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TRANSCRIPT
![Page 1: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/1.jpg)
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
![Page 2: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/2.jpg)
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
![Page 3: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/3.jpg)
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
![Page 4: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/4.jpg)
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
![Page 5: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/5.jpg)
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
![Page 6: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/6.jpg)
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
![Page 7: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/7.jpg)
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
![Page 8: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/8.jpg)
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
![Page 9: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/9.jpg)
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
![Page 10: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/10.jpg)
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
![Page 11: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/11.jpg)
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
![Page 12: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/12.jpg)
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
![Page 13: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/13.jpg)
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
![Page 14: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/14.jpg)
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
![Page 15: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/15.jpg)
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
![Page 16: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/16.jpg)
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
![Page 17: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/17.jpg)
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
![Page 18: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/18.jpg)
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
![Page 19: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/19.jpg)
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
![Page 20: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/20.jpg)
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
![Page 21: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/21.jpg)
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
![Page 22: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/22.jpg)
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
![Page 23: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/23.jpg)
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
![Page 24: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/24.jpg)
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
![Page 25: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/25.jpg)
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
![Page 26: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/26.jpg)
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
![Page 27: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/27.jpg)
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
![Page 28: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/28.jpg)
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
![Page 29: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/29.jpg)
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
![Page 30: Pictures of Commutative Algebra - Loras Collegemyweb.loras.edu/ak265390/gsstalk.pdf · Pictures of Commutative Algebra Combinatorics of Monomial Ideals Angela Kohlhaas Department](https://reader036.vdocument.in/reader036/viewer/2022062603/5f0fd0587e708231d4460318/html5/thumbnails/30.jpg)
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
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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
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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)
Kohlhaas Pictures of Comm Alg
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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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Answers to questions
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
Answers to questions
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
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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).
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
Irreducible decompositionHilbert functionMinimal free resolution
I
m
x3y ·mI
xy4 ·m
I
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
Irreducible decompositionHilbert functionMinimal free resolution
I
m
x3y ·mI
xy4 ·m
I
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
Irreducible decompositionHilbert functionMinimal free resolution
I
m
x3y ·mI
xy4 ·m
I
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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.
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Answers to questions
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
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Answers to questions
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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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)
Kohlhaas Pictures of Comm Alg
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Questions in commutative algebraImportance of monomial ideals
Answers to questions
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
Answers to questions
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
Answers to questions
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
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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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!
Kohlhaas Pictures of Comm Alg