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

Transcription

1 On the stable set of associated prime ideals of monomial ideals and square-free monomial ideals Kazem Khashyarmanesh and Mehrdad Nasernejad The 10th Seminar on Commutative Algebra and Related Topics, December 2013 (In honor of Professor Hossein Zakeri)

2 Cover ideals Let R be a commutative Noetherian ring and I be an ideal of R Brodmann showed that Ass(R/I s ) = Ass(R/I s+1 ) for all sufficiently large s A natural question arises in the context of Brodmann s Theorem: ( ) Is it true that Ass R (R/I) Ass R (R/I 2 ) Ass R (R/I k )? McAdam a presented an example which says, in general, the above question has negative answer a McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103, Springer-Verlag, New York, 1983

3 Cover ideals Let R be a commutative Noetherian ring and I be an ideal of R Brodmann showed that Ass(R/I s ) = Ass(R/I s+1 ) for all sufficiently large s A natural question arises in the context of Brodmann s Theorem: ( ) Is it true that Ass R (R/I) Ass R (R/I 2 ) Ass R (R/I k )? McAdam a presented an example which says, in general, the above question has negative answer a McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103, Springer-Verlag, New York, 1983

4 Cover ideals The ideal I is said to have the persistence property if Ass(R/I s ) Ass(R/I s+1 ) for all s 1 Let k be a fixed field and R = k[x 1,, x n ] a polynomial ring over k An ideal in R is monomial if it is generated by a set of monomials A monomial ideal is square-free if it has a generating set of monomials, where the exponent of each variable is at most 1 Problem : Do all square-free monomial ideals have the persistence property?

5 Cover ideals The ideal I is said to have the persistence property if Ass(R/I s ) Ass(R/I s+1 ) for all s 1 Let k be a fixed field and R = k[x 1,, x n ] a polynomial ring over k An ideal in R is monomial if it is generated by a set of monomials A monomial ideal is square-free if it has a generating set of monomials, where the exponent of each variable is at most 1 Problem : Do all square-free monomial ideals have the persistence property?

6 Cover ideals We recall the following definitions and construction A graph G is said to be critically s-chromatic if χ(g) = s but χ(g\x) = s 1 for every x V (G), where G\x denotes the graph obtained from G by removing the vertex x and all edges incident to x A graph that is critically s-chromatic for some s is called critical For any vertex x i V (G), the expansion of G at the vertex x i is the graph G = G[{x i }] whose vertex set is given by V (G ) = (V (G)\{x i }) {x i,1, x i,2 } and whose edge set has form E(G ) = {{u, v} E(G) u x i and v x i } {{u, x i,1 }, {u, x i,2 } {u, x i } E(G)} {{x i,1, x i,2 }}

7 Cover ideals We recall the following definitions and construction A graph G is said to be critically s-chromatic if χ(g) = s but χ(g\x) = s 1 for every x V (G), where G\x denotes the graph obtained from G by removing the vertex x and all edges incident to x A graph that is critically s-chromatic for some s is called critical For any vertex x i V (G), the expansion of G at the vertex x i is the graph G = G[{x i }] whose vertex set is given by V (G ) = (V (G)\{x i }) {x i,1, x i,2 } and whose edge set has form E(G ) = {{u, v} E(G) u x i and v x i } {{u, x i,1 }, {u, x i,2 } {u, x i } E(G)} {{x i,1, x i,2 }}

8 Cover ideals Equivalently, G[{x i }] is formed by replacing the vertex x i with the clique K 2 on the vertex set {x i,1, x i,2 } For any W V (G), the expansion of G at W, denoted G[W ], is formed by successively expanding all the vertices of W (in any order) C A Francisco, H T Há and A Van Tuyl a Conjecture Let s be a positive integer, and let G be a finite simple graph that is critically s-chromatic Then there exists a subset W V (G) such that G[W ] is a critically (s + 1)-chromatic graph a C A Francisco, H T Há and A Van Tuyl, A conjecture on critical graphs and connections to the persistence of associated primes, Discrete Math 310 (2010),

