], with a1 > . . . > an ≥ 0 and0 ≤ b1 < . . . < bn. Then

I = (yb1) ∩ (xa1 , yb2) ∩ . . . ∩ (xan),

with the first or last components deleted if b1 = 0 or an = 0.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Irreducible decomposition

Fact: Every ideal I of R can be written as the irredundantintersection of finitely many irreducible ideals, called anirreducible decomposition of I.

Remark: If I is monomial, this decomposition is unique.

Proposition

Let I = (xa1yb1 , . . . , xanybn) in k[x, y], with a1 > . . . > an ≥ 0 and0 ≤ b1 < . . . < bn. Then

I = (yb1) ∩ (xa1 , yb2) ∩ . . . ∩ (xan),

with the first or last components deleted if b1 = 0 or an = 0.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Irreducible decomposition

Fact: Every ideal I of R can be written as the irredundantintersection of finitely many irreducible ideals, called anirreducible decomposition of I.

Remark: If I is monomial, this decomposition is unique.

Proposition

Let I = (xa1yb1 , . . . , xanybn) in k[x, y], with a1 > . . . > an ≥ 0 and0 ≤ b1 < . . . < bn. Then

I = (yb1) ∩ (xa1 , yb2) ∩ . . . ∩ (xan),

with the first or last components deleted if b1 = 0 or an = 0.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Irreducible decomposition

Fact: Every ideal I of R can be written as the irredundantintersection of finitely many irreducible ideals, called anirreducible decomposition of I.

Remark: If I is monomial, this decomposition is unique.

Proposition

Let I = (xa1yb1 , . . . , xanybn) in k[x, y], with a1 > . . . > an ≥ 0 and0 ≤ b1 < . . . < bn. Then

I = (yb1) ∩ (xa1 , yb2) ∩ . . . ∩ (xan),

with the first or last components deleted if b1 = 0 or an = 0.

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y]

Example: The irreducible decomposition for I = (x4, x2y2, y5) isI = (x4, y2) ∩ (x2, y5).

I=

(2, 5)

(4, 2)

(x4, y2)

∩(x2, y5)

Note: irreducible components←→ outer corners (white dots)

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y]

Example: The irreducible decomposition for I = (x4, x2y2, y5) isI = (x4, y2) ∩ (x2, y5).

I=

(2, 5)

(4, 2)

(x4, y2)

∩(x2, y5)

Note: irreducible components←→ outer corners (white dots)

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y]

Example: The irreducible decomposition for I = (x4, x2y2, y5) isI = (x4, y2) ∩ (x2, y5).

I=

(2, 5)

(4, 2)

(x4, y2)

∩(x2, y5)

Note: irreducible components←→ outer corners (white dots)

Kohlhaas Pictures of Comm Alg

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

I =(x4, y, z4) ∩ (x2, y3, z4)

∩ (x, y4, z4) ∩ (x4, y2, z3)

∩ (x3, y3, z3) ∩ (x3, y4, z2)

∩ (x4, y4, z)

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

I =(x4, y, z4) ∩ (x2, y3, z4)

∩ (x, y4, z4) ∩ (x4, y2, z3)

∩ (x3, y3, z3) ∩ (x3, y4, z2)

∩ (x4, y4, z)

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

z4

x4 y4

x2yz3

x3y2z

xy3z2

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

z4

x4 y4

x2yz3

x3y2z

xy3z2

404

440

044

214

323

341421

332

134

142

233

413

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

z4

x4 y4

x2yz3

x3y2z

xy3z2

404

440

044

214

323

341421

332

134

142

233

413

234 144

423

342

441

414

333

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Decomposition in k[x, y, z]

I = (x4, y, z4) ∩ (x2, y3, z4) ∩ (x, y4, z4) ∩ (x4, y2, z3) ∩ (x3, y3, z3) ∩(x3, y4, z2) ∩ (x4, y4, z)

z4

x4 y4

x2yz3

x3y2z

xy3z2

404

440

044

214

323

341421

332

134

142

233

413

234 144

423

342

441

414

333

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Buchberger graph

Definition

Given monomials m and m′, m′ strictly divides m (m′|sm) if m′ dividesm/xi for every xi which divides m.

