On a class of squarefree monomial ideals of linear typemath.sjtu.edu.cn/conference/Bannai/2013/data/20131102A/slides.pdf · Two ways to connect squarefree monomial ideals to combinatorial

On a class of squarefree monomial

ideals of linear type

Yi-Huang Shen

University of Science and Technology of China

Shanghai / November 2, 2013

Basic definition

Let K be a field and S = K[x1, . . . , xn] a polynomial ring of nvariables. A monomial xa := xa11 xa22 · · · xann ∈ S is squarefree ifeach ai ∈ { 0, 1 }. Its degree is deg(xa) = a1 + · · ·+ an. An ideal Iof S is squarefree if it can be (minimally) generated by a (finiteand unique) set of squarefree monomials. A squarefree monomialideal of degree 2 (i.e., a quadratic monomial ideal) is a squarefreemonomial ideal whose minimal monomial generators are all ofdegree 2.

Yi-Huang Shen On a class of squarefree monomial ideals of linear type

Two ways to connect squarefree monomial ideals to

combinatorial objects

1 I is the Stanley-Reisner ideal of some simplicial complex.

2 I is the facet ideal of another simplicial complex. Equivalently,I is the (hyper)edge ideal of some clutter.

Definition

Let V be a finite set. A clutter C with vertex set V (C) = Vconsists of a set E (C) of subsets of V , called the edges of C, withthe property that no edge contains another. Clutters are specialhypergraphs.

Squarefree ideals of degree 2 ⇔ (finite simple) graphs.

Squarefree ideals of higher degree ⇔ clutters of higherdimension.


Examples

Example (1)

5

3

1

24

〈x1x2, x2x5, x3x5, x1x3, x1x4〉

⊂ K[x1, . . . , x5].

Example (2)

3

7

6 9

10

8

5

12 11

2

1 4F3

F4

F2 G

F1〈x1x2x5x6, x2x3x7x8, x3x4x9x10, x1x4x11x12,

x3x8x9〉 ⊂ K[x1, . . . , x12].


Interplay between combinatorics and commutative algebra

Commutative algebra ⇒ combinatorics

E.g., Richard Stanley’s proof of the Upper Bound Conjecture

for simplicial spheres bymeans of the theory of Cohen-Macaulay rings.

Combinatorics ⇒ commutative algebra

E.g., if G is a graph and each of its connected components has atmost one odd cycle (i.e., each component either has no cycle, orhas no even cycle), then its edge ideal I (G ) is of linear type.


Commutative algebra background: the harder way

Let S be a Noetherian ring and I an S-ideal. The Rees algebra of Iis the subring of the ring of polynomials S [t]

R(I ) := S [It] = ⊕i≥0Ii t i .

Analogously, one has Sym(I ), the symmetric algebra of I which isobtained from the tensor algebra of I by imposing thecommutative law.

There is a canonical surjection Φ: Sym(I ) ։ R(I ). When thecanonical map Φ is an isomorphism, I is called an ideal of lineartype.


Commutative algebra background: the harder way

The symmetric algebra Sym(I ) is equipped with an S-Modulehomomorphism π : I → Sym(I ) which solves the following universalproblem. For a commutative S-algebra B and any S-modulehomomorphism ϕ : I → B , there exists a unique S-algebrahomomorphism Φ: Sym(I ) → B such that the diagram

Iϕ

//

π

��

B

Sym(I )

Φ

;;①①①①①①①①①

is commutative.


Commutative algebra background: the easier way

Suppose I = 〈f1, . . . , fs〉 and consider the S-linear presentation

ψ : S [T ] := S [T1, . . . ,Ts ] → S [It]

defined by setting ψ(Ti ) = fi t. Since this map is homogeneous,the kernel J =

⊕i≥1 Ji is a graded ideal; it will be called the

defining ideal of R(I ) (with respect to this presentation). Sincethe linear part J1 generates the defining ideal of Sym(R), I is oflinear type if and only if J = 〈J1〉.

The maximal degree in T of the minimal generators of the definingideal J is called the relation type of I .