9 Cover ideals Equivalently, G[{x i }] is formed by replacing the vertex x i with the clique K 2 on the vertex set {x i,1, x i,2 } For any W V (G), the expansion of G at W, denoted G[W ], is formed by successively expanding all the vertices of W (in any order) C A Francisco, H T Há and A Van Tuyl a Conjecture Let s be a positive integer, and let G be a finite simple graph that is critically s-chromatic Then there exists a subset W V (G) such that G[W ] is a critically (s + 1)-chromatic graph a C A Francisco, H T Há and A Van Tuyl, A conjecture on critical graphs and connections to the persistence of associated primes, Discrete Math 310 (2010),

10 Cover ideals Definition Let G be a finite simple graph on the vertex set V (G) = {x 1,, x n } The cover ideal of G is the monomial ideal J = J(G) = {x i,x j } E(G) (x i, x j ) R = k[x 1,, x n ] It is not hard to see that J(G) = (x i1 x ir W = {x i1,, x ir } is a minimal vertex cover of G)

11 Cover ideals Definition Let G be a finite simple graph on the vertex set V (G) = {x 1,, x n } The cover ideal of G is the monomial ideal J = J(G) = {x i,x j } E(G) (x i, x j ) R = k[x 1,, x n ] It is not hard to see that J(G) = (x i1 x ir W = {x i1,, x ir } is a minimal vertex cover of G)

12 Cover ideals (Francisco et al(2010)) Theorem Let G be a finite simple graph with cover ideal J = J(G) Let s 1 and assume that the conjecture holds for (s + 1) Then Ass(R/J s ) Ass(R/J s+1 ) In particular, if the conjecture holds for all s, then J has the persistence property

13 Cover ideals Theorem (T Kaiser, M Stehlik and R Skrekovski) a The cover ideal of the following 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)

14 Cover ideals Finally, Morey and Villarreal a prove persistence for edge ideals I of any graphs containing a leaf (a vertex of degree 1) a S Morey and R H Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in Commutative Algebra, Combinatorics and Homology, Vol 1, 2012,

15 persistence property Definition The smallest integer k 0 such integerass(r/i k ) = Ass(R/I k+1 ) for all k k 0, denoted astab(i), is called the index of stability for the associated prime ideals of I Also the set Ass R (R/I k 0) is called the stable set of associated prime ideals of I, which is denoted by Ass (I)

16 persistence property It has been shown, see McAdam a and Bandari, Herzog and Hibi, b that given any numbern there exists an ideal I in a suitable graded ring R and a prime ideal p of R such that, for all k n, p Ass(R/I k ) if k is even and p Ass(R/I k ) if k is odd a S McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103, Springer-Verlag, New York, 1983 b S Bandari, J Herzog, T Hibi, Monomial ideals whose depth function has any given number of strict local maxima, Preprint 2011

17 persistence property Definition The edge ideal of a simple graph G, denoted by I(G), is the ideal of R generated by all square-free monomials x i x j such that {x i, x j } E(G) The assignment G I(G) gives a natural one to one correspondence between the family of graphs and the family of monomial ideals generated by square-free monomials of degree 2

18 persistence property Theorem Martinez-Bernal, Morey and Villarreal a Let G be a graph and let I = I(G) be its edge ideal Then for all k Ass(R/I k ) Ass(R/I k+1 ) a J Martinez-Bernal, S Morey and R H Villarreal, Associated primes of powers of edge ideals, Collect Math 63 (2012),

19 persistence property Definition J Herzog, A Qureshi a, Let p V (I) We say that I satisfies the strong persistence property with respect to p if, for all k and all f (Ip k : pr p )\Ip k, there exists g I p such that fg Ip k+1 The ideal I is said to satisfy the strong persistence property if it satisfies the strong persistence property for all p V (I) a J Herzog, A Qureshi, Persistence and stability properties of powers of ideals (2012) Theorem The ideal I of R satisfies the strong persistence property if and only if I k+1 : I = I k for all k