Definition

The Buchberger graph Buch(I) of a monomial ideal I = (m1, . . . , mn)has vertices 1, . . . , n and an edge (i, j) whenever there is nomonomial generator mk which strictly divides lcm(mi, mj).

Example:1 In I = (x4, y4, z4, x3y2z, x2yz3, xy3z2), x4 to x3y2z is an edge, but x4 to

xy3z2 is not, since x3y2z|sx4y3z2 = lcm(x4, xy3z2).2 In k[x, y], if I = (xa1 yb1 , . . . , xan ybn), the edges of Buch(I) are the

n− 1 consecutive pairs of generators.

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Irreducible decompositionHilbert functionMinimal free resolution

Buchberger graph

Definition

Given monomials m and m′, m′ strictly divides m (m′|sm) if m′ dividesm/xi for every xi which divides m.

Definition

The Buchberger graph Buch(I) of a monomial ideal I = (m1, . . . , mn)has vertices 1, . . . , n and an edge (i, j) whenever there is nomonomial generator mk which strictly divides lcm(mi, mj).

Example:1 In I = (x4, y4, z4, x3y2z, x2yz3, xy3z2), x4 to x3y2z is an edge, but x4 to

xy3z2 is not, since x3y2z|sx4y3z2 = lcm(x4, xy3z2).2 In k[x, y], if I = (xa1 yb1 , . . . , xan ybn), the edges of Buch(I) are the

n− 1 consecutive pairs of generators.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Buchberger graph

Definition

Given monomials m and m′, m′ strictly divides m (m′|sm) if m′ dividesm/xi for every xi which divides m.

Definition

The Buchberger graph Buch(I) of a monomial ideal I = (m1, . . . , mn)has vertices 1, . . . , n and an edge (i, j) whenever there is nomonomial generator mk which strictly divides lcm(mi, mj).

Example:1 In I = (x4, y4, z4, x3y2z, x2yz3, xy3z2), x4 to x3y2z is an edge, but x4 to

xy3z2 is not, since x3y2z|sx4y3z2 = lcm(x4, xy3z2).2 In k[x, y], if I = (xa1 yb1 , . . . , xan ybn), the edges of Buch(I) are the

n− 1 consecutive pairs of generators.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Buchberger graph

Definition

Given monomials m and m′, m′ strictly divides m (m′|sm) if m′ dividesm/xi for every xi which divides m.

Definition

The Buchberger graph Buch(I) of a monomial ideal I = (m1, . . . , mn)has vertices 1, . . . , n and an edge (i, j) whenever there is nomonomial generator mk which strictly divides lcm(mi, mj).

Example:1 In I = (x4, y4, z4, x3y2z, x2yz3, xy3z2), x4 to x3y2z is an edge, but x4 to

xy3z2 is not, since x3y2z|sx4y3z2 = lcm(x4, xy3z2).2 In k[x, y], if I = (xa1 yb1 , . . . , xan ybn), the edges of Buch(I) are the

n− 1 consecutive pairs of generators.

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Irreducible decompositionHilbert functionMinimal free resolution

Problem!

Let I = (x2z, xyz, y2z, x3y5, x4y4, x5y3).

Buch(I) contains K3,3 as a subgraph:

DefinitionA monomial ideal I = (m1, . . . , mn) is generic if wheneverdistinct mi and mj have the same nonzero degree in some x`, athird mk strictly divides lcm(mi, mj).

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Irreducible decompositionHilbert functionMinimal free resolution

Problem!

Let I = (x2z, xyz, y2z, x3y5, x4y4, x5y3).

Buch(I) contains K3,3 as a subgraph:

DefinitionA monomial ideal I = (m1, . . . , mn) is generic if wheneverdistinct mi and mj have the same nonzero degree in some x`, athird mk strictly divides lcm(mi, mj).

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Irreducible decompositionHilbert functionMinimal free resolution

Problem!

Let I = (x2z, xyz, y2z, x3y5, x4y4, x5y3).

Buch(I) contains K3,3 as a subgraph:

DefinitionA monomial ideal I = (m1, . . . , mn) is generic if wheneverdistinct mi and mj have the same nonzero degree in some x`, athird mk strictly divides lcm(mi, mj).