Example of defining ideals

Example (3)

Let S = K[x1, . . . , x7] and I be the ideal of S generated byf1 = x1x2x3, f2 = x2x4x5, f3 = x5x6x7 and f4 = x3x6x7. Then thedefining ideal is minimally generated by x3T3 − x5T4,x6x7T1 − x1x2T4, x6x7T2 − x2x4T3, x4x5T1 − x1x3T2 andx4T1T3 − x1T2T4.

Check for x4T1T3 − x1T2T4:

x4T1T3 7→ x4(x1x2x3t)(x5x6x7t),x1T2T4 7→ x1(x2x4x5t)(x3x6x7t).

This minimal generator of the defining ideal is of degree 2 in T .Thus the ideal I is not of linear type. Indeed, its relation type is 2.


The defining ideal is binomial

The defining ideal of squarefree monomial ideals are alwaysbinomial, i.e., are generated by binomials.

Theorem (Taylor)

Suppose I is minimally generated by monomials f1, . . . , fs . Let Ikbe the set of non-decreasing sequence of integers in { 1, 2, . . . , s }of length k. If α = (i1, i2, . . . , ik) ∈ Ik , set f α = fi1 · · · fik andTα = Ti1 · · ·Tik . For every α,β ∈ Ik , set

Tα,β =f β

gcd(f α, f β)Tα −

f α

gcd(f α, f β)Tβ.

Then the defining ideal J is generated by these Tα,β’s withα,β ∈ Ik and k ≥ 1.


How to compute?

Q: How to compute the defining ideal? A: Grobner basistheory.

Q: How to check the minimality? A: Grobner basis theory.

Websites:

Macaulay2 → http://www.math.uiuc.edu/Macaulay2/

Singular → http://www.singular.uni-kl.de/

CoCoA System → http://cocoa.dima.unige.it/

Example (2, continued)

〈x1x2x5x6, x2x3x7x8, x3x4x9x10, x1x4x11x12, x3x8x9〉 ⊂ K[x1, . . . , x12]is of linear type.


Macaulay 2 codes for Example 2

[10:31:27][2013SJTU]$ M2

Macaulay2, version 1.6

with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,

PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[x_1..x_12]

o1 = S

o1 : PolynomialRing

i2 : I = monomialIdeal(x_1*x_2*x_5*x_6,x_2*x_3*x_7*x_8,x_3*x_4*x_9*x_10,

x_1*x_4*x_11*x_12,x_3*x_8*x_9)

o2 = monomialIdeal (x x x x , x x x x , x x x , x x x x , x x x x )

1 2 5 6 2 3 7 8 3 8 9 3 4 9 10 1 4 11 12

o2 : MonomialIdeal of S

i3 : isLinearType ideal I

o3 = true


Singular codes for Example 3

> LIB "reesclos.lib";

> ring S=0,(x(1..7)),dp;

> ideal I=x(1)*x(2)*x(3),x(2)*x(4)*x(5), x(5)*x(6)*x(7),

x(3)*x(6)*x(7);

> list L=ReesAlgebra(I);

> def Rees=L[1];

> setring Rees;

> Rees;

// characteristic : 0

// number of vars : 11

// block 1 : ordering dp

// : names x(1) x(2) x(3) x(4) x(5) x(6) x(7)

U(1) U(2) U(3) U(4)

// block 2 : ordering C

> ker;

ker[1]=x(3)*U(3)-x(5)*U(4)

ker[2]=x(4)*U(1)*U(3)-x(1)*U(2)*U(4)

ker[3]=x(6)*x(7)*U(2)-x(2)*x(4)*U(3)

ker[4]=x(6)*x(7)*U(1)-x(1)*x(2)*U(4)

ker[5]=x(4)*x(5)*U(1)-x(1)*x(3)*U(2)


Singular codes for Example 3, continued

> ideal NewVars=U(1),U(2),U(3),U(4);

> ideal LI=reduce(ker,std(NewVars^2));

> LI;

LI[1]=x(3)*U(3)-x(5)*U(4)