20 persistence property Definition J Herzog, A Qureshi a, Let p V (I) We say that I satisfies the strong persistence property with respect to p if, for all k and all f (Ip k : pr p )\Ip k, there exists g I p such that fg Ip k+1 The ideal I is said to satisfy the strong persistence property if it satisfies the strong persistence property for all p V (I) a J Herzog, A Qureshi, Persistence and stability properties of powers of ideals (2012) Theorem The ideal I of R satisfies the strong persistence property if and only if I k+1 : I = I k for all k

21 persistence property Definition An ideal I is polymatroidal if the following exchange condition is satisfied: For monomials u = x a 1 1 x n an and v = x b 1 1 x n bn belonging to G(I) and, for each i with a i > b i, one has j with a j < b j such that x j u/x i G(I) Proposition Let I be a polymatroidal ideal Then I satisfies the strong per- sistence property

22 persistence property Definition An ideal I is polymatroidal if the following exchange condition is satisfied: For monomials u = x a 1 1 x n an and v = x b 1 1 x n bn belonging to G(I) and, for each i with a i > b i, one has j with a j < b j such that x j u/x i G(I) Proposition Let I be a polymatroidal ideal Then I satisfies the strong per- sistence property

23 persistence property Definition A graph G = (V (G), E(G)) is perfect if, for every induced subgraph G S, with S V (G), we have χ(g S ) = ω(g S )

24 persistence property Theorem Francisco, Há and Van Tuyl a Let G be a perfect graph with cover ideal J Then (1) Ass(R/J s ) Ass(R/J s+1 ) for all integers s 1 (2) χ(g) 1 Ass(R/J s ) = Ass(R/J s ) s=1 s=1 a C A Francisco, H T Há, and A Van Tuyl, Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals, J Algebra 331 (2011),

25 persistence property Lemma Ḷet I be a monomial ideal Then Ass(I t 1 /I t ) = Ass(R/I t )

26 Stable set Theorem Hoa a Let Then we have for all n B B = max{d(rs + s + d)( r) r+1 ( 2d) (r+1)(s 1) Math 34 (2006), no 4, , s(s + r) 4 s r+2 d 2 (2d 2 ) s2 s+1 } Ass(I n /I n+1 ) = Ass(I B /I B+1 ) a LT Hoa, Stability of associated primes of monomial ideals, Vietnam J

27 Stable set Example Let d 4 and I = (x d, x d 1 y, xy d 1, y d, x 2 y d 2 z) K [x, y, z] Then Ass(I n 1 /I n ) = {(x, y, z), (x, y)} if n < d 2, and Ass(I n 1 /I n ) = {(x, y)} if n d 2

28 Stable set Theorem Bayati, Herzog and Rinaldo a Let p 1,, p m R be an arbitrary collection of nonzero monomial prime ideals Then there exists a monomial ideal I of R such that Ass (I) = {p 1,, p m } a Sh Bayati, J Herzog and G Rinaldo, On the stable set of associated prime ideals of a monomial ideal, Arch Math 98, No 3, (2012)

29 Stable set Question suppose we are given two sets A = {p 1,, p l } and B = {p 1,, p m} of monomial prime ideals such that the minimal elements of these sets with respect to inclusion are the same For which such sets does exist a monomial ideal I such that Ass(R/I) = A and Ass (I) = B? For example, there is no monomial ideal I with Ass(R/I) = {(x 1 ), (x 2 )} and Ass (I) = {(x 1 ), (x 2 ), (x 1, x 2 )}

30 Results Remark Let p 1,, p m be non-zero monomial prime ideals of R such that G(p i ) G(p j ) for all 1 i < j m Then, for all d N, Ass R (R/p d 1 p2d 2 p4d 3 p2m 1 d m ) = {p 1,, p m } This means that there exist infinite monomial ideals with associated prime {p 1,, p m }

31 Results Theorem Let A = {p 1,, p m } and B = {p 1,, p t } be two arbitrary sets of monomial prime ideals of R Then there exist monomial ideals I and J of R with the following properties: (i) Ass R (R/I) = A B, Ass R (R/J) = B and (ii) I J, Ass R (J/I) = A\B