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert function

DefinitionLet R = k[x1, . . . , xd] be a polynomial ring over a field k. TheHilbert function of a finitely generated graded R-module M isthe function

HF(M, s) := dimk(Ms).

The Hilbert function can be encoded in the formal power series ring

HS(M, t) :=∑s∈Z

HF(M, s)ts.

Fact: HS(M, t) = P(M, t)/(1− t)d, where P(M, t) ∈ Z[t, t−1]. P(M, t) issometimes called the Poincaré polynomial and encodes the Bettinumbers of M.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert function

DefinitionLet R = k[x1, . . . , xd] be a polynomial ring over a field k. TheHilbert function of a finitely generated graded R-module M isthe function

HF(M, s) := dimk(Ms).

The Hilbert function can be encoded in the formal power series ring

HS(M, t) :=∑s∈Z

HF(M, s)ts.

Fact: HS(M, t) = P(M, t)/(1− t)d, where P(M, t) ∈ Z[t, t−1]. P(M, t) issometimes called the Poincaré polynomial and encodes the Bettinumbers of M.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert function

DefinitionLet R = k[x1, . . . , xd] be a polynomial ring over a field k. TheHilbert function of a finitely generated graded R-module M isthe function

HF(M, s) := dimk(Ms).

The Hilbert function can be encoded in the formal power series ring

HS(M, t) :=∑s∈Z

HF(M, s)ts.

Fact: HS(M, t) = P(M, t)/(1− t)d, where P(M, t) ∈ Z[t, t−1]. P(M, t) issometimes called the Poincaré polynomial and encodes the Bettinumbers of M.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

Our case: M = R/I, Ms = forms of degree s in R/I

Fact: For any homogeneous R-ideal I, the monomials ofdegree s not in I form a k-basis for (R/I)s.

How do we decide whether a monomial is in an ideal?

Use a Gröbner basis.

Theorem (Macaulay)Given any homogeneous ideal I in R,HF(R/I, s) = HF(R/in(I), s), where in(I) is the ideal generatedby the initial terms of a Gröbner basis for R/I.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

Our case: M = R/I, Ms = forms of degree s in R/I

Fact: For any homogeneous R-ideal I, the monomials ofdegree s not in I form a k-basis for (R/I)s.

How do we decide whether a monomial is in an ideal?

Use a Gröbner basis.

Theorem (Macaulay)Given any homogeneous ideal I in R,HF(R/I, s) = HF(R/in(I), s), where in(I) is the ideal generatedby the initial terms of a Gröbner basis for R/I.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

Our case: M = R/I, Ms = forms of degree s in R/I

Fact: For any homogeneous R-ideal I, the monomials ofdegree s not in I form a k-basis for (R/I)s.

How do we decide whether a monomial is in an ideal?

Use a Gröbner basis.

Theorem (Macaulay)Given any homogeneous ideal I in R,HF(R/I, s) = HF(R/in(I), s), where in(I) is the ideal generatedby the initial terms of a Gröbner basis for R/I.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

Our case: M = R/I, Ms = forms of degree s in R/I

Fact: For any homogeneous R-ideal I, the monomials ofdegree s not in I form a k-basis for (R/I)s.

How do we decide whether a monomial is in an ideal?

Use a Gröbner basis.

Theorem (Macaulay)Given any homogeneous ideal I in R,HF(R/I, s) = HF(R/in(I), s), where in(I) is the ideal generatedby the initial terms of a Gröbner basis for R/I.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

Our case: M = R/I, Ms = forms of degree s in R/I

Fact: For any homogeneous R-ideal I, the monomials ofdegree s not in I form a k-basis for (R/I)s.

How do we decide whether a monomial is in an ideal?

Use a Gröbner basis.

Theorem (Macaulay)Given any homogeneous ideal I in R,HF(R/I, s) = HF(R/in(I), s), where in(I) is the ideal generatedby the initial terms of a Gröbner basis for R/I.

