on the stable set of associated prime ideals of monomial...

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On the stable set of associated prime ideals ofmonomial ideals and square-free monomial

ideals

Kazem Khashyarmanesh and Mehrdad Nasernejad

The 10th Seminar on Commutative Algebra and Related Topics,18-19 December 2013

(In honor of Professor Hossein Zakeri)

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Cover ideals

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Let R be a commutative Noetherian ring and I be an ideal of R.Brodmann showed that Ass(R/Is) = Ass(R/Is+1) for allsufficiently large s.

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A natural question arises in the context of Brodmann’sTheorem:(∗) Is it true thatAssR(R/I) ⊆ AssR(R/I2) ⊆ · · · ⊆ AssR(R/Ik ) ⊆ · · ·?McAdama presented an example which says, in general, theabove question has negative answer.

a McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103,Springer-Verlag, New York, 1983.

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Cover ideals

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......The ideal I is said to have the persistence property ifAss(R/Is) ⊆ Ass(R/Is+1) for all s ≥ 1.

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Let k be a fixed field and R = k [x1, . . . , xn] a polynomial ringover k . An ideal in R is monomial if it is generated by a set ofmonomials. A monomial ideal is square-free if it has agenerating set of monomials, where the exponent of eachvariable is at most 1.Problem : Do all square-free monomial ideals have thepersistence property?

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Cover ideals

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We recall the following definitions and construction.A graph G is said to be critically s-chromatic if χ(G) = s butχ(G\x) = s − 1 for every x ∈ V (G), where G\x denotes thegraph obtained from G by removing the vertex x and all edgesincident to x . A graph that is critically s-chromatic for some s iscalled critical.

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For any vertex xi ∈ V (G), the expansion of G at the vertex xi isthe graph G′ = G[{xi}] whose vertex set is given byV (G′) = (V (G)\{xi}) ∪ {xi,1, xi,2} and whose edge set has formE(G′) = {{u, v} ∈ E(G)|u = xi and v = xi}∪{{u, xi,1}, {u, xi,2}|{u, xi} ∈ E(G)} ∪ {{xi,1, xi,2}}.

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Cover ideals

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Equivalently, G[{xi}] is formed by replacing the vertex xi withthe clique K2 on the vertex set {xi,1, xi,2}. For any W ⊆ V (G),the expansion of G at W , denoted G[W ], is formed bysuccessively expanding all the vertices of W (in any order).

.C. A. Francisco, H. T. Ha and A. Van Tuyl..

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a Conjecture . Let s be a positive integer, and let G be a finitesimple graph that is critically s-chromatic. Then there exists asubset W ⊆ V (G) such that G[W ] is a critically(s + 1)-chromatic graph.

a C. A. Francisco, H. T. Ha and A. Van Tuyl, A conjecture on critical graphsand connections to the persistence of associated primes, Discrete Math. 310(2010), 2176-2182.

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Cover ideals

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.Definition..

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Let G be a finite simple graph on the vertex setV (G) = {x1, . . . , xn}. The cover ideal of G is the monomial idealJ = J(G) =

∩{xi ,xj}∈E(G)(xi , xj) ⊆ R = k [x1, . . . , xn].

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It is not hard to see that

J(G) = (xi1 . . . xir | W = {xi1 , . . . , xir } is a minimal vertex cover of G).

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Cover ideals

.(Francisco et al.(2010))..

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Theorem. Let G be a finite simple graph with cover idealJ = J(G). Let s ⩾ 1 and assume that the conjecture holds for(s + 1). Then

Ass(R/Js) ⊆ Ass(R/Js+1).

In particular, if the conjecture holds for all s, then J has thepersistence property.

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Cover ideals.Theorem..

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(T. Kaiser, M. Stehlik and R. Skrekovski)a The cover ideal of thefollowing graph does not have the persistence property.

a Replication in critical graphs and the persistence of monomial ideals, J.Combin. Theory, Ser. A, (to appear).

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Cover ideals

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Finally, Morey and Villarreala prove persistence for edge idealsI of any graphs containing a leaf (a vertex of degree 1).

a S. Morey and R. H. Villarreal, Edge ideals: algebraic and combinatorialproperties, Progress in Commutative Algebra, Combinatorics and Homology,Vol. 1, 2012, 85-126.

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persistence property

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.Definition..

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The smallest integer k0 such integerAss(R/Ik ) = Ass(R/Ik+1)for all k ≥ k0, denoted astab(I), is called the index of stability forthe associated prime ideals of I. Also the set AssR(R/Ik0) iscalled the stable set of associated prime ideals of I, which isdenoted by Ass∞(I).