32 Results Theorem Let A = {p 1,, p m } and B = {p 1,, p t } be two arbitrary sets of monomial prime ideals of R Then there exist monomial ideals I and J of R such that (i) Ass (I) = A B, Ass R (R/J) = B and (ii) I J, Ass R (J/I) = A\B

33 Results Theorem Let A = {p 1,, p m } be a set of non-zero monomial prime ideals of R such that they are generated by disjoint non-empty subsets of {x 1,, x n } Also, suppose that {A 1,, A r } is a partition of A Then there exist square-free monomial ideals I 1,, I r such that, for all positive integers k 1,, k r, d, (i) Ass R (R/I k i i ) = A i, (ii) Ass R (R/I k 1d 1 Ir kr d ) = {p 1,, p m }, and (iii) Ass (I k 1 1 Ikr r ) = Ass (I k 1 1 ) Ass (Ir kr )

34 Results Suppose that I is a monomial ideal of R with minimal generating set {u 1,, u m } We say that I satisfies the condition ( ) if there exists a nonnegative integer i with 1 i m such that (u α 1 1 uα i 1 i 1 ûα i i u α i+1 i+1 uα m m u j : R u i ) = u α 1 1 uα i 1 i 1 ûα i i u α i+1 i+1 uα m m (u j : R u i ) for all j = 1,, m with j i and α 1,, α m 0, where ûα i i means that this term is omitted

35 Results Theorem Every ideal satisfies the condition ( ) has the persistence property

36 Results Definition Let I be a monomial ideal of R with the unique minimal set of monomial generators G(I) = {u 1,, u m } Then we say that I is a weakly monomial ideal if there exists i N with 1 i m such that each monomial u j has no common factor with u i for all j N with 1 j m and j i Example Consider the ideal I = (x3 2x 5x6 3, x 1 3x 2 2x 4 4, x 1 6x 2 3x 7 4, x 2 2x 7 4x 4 5 ) in the polynomial ring R = K [x 1, x 2, x 3, x 4, x 5, x 6, x 7 ] It is easy to see that I is a weakly monomial ideal of R

37 Results Definition Let I be a monomial ideal of R with the unique minimal set of monomial generators G(I) = {u 1,, u m } Then we say that I is a weakly monomial ideal if there exists i N with 1 i m such that each monomial u j has no common factor with u i for all j N with 1 j m and j i Example Consider the ideal I = (x3 2x 5x6 3, x 1 3x 2 2x 4 4, x 1 6x 2 3x 7 4, x 2 2x 7 4x 4 5 ) in the polynomial ring R = K [x 1, x 2, x 3, x 4, x 5, x 6, x 7 ] It is easy to see that I is a weakly monomial ideal of R

38 Results Definition Let I be a monomial ideal of R with the unique minimal set of monomial generators G(I) = {u 1,, u m } Then we say that I is a strongly monomial ideal if there exist i N with 1 i m and monomials g and w in R such that u i = wg, gcd(w, g) = 1, and for all j N with 1 j i m, gcd(u j, u i ) = w

39 Results Example Consider the ideal I = (x 1 x 2 x 3 3 x 5 4, x 2 1 x 3 2 x 4 3 x 3 5 x 5 6, x 2 1 x 2x 5 3 x 2 5 x 6, x 3 1 x 2 2 x 6 3 x 4 6 ) in the polynomial ring R = K [x 1, x 2, x 3, x 4, x 5, x 6 ] Then, by setting u 1 := x 1 x 2 x 3 3 x 5 4, u 2 := x 2 1 x 3 2 x 4 3 x 3 5 x 5 6, u 3 := x 2 1 x 2x 5 3 x 2 5 x 6, u 4 := x 3 1 x 2 2 x 6 3 x 4 6, i := 1 and w := x 1 x 2 x3 3, clearly that I is a strongly monomial ideal of R

40 Results Theorem Every strongly (or weakly) monomial ideal of R satisfies condition ( )

41 Thanks For Your Patience