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Irreducible decompositionHilbert functionMinimal free resolution

HF in k[x, y]

I = (x4, x2y2, y5) : HS(R/I, t) = 1 + 2t + 3t2 + 4t3 + 3t4 + t5

P(R/I, t) = (1− t)2(1+2t+3t2 +4t3 +3t4 + t5) = 1−2t4− t5 + t6 + t7

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Irreducible decompositionHilbert functionMinimal free resolution

HF in k[x, y]

I = (x4, x2y2, y5) : HS(R/I, t) = 1 + 2t + 3t2 + 4t3 + 3t4 + t5

P(R/I, t) = (1− t)2(1+2t+3t2 +4t3 +3t4 + t5) = 1−2t4− t5 + t6 + t7

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Irreducible decompositionHilbert functionMinimal free resolution

HF in k[x, y]

I = (x4, x2y2, y5) : HS(R/I, t) = 1 + 2t + 3t2 + 4t3 + 3t4 + t5

P(R/I, t) = (1− t)2(1+2t+3t2 +4t3 +3t4 + t5) = 1−2t4− t5 + t6 + t7

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Irreducible decompositionHilbert functionMinimal free resolution

HF in k[x, y]

I = (x4, x2y2, y5) : HS(R/I, t) = 1 + 2t + 3t2 + 4t3 + 3t4 + t5

P(R/I, t) = (1− t)2(1+2t+3t2 +4t3 +3t4 + t5) = 1−2t4− t5 + t6 + t7

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Irreducible decompositionHilbert functionMinimal free resolution

HF in k[x, y]

I = (x4, x2y2, y5) : HS(R/I, t) = 1 + 2t + 3t2 + 4t3 + 3t4 + t5

P(R/I, t) = (1− t)2(1+2t+3t2 +4t3 +3t4 + t5) = 1−2t4− t5 + t6 + t7

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

P(R/I, t) = 1−2t4−t5+t6+t7 = 1− inner corners + outer corners

I

Multi-graded Poincaré polynomial:

P(R/I, x, y)

= 1− x4 − x2y2 − y5 + x4y2 + x2y5

= 1− inner + outer

Specializing to t = x = y gives theresult.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

P(R/I, t) = 1−2t4−t5+t6+t7 = 1− inner corners + outer corners

I

Multi-graded Poincaré polynomial:

P(R/I, x, y)

= 1− x4 − x2y2 − y5 + x4y2 + x2y5

= 1− inner + outer

Specializing to t = x = y gives theresult.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

P(R/I, t) = 1−2t4−t5+t6+t7 = 1− inner corners + outer corners

I

Multi-graded Poincaré polynomial:

P(R/I, x, y)

= 1− x4 − x2y2 − y5 + x4y2 + x2y5

= 1− inner + outer

Specializing to t = x = y gives theresult.

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Irreducible decompositionHilbert functionMinimal free resolution

Poincaré polynomial

P(R/I, t) = 1−2t4−t5+t6+t7 = 1− inner corners + outer corners

I

Multi-graded Poincaré polynomial:

P(R/I, x, y)

= 1− x4 − x2y2 − y5 + x4y2 + x2y5

= 1− inner + outer

Specializing to t = x = y gives theresult.

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Irreducible decompositionHilbert functionMinimal free resolution

Minimal free resolution

DefinitionA free resolution of an R-module M is a complex

F• : · · · ϕi+1−−−→ Fiϕi−−→ · · · ϕ2−−→ F1

ϕ1−−→ F0

where Fi ∼= Rni , some i, so that

· · · ϕi+1−−−→ Fiϕi−−→ · · · ϕ2−−→ F1

ϕ1−−→ F0 −→ M −→ 0

is an exact sequence. That is, ker ϕi = im ϕi+1 for every i andcoker ϕ1 = M.

DefinitionIf F• is a free resolution for M, the kernel of ϕi is called the i-thsyzygy module of M.

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Irreducible decompositionHilbert functionMinimal free resolution

Minimal free resolution

DefinitionA free resolution of an R-module M is a complex

F• : · · · ϕi+1−−−→ Fiϕi−−→ · · · ϕ2−−→ F1

ϕ1−−→ F0

where Fi ∼= Rni , some i, so that

· · · ϕi+1−−−→ Fiϕi−−→ · · · ϕ2−−→ F1

ϕ1−−→ F0 −→ M −→ 0

is an exact sequence. That is, ker ϕi = im ϕi+1 for every i andcoker ϕ1 = M.

DefinitionIf F• is a free resolution for M, the kernel of ϕi is called the i-thsyzygy module of M.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert Syzygy Theorem

If R is a polynomial ring in d variables over a field k, then everymodule over R has a free resolution of length ≤ d.