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It has been shown, see McAdama and Bandari, Herzog andHibi,b that given any numbern there exists an ideal I in asuitable graded ring R and a prime ideal p of R such that, for allk ≤ n, p ∈ Ass(R/Ik ) if k is even and p ∈ Ass(R/Ik ) if k is odd.

a S McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics103, Springer-Verlag, New York, 1983.

bS. Bandari, J. Herzog, T. Hibi, Monomial ideals whose depth function hasany given number of strict local maxima, Preprint 2011

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.Definition..

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The edge ideal of a simple graph G, denoted by I(G), is theideal of R generated by all square-free monomials xixj suchthat {xi , xj} ∈ E(G). The assignment G −→ I(G) gives a naturalone to one correspondence between the family of graphs andthe family of monomial ideals generated by square-freemonomials of degree 2.

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.Theorem..

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Martinez-Bernal, Morey and Villarreala Let G be a graph and letI = I(G) be its edge ideal. Then

Ass(R/Ik ) ⊆ Ass(R/Ik+1)

for all k .a J. Martinez-Bernal, S. Morey and R. H. Villarreal, Associated primes of

powers of edge ideals, Collect. Math. 63 (2012), 361-374.

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persistence property.

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.Definition..

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J. Herzog, A. Qureshia, Let p ∈ V (I). We say that I satisfies thestrong persistence property with respect to p if, for all k and allf ∈ (Ik

p : pRp)\Ikp , there exists g ∈ Ip such that fg ∈ Ik+1

p . Theideal I is said to satisfy the strong persistence property if itsatisfies the strong persistence property for all p ∈ V (I).

a J. Herzog, A. Qureshi, Persistence and stability properties of powers ofideals (2012)

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.Theorem..

......The ideal I of R satisfies the strong persistence property if andonly if Ik+1 : I = Ik for all k.

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.Definition..

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An ideal I is polymatroidal if the following “exchange condition”is satisfied: For monomials u = xa1

1 . . . xann and v = xb1

1 . . . xbnn

belonging to G(I) and, for each i with ai > bi , one has j withaj < bj such that xju/xi ∈ G(I).

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.Proposition..

......Let I be a polymatroidal ideal. Then I satisfies the strong per-sistence property.

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.Definition..

......A graph G = (V (G),E(G)) is perfect if, for every inducedsubgraph GS, with S ⊆ V (G), we have χ(GS) = ω(GS).

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.Theorem..

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Francisco, Ha and Van Tuyla Let G be a perfect graph withcover ideal J. Then(1) Ass(R/Js) ⊆ Ass(R/Js+1) for all integers s ≥ 1.(2)

∞∪s=1

Ass(R/Js) =

χ(G)−1∪s=1

Ass(R/Js).

a C. A. Francisco, H. T. Ha, and A. Van Tuyl, Colorings of hypergraphs,perfect graphs, and associated primes of powers of monomial ideals, J.Algebra 331 (2011), 224-242.

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.Lemma........Let I be a monomial ideal. Then Ass(I t−1/I t) = Ass(R/I t).

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Stable set.

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.Theorem..

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Hoaa Let

B = max{d(rs + s + d)(√

r)r+1(√

2d)(r+1)(s−1)

, s(s + r)4sr+2d2(2d2)s2−s+1}.

Then we have

Ass(In/In+1) = Ass(IB/IB+1)

for all n ≥ B.aL.T. Hoa, Stability of associated primes of monomial ideals, Vietnam J.

Math. 34 (2006), no. 4, 473-487.

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Stable set

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.Example..

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Let d ≥ 4 and

I = (xd , xd−1y , xyd−1, yd , x2yd−2z) ⊂ K [x , y , z].

ThenAss(In−1/In) = {(x , y , z), (x , y)} if n < d − 2, andAss(In−1/In) = {(x , y)} if n ≥ d − 2.

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Stable set

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.Theorem..

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Bayati, Herzog and Rinaldoa Let p1, ..., pm ⊆ R be an arbitrarycollection of nonzero monomial prime ideals. Then there existsa monomial ideal I of R such that Ass∞(I) = {p1, ..., pm}.

aSh. Bayati, J. Herzog and G. Rinaldo, On the stable set of associatedprime ideals of a monomial ideal, Arch. Math. 98, No. 3, 213-217 (2012).

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Stable set

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.Question..

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suppose we are given two sets A = {p1, ..., pℓ} andB = {p′1, . . . , p′m} of monomial prime ideals such that theminimal elements of these sets with respect to inclusion are thesame. For which such sets does exist a monomial ideal I suchthat Ass(R/I) = A and Ass∞(I) = B?