LI[2]=0

LI[3]=x(6)*x(7)*U(2)-x(2)*x(4)*U(3)

LI[4]=x(6)*x(7)*U(1)-x(1)*x(2)*U(4)

LI[5]=x(4)*x(5)*U(1)-x(1)*x(3)*U(2)

> reduce(ker,std(LI));

_[1]=0

_[2]=x(4)*U(1)*U(3)-x(1)*U(2)*U(4)

_[3]=0

_[4]=0

_[5]=0


Villarreal’s result

Theorem (Villarreal)

Let G be a connected graph and I = I (G) its edge ideal. Then I isan ideal of linear type if and only if G is a tree or G has a uniquecycle of odd length. This result is independent of the characteristicof the base field K.

Example (4)

5

1

2

4

3

6

3

2

1

The Stanley-Reisner ring of the realprojective plane is Cohen-Macaulay ifand only if the characteristic of thebase field is not 2.


Fouli and Lin’s result

Definition (Generator graph)

Let I be a squarefree monomial ideal whose minimal monomialgenerating set is { f1, . . . , fs }. Let G be a graph whose vertices vicorresponds to fi respectively and two vertices vi and vj areadjacent if and only if the two monomials fi and fj have anon-trivial GCD. This graph G is called the generator graph of I .

Theorem (Fouli and Lin)

When I is a squarefree monomial ideal and the generator graph ofI is the graph of a disjoint union of trees and graphs with a uniqueodd cycle, then I is an ideal of linear type.


New idea

Observation

Let I be a monomial ideal in S = K[x1, . . . , xn]. Let xn+1 be a newvariable with S ′ = K[x1, . . . , xn, xn+1]. Then I is a squarefreemonomial ideal if and only if I ′ = I · xn+1 is so. And I is of lineartype if and only if I ′ is so. Indeed, I ′ and I will have essentiallyidentical defining ideals. However, the generator graph of I ′ is acomplete graph.


Leaves and quasi-forests

Definition

Let ∆ be a clutter. The edge F of ∆ is a leaf of ∆ if there existsan edge G such that (H ∩ F ) ⊆ (G ∩ F ) for all edges H ∈ ∆. Theedge G is called a branch or joint of F .

Definition

A clutter ∆ is called a quasi-forest if there exists a total order ofthe edges { F1, . . . ,Fm } such that Fi is a leaf of the sub-clutter〈F1, . . . ,Fi 〉 for all i = 1, . . . ,m. This order is called a leaf order ofthe quasi-forest. A connected quasi-forest is called a quasi-tree.


Forests

Definition

A (simplicial) forest is a clutter ∆ which enjoys the property thatfor every subset

{Fi1 , . . . ,Fiq

}of F(∆) the sub-clutter

〈Fi1 , . . . ,Fiq〉 of ∆ has a leaf. A tree is a forest which is connected.

Facts

1 Edge ideals of forests are always of linear type.

2 Edge ideals of quasi-forests are not necessarily of linear type.


Example

Example (5)

3

4

6

51

2

8

7

This is a quasi-tree, but not a tree. Its edge ideal

〈x1x2x3x4, x1x4x5, x1x2x8, x2x3x7, x3x4x6〉

is not of linear type.


Good leaves

Definition

An edge F of the clutter ∆ is called a good leaf if this F is a leafof each sub-clutter Γ of ∆ to which F belongs. An orderF1, . . . ,Fs of the edges is called a good leaf order if Fi is a goodleaf of 〈F1, . . . ,Fi 〉 for each i = 1, . . . , s.

It is known that a clutter is a forest if and only if it has a good leaforder.


Related result

Theorem (Shen)

Suppose ∆ is a clutter which is obtained from the clutter ∆′ byadding a good leaf. If the edge ideal of ∆′ is of linear type, thenthe edge ideal of ∆ also shares this property.

Tools: Grobner basis. This result reproves the fact that the edgeideals of forests are of linear type.

Question: Is the converse true?

Suppose ∆ is a clutter which is obtained from the clutter ∆′ byadding a good leaf. If the edge ideal of ∆ is of linear type, doesthe edge ideal of ∆′ also share this property?