Example: If I is an ideal in k[x, y], then I has a resolution

0 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0.

Remark: The ni are called the Betti numbers of R/I.

Let I = (xa1yb1 , . . . , xanybn), with a1 > . . . > an and b1 < . . . < bn.

Let’s build a resolution for I.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert Syzygy Theorem

If R is a polynomial ring in d variables over a field k, then everymodule over R has a free resolution of length ≤ d.

Example: If I is an ideal in k[x, y], then I has a resolution

0 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0.

Remark: The ni are called the Betti numbers of R/I.

Let I = (xa1yb1 , . . . , xanybn), with a1 > . . . > an and b1 < . . . < bn.

Let’s build a resolution for I.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert Syzygy Theorem

If R is a polynomial ring in d variables over a field k, then everymodule over R has a free resolution of length ≤ d.

Example: If I is an ideal in k[x, y], then I has a resolution

0 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0.

Remark: The ni are called the Betti numbers of R/I.

Let I = (xa1yb1 , . . . , xanybn), with a1 > . . . > an and b1 < . . . < bn.

Let’s build a resolution for I.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert Syzygy Theorem

If R is a polynomial ring in d variables over a field k, then everymodule over R has a free resolution of length ≤ d.

Example: If I is an ideal in k[x, y], then I has a resolution

0 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0.

Remark: The ni are called the Betti numbers of R/I.

Let I = (xa1yb1 , . . . , xanybn), with a1 > . . . > an and b1 < . . . < bn.

Let’s build a resolution for I.

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Irreducible decompositionHilbert functionMinimal free resolution

Hilbert Syzygy Theorem

If R is a polynomial ring in d variables over a field k, then everymodule over R has a free resolution of length ≤ d.

Example: If I is an ideal in k[x, y], then I has a resolution

0 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0.

Remark: The ni are called the Betti numbers of R/I.

Let I = (xa1yb1 , . . . , xanybn), with a1 > . . . > an and b1 < . . . < bn.

Let’s build a resolution for I.

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y]

Let I = (x4, x2y2, y5). Then n = 3, so we have a resolution

0 −→ R2 −→ R3 −→ R −→ 0.

I

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y]

Let I = (x4, x2y2, y5). Then n = 3, so we have a resolution

0 −→ R2 −→ R3 −→ R −→ 0.

I

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Irreducible decompositionHilbert functionMinimal free resolution

Socle

DefinitionThe socle of a module M is the ideal soc(M) = (0 :M m), theset of elements annihilated by every variable.

Remark: The last Betti number of a free resolution for M isequal to dimk soc(M).

Our case: soc(R/I) = (0 :R/I m) = (I : m)

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Irreducible decompositionHilbert functionMinimal free resolution

Socle

DefinitionThe socle of a module M is the ideal soc(M) = (0 :M m), theset of elements annihilated by every variable.

Remark: The last Betti number of a free resolution for M isequal to dimk soc(M).

Our case: soc(R/I) = (0 :R/I m) = (I : m)

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Irreducible decompositionHilbert functionMinimal free resolution

Socle

DefinitionThe socle of a module M is the ideal soc(M) = (0 :M m), theset of elements annihilated by every variable.

Remark: The last Betti number of a free resolution for M isequal to dimk soc(M).

Our case: soc(R/I) = (0 :R/I m) = (I : m)

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Irreducible decompositionHilbert functionMinimal free resolution

Socle

DefinitionThe socle of a module M is the ideal soc(M) = (0 :M m), theset of elements annihilated by every variable.

Remark: The last Betti number of a free resolution for M isequal to dimk soc(M).

Our case: soc(R/I) = (0 :R/I m) = (I : m)

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Irreducible decompositionHilbert functionMinimal free resolution

I

m

x3y ·mI

xy4 ·m

I

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Irreducible decompositionHilbert functionMinimal free resolution

I

m

x3y ·mI

xy4 ·m

I

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Irreducible decompositionHilbert functionMinimal free resolution

I

m

x3y ·mI

xy4 ·m

I

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Irreducible decompositionHilbert functionMinimal free resolution

soc(R/I) = (x3y + I, xy4 + I)dimk soc(R/I) = 2

x3y ·mI

xy4 ·m

I

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Irreducible decompositionHilbert functionMinimal free resolution

Proposition

If I is a monomial ideal, xa11 · · · x

add is a minimal generator of

soc(R/I) if and only if (xa1+11 , . . . , xad+1

d ) is an irreduciblecomponent of I. That is, generators of the socle are in one toone correspondence with irreducible components.