For example, there is no monomial ideal I withAss(R/I) = {(x1), (x2)} and Ass∞(I) = {(x1), (x2), (x1, x2)}.

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Results

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.Remark..

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Let p1, . . . , pm be non-zero monomial prime ideals of R suchthat |G(pi)| ≤ |G(pj)| for all 1 ≤ i < j ≤ m. Then, for all d ∈ N,

AssR(R/pd1 ∩ p2d

2 ∩ p4d3 ∩ · · · ∩ p2m−1d

m ) = {p1, . . . , pm}.

This means that there exist infinite monomial ideals withassociated prime {p1, . . . , pm}.

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Results

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.Theorem..

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Let A = {p1, . . . , pm} and B = {p′1, . . . , p′t} be two arbitrary setsof monomial prime ideals of R. Then there exist monomialideals I and J of R with the following properties:

(i) AssR(R/I) = A ∪B, AssR(R/J) = B and(ii) I ⊆ J, AssR(J/I) = A\B.

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Results

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.Theorem..

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Let A = {p1, . . . , pm} and B = {p′1, . . . , p′t} be two arbitrary setsof monomial prime ideals of R. Then there exist monomialideals I and J of R such that

(i) Ass∞(I) = A ∪B, AssR(R/J) = B and(ii) I ⊆ J, AssR(J/I) = A\B.

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Results

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.Theorem..

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Let A = {p1, . . . , pm} be a set of non-zero monomial primeideals of R such that they are generated by disjoint non-emptysubsets of {x1, . . . , xn}. Also, suppose that {A1, . . . ,Ar} is apartition of A. Then there exist square-free monomial idealsI1, . . . , Ir such that, for all positive integers k1, . . . , kr ,d,

(i) AssR(R/Ikii ) = Ai ,

(ii) AssR(R/Ik1d1 . . . Ikr d

r ) = {p1, . . . , pm}, and

(iii) Ass∞(Ik11 . . . Ikr

r ) = Ass∞(Ik11 ) ∪ · · · ∪ Ass∞(Ikr

r ).

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Results

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Suppose that I is a monomial ideal of R with minimal generatingset {u1, . . . , um}. We say that I satisfies the condition (♯) if thereexists a nonnegative integer i with 1 ≤ i ≤ m such that

(uα11 . . . uαi−1

i−1 uαii uαi+1

i+1 . . . uαmm uj :R ui) =

uα11 . . .uαi−1

i−1 uαii uαi+1

i+1 . . . uαmm (uj :R ui)

for all j = 1, . . . ,m with j = i and α1, . . . , αm ≥ 0, where uαii

means that this term is omitted.

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Results

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.Theorem..

......Every ideal satisfies the condition (♯) has the persistenceproperty.

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Results

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.Definition..

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Let I be a monomial ideal of R with the unique minimal set ofmonomial generators G(I) = {u1, . . . , um}. Then we say that Iis a weakly monomial ideal if there exists i ∈ N with 1 ≤ i ≤ msuch that each monomial uj has no common factor with ui forall j ∈ N with 1 ≤ j ≤ m and j = i .

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.Example..

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Consider the ideal I = (x23 x5x3

6 , x31 x2

2 x44 , x

61 x3

2 x47 , x

22 x4

7 x54 ) in the

polynomial ring R = K [x1, x2, x3, x4, x5, x6, x7]. It is easy to seethat I is a weakly monomial ideal of R.

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Results

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.Definition..

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Let I be a monomial ideal of R with the unique minimal set ofmonomial generators G(I) = {u1, . . . , um}. Then we say that Iis a strongly monomial ideal if there exist i ∈ N with 1 ≤ i ≤ mand monomials g and w in R such that ui = wg, gcd(w ,g) = 1,and for all j ∈ N with 1 ≤ j = i ≤ m, gcd(uj ,ui) = w .

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Results.

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.Example..

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Consider the ideal

I = (x1x2x33 x5

4 , x21 x3

2 x43 x3

5 x56 , x

21 x2x5

3 x25 x6, x3

1 x22 x6

3 x46 )

in the polynomial ring R = K [x1, x2, x3, x4, x5, x6]. Then, bysetting

u1 := x1x2x33 x5

4 ,

u2 := x21 x3

2 x43 x3

5 x56 ,

u3 := x21 x2x5

3 x25 x6,

u4 := x31 x2

2 x63 x4

6 ,

i := 1 and w := x1x2x33 , clearly that I is a strongly monomial

ideal of R.

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Results

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.Theorem..

......Every strongly (or weakly) monomial ideal of R satisfiescondition (♯).

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Thanks For Your Patience

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