More details

Theorem (Conca and De Negri)

If I is a monomial ideal which is generated by an M-sequence, thenI is of linear type.

Facts

The monomial ideal I is generated by an M-sequence if and only ifthis I is of forest type (in the sense of Soleyman Jahan and Zheng).In particular, when I is squarefree, I is generated by an M-sequenceif and only if it is the edge ideal of a simplicial forest, and thisM-sequence corresponds to the good leaf order of the forest.


Simplicial cycles

Definition

A clutter ∆ is called a simplicial cycle or simply a cycle if ∆ has noleaf but every nonempty proper sub-clutter of ∆ has a leaf.

This definition is more restrictive than the classic definition of(hyper)cycles of hypergraphs due to Berge.

Fact

If ∆ is a simplicial cycle. Then

1 either the generator graph of the edge ideal of ∆ is a cycle, or,

2 ∆ is a cone over such a structure.


Villarreal class

Definition

Let V be the class of clutters minimal with respect to the followingproperties:

Disjoint simplicial cycles of odd lengths are in V , withsimplexes being considered as simplicial cycles of length 1.

V is closed under the operation of attaching good leaves.

We shall call V the Villarreal class. When a clutter ∆ is in V , wesay ∆ and its edge ideal I (∆) are of Villarreal type.

Theorem (Shen)

Squarefree monomial ideals of Villarreal type are of linear type (butnot vice versa).


Example

Example (6)

∆ =3

7

6 9

10

8

5

12 11

2

1 4F3

F4

F2 G

F1 ∆ =

87

6 9

105

12 11

2

1 4

3

13

F3

F4

F2

F1

G

∆ is not a simplicial cycle and has no leaves. It is not of Villarrealtype, but its edge ideal is of linear type. On the other hand, theideal of ∆ is not of linear type.


With a patch?

Example (6, continued)

∆′ =3

7

6 9

10

8

5

12 11

2

1 4F3

F4

F2

F1

This ∆′ is a simplicial cycleof length 4. Both ∆ and ∆are obtained from ∆ byattaching new edges. The Gin ∆ introduces a new vertexwhile the Γ in ∆ does not.The Γ is a patch attached to∆′ connecting the adjacentedges F2 and F3.

Theorem (Shen)

Suppose ∆′ is a simplicial cycle of even length and ∆ is obtainedfrom ∆′ by attaching a patch. Then the edge ideal of ∆ is oflinear type.


References

1 R. H. Villarreal, Rees algebras of edge ideals, Comm. Algebra23 (1995), 3513–3524.

2 L. Fouli and K.-N. Lin, Rees algebras of square-free monomialideals (2012), available at arXiv:1205.3127.

3 Y.-H. Shen, On a class of squarefree monomial ideals of lineartype (2013), available at arXiv:1309.1072.

4 A. Conca and E. De Negri, M-sequences, graph ideals, andladder ideals of linear type, J. Algebra 211 (1999), 599–624.

5 A. Alilooee and S. Faridi, When is a squarefree monomialideal of linear type? (2013), available at arXiv:1309.1771.


Further reading

1 Takayuki Hibi, Algebraic Combinatorics on Convex Polytopes,Carslaw Publications.

2 Richard Stanley, Combinatorics and Commutative Algebra,Birkhauser.

3 Ezra Miller and Bernd Sturmfels, Combinatorial CommutativeAlgebra, GTM 227, Springer.

4 Susan Morey and Rafael H. Villarreal, Edge Ideals: Algebraicand Combinatorial Properties, in Progress in CommutativeAlgebra 1: Combinatorics and Homology, De Gruyter.(available in arXiv:1012.5329)

5 Gert-Martin Greuel and Gerhard Pfister, A SingularIntroduction to Commutative Algebra, Springer.


Thank you!

Yi-Huang Shen ([email protected])


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On a class of squarefree monomial ideals of linear typemath.sjtu.edu.cn/conference/Bannai/2013/data/20131102A/slides.pdf · Two ways to connect squarefree monomial ideals to combinatorial