Remark: The ring R/I is Gorenstein if soc(M) is onedimensional.

CorollaryIf I is an m-primary monomial ideal, then

I Gorenstein ⇐⇒ I complete intersection.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Proposition

If I is a monomial ideal, xa11 · · · x

add is a minimal generator of

soc(R/I) if and only if (xa1+11 , . . . , xad+1

d ) is an irreduciblecomponent of I. That is, generators of the socle are in one toone correspondence with irreducible components.

Remark: The ring R/I is Gorenstein if soc(M) is onedimensional.

CorollaryIf I is an m-primary monomial ideal, then

I Gorenstein ⇐⇒ I complete intersection.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Proposition

If I is a monomial ideal, xa11 · · · x

add is a minimal generator of

soc(R/I) if and only if (xa1+11 , . . . , xad+1

d ) is an irreduciblecomponent of I. That is, generators of the socle are in one toone correspondence with irreducible components.

Remark: The ring R/I is Gorenstein if soc(M) is onedimensional.

CorollaryIf I is an m-primary monomial ideal, then

I Gorenstein ⇐⇒ I complete intersection.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Proposition

If I is a monomial ideal, xa11 · · · x

add is a minimal generator of

soc(R/I) if and only if (xa1+11 , . . . , xad+1

d ) is an irreduciblecomponent of I. That is, generators of the socle are in one toone correspondence with irreducible components.

Remark: The ring R/I is Gorenstein if soc(M) is onedimensional.

CorollaryIf I is an m-primary monomial ideal, then

I Gorenstein ⇐⇒ I complete intersection.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

0 −→ Rn3 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0

0 −→ Rn3 −→ Rn2 −→ R6 −→ R −→ R/I −→ 0

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

0 −→ Rn3 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0

0 −→ Rn3 −→ Rn2 −→ R6 −→ R −→ R/I −→ 0

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

0 −→ Rn3 −→ Rn2 −→ Rn1 −→ Rn0 −→ 0

0 −→ Rn3 −→ Rn2 −→ R6 −→ R −→ R/I −→ 0

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)z4

x4 y4

x2yz3

x3y2z

xy3z2

404

440

044

214

323

341421

332

134

142

233

413

234 144

423

342

441

414

333

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Example in k[x, y, z]

Example: I = (x4, y4, z4, x3y2z, x2yz3, xy3z2)

0 −→ R7 −→ R12 −→R6 −→ R −→ 0

P(R/I, s) = 1− vertices+ edges− faces

= 1− 3t4 − 3t6

+ 3t7 + 9t8 − 7t9

z4

x4 y4

x2yz3

x3y2z

xy3z2

404

440

044

214

323

341421

332

134

142

233

413

234 144

423

342

441

414

333

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Resolution by planar graph

TheoremEvery monomial ideal I in k[x, y, z] has a minimal free resolutionby some planar graph.

Proof idea: Create a "strongly generic deformation" ideal Iε of Iso that Buch(Iε) resolves I.

DefinitionA deformation ideal of I = (m1, . . . , mn) is a formal idealIε = (m1 · xε1 , . . . , mn · xεn).

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Irreducible decompositionHilbert functionMinimal free resolution

Resolution by planar graph

TheoremEvery monomial ideal I in k[x, y, z] has a minimal free resolutionby some planar graph.

Proof idea: Create a "strongly generic deformation" ideal Iε of Iso that Buch(Iε) resolves I.

DefinitionA deformation ideal of I = (m1, . . . , mn) is a formal idealIε = (m1 · xε1 , . . . , mn · xεn).

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Irreducible decompositionHilbert functionMinimal free resolution

Resolution by planar graph

TheoremEvery monomial ideal I in k[x, y, z] has a minimal free resolutionby some planar graph.

Proof idea: Create a "strongly generic deformation" ideal Iε of Iso that Buch(Iε) resolves I.

