Post on 13-Jul-2020
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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].
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals 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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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 .
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals 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
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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)
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals 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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals 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.
Yi-Huang Shen On a class of squarefree monomial ideals 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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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?
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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).
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals 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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear 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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
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.
Yi-Huang Shen On a class of squarefree monomial ideals of linear type
Thank you!
Yi-Huang Shen (yhshen@ustc.edu.cn)
Yi-Huang Shen On a class of squarefree monomial ideals of linear type