DefinitionA deformation ideal of I = (m1, . . . , mn) is a formal idealIε = (m1 · xε1 , . . . , mn · xεn).

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Irreducible decompositionHilbert functionMinimal free resolution

Higher dimensions

Remark: If I is a generic ideal, Buch(I) is a special example ofthe Scarf complex.

DefinitionLet I = (m1, . . . , mn) be a monomial ideal in k[x1, . . . , xd]. TheScarf complex ∆I is the collection of all subsets of{m1, . . . , mn} whose least common multiple is unique.

Remark: In general, edges(∆I) ⊆ Buch(I).

Example: I = (xy, xz, yz)

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Irreducible decompositionHilbert functionMinimal free resolution

Higher dimensions

Remark: If I is a generic ideal, Buch(I) is a special example ofthe Scarf complex.

DefinitionLet I = (m1, . . . , mn) be a monomial ideal in k[x1, . . . , xd]. TheScarf complex ∆I is the collection of all subsets of{m1, . . . , mn} whose least common multiple is unique.

Remark: In general, edges(∆I) ⊆ Buch(I).

Example: I = (xy, xz, yz)

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Higher dimensions

Remark: If I is a generic ideal, Buch(I) is a special example ofthe Scarf complex.

DefinitionLet I = (m1, . . . , mn) be a monomial ideal in k[x1, . . . , xd]. TheScarf complex ∆I is the collection of all subsets of{m1, . . . , mn} whose least common multiple is unique.

Remark: In general, edges(∆I) ⊆ Buch(I).

Example: I = (xy, xz, yz)

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Higher dimensions

Remark: If I is a generic ideal, Buch(I) is a special example ofthe Scarf complex.

DefinitionLet I = (m1, . . . , mn) be a monomial ideal in k[x1, . . . , xd]. TheScarf complex ∆I is the collection of all subsets of{m1, . . . , mn} whose least common multiple is unique.

Remark: In general, edges(∆I) ⊆ Buch(I).

Example: I = (xy, xz, yz)

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Irreducible decompositionHilbert functionMinimal free resolution

Let R = k[x1, . . . , xd].

LemmaIf I is a generic monomial ideal, then edges(∆I) = Buch(I).

TheoremIf I is a generic monomial ideal, then the algebraic Scarfcomplex (Taylor complex F∆I supported on ∆I) minimallyresolves R/I.

TheoremFix a monomial ideal I and a generic deformation ideal Iε. TheTaylor complex F∆ε

Igives a resolution for R/I of length ≤ d.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Let R = k[x1, . . . , xd].

LemmaIf I is a generic monomial ideal, then edges(∆I) = Buch(I).

TheoremIf I is a generic monomial ideal, then the algebraic Scarfcomplex (Taylor complex F∆I supported on ∆I) minimallyresolves R/I.

TheoremFix a monomial ideal I and a generic deformation ideal Iε. TheTaylor complex F∆ε

Igives a resolution for R/I of length ≤ d.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

Let R = k[x1, . . . , xd].

LemmaIf I is a generic monomial ideal, then edges(∆I) = Buch(I).

TheoremIf I is a generic monomial ideal, then the algebraic Scarfcomplex (Taylor complex F∆I supported on ∆I) minimallyresolves R/I.

TheoremFix a monomial ideal I and a generic deformation ideal Iε. TheTaylor complex F∆ε

Igives a resolution for R/I of length ≤ d.

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Questions in commutative algebraImportance of monomial ideals

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Irreducible decompositionHilbert functionMinimal free resolution

References

Ezra Miller and Bernd Sturmfels, CombinatorialCommutative Algebra, Springer, New York (2005).

David Eisenbud, Commutative Algebra with a View TowardAlgebraic Geometry, Springer, New York (2004).

Thank you!

Kohlhaas Pictures of Comm Alg

Questions in commutative algebraImportance of monomial ideals

Answers to questions

Irreducible decompositionHilbert functionMinimal free resolution

References

Ezra Miller and Bernd Sturmfels, CombinatorialCommutative Algebra, Springer, New York (2005).

David Eisenbud, Commutative Algebra with a View TowardAlgebraic Geometry, Springer, New York (2004).

Thank you!

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