Regularity of powers of edge ideals Huy Ti H Tulane University - - PowerPoint PPT Presentation

Regularity of powers of edge ideals

Huy Tài Hà Tulane University Joint with Selvi Beyarslan and Trân Nam Trung

Huy Tài Hà Regularity of powers of edge ideals

1

Problem and Motivation Asymptotic linearity of regularity Stabilization index and free constant

2

Literature Reviews Polynomial ideals Edge ideals Known answers

3

Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds

Huy Tài Hà Regularity of powers of edge ideals

Asymptotic linearity of regularity

R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R-module end(M) := max{l | Ml = 0} The regularity of M is reg(M) = max{end(Hi

m(M)) + i}.

Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg(IqM) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0, reg(IqM) = aq + b.

Huy Tài Hà Regularity of powers of edge ideals

Asymptotic linearity of regularity

R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R-module end(M) := max{l | Ml = 0} The regularity of M is reg(M) = max{end(Hi

m(M)) + i}.

Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg(IqM) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0, reg(IqM) = aq + b.

Huy Tài Hà Regularity of powers of edge ideals

Asymptotic linearity of regularity

R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R-module end(M) := max{l | Ml = 0} The regularity of M is reg(M) = max{end(Hi

m(M)) + i}.

Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg(IqM) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0, reg(IqM) = aq + b.

Huy Tài Hà Regularity of powers of edge ideals

Stabilization index and free constant

The constant a is well understood (Trung-Wang (2005)) The free constant b and the stabilization index q0 = min{q′ | reg(IqM) = aq + b ∀ q ≥ q′} are not known. Problem Understand b and q0 from invariants and properties of I and M Explicitly compute b and q0 for special classes of ideals and modules.

Huy Tài Hà Regularity of powers of edge ideals

Stabilization index and free constant

The constant a is well understood (Trung-Wang (2005)) The free constant b and the stabilization index q0 = min{q′ | reg(IqM) = aq + b ∀ q ≥ q′} are not known. Problem Understand b and q0 from invariants and properties of I and M Explicitly compute b and q0 for special classes of ideals and modules.

Huy Tài Hà Regularity of powers of edge ideals

1

Problem and Motivation Asymptotic linearity of regularity Stabilization index and free constant

2

Literature Reviews Polynomial ideals Edge ideals Known answers

3

Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds

Huy Tài Hà Regularity of powers of edge ideals

Polynomial ideals

R = k[x1, . . . , xn] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m-primary then q0 can be related to “partial” regularity of the Rees algebra of I. In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute.

Huy Tài Hà Regularity of powers of edge ideals

Polynomial ideals

R = k[x1, . . . , xn] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m-primary then q0 can be related to “partial” regularity of the Rees algebra of I. In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute.

Huy Tài Hà Regularity of powers of edge ideals

Polynomial ideals

R = k[x1, . . . , xn] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m-primary then q0 can be related to “partial” regularity of the Rees algebra of I. In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute.

Huy Tài Hà Regularity of powers of edge ideals

Edge ideals of graphs

V = {x1, . . . , xn} ← → R = k[x1, . . . , xn] G = (V, E) a simple graph The edge ideal of G is I(G) =

• xixj
• {xi, xj} ∈ E
• .

Example

✈

x2

✈ x4 ✈ x5 ✈ x6 ✈

x3

✈

x1

❅ ❅ ❅

• ❅

❅ ❅

I(G) = (x1x2, x2x3, x2x4, x4x5, x4x6) ⊆ k[x1, . . . , x6].

Huy Tài Hà Regularity of powers of edge ideals

Edge ideals of graphs

V = {x1, . . . , xn} ← → R = k[x1, . . . , xn] G = (V, E) a simple graph The edge ideal of G is I(G) =

• xixj
• {xi, xj} ∈ E
• .

Example

✈

x2

✈ x4 ✈ x5 ✈ x6 ✈

x3

✈

x1

❅ ❅ ❅

• ❅

❅ ❅

I(G) = (x1x2, x2x3, x2x4, x4x5, x4x6) ⊆ k[x1, . . . , x6].

Huy Tài Hà Regularity of powers of edge ideals

Asymptotic linearity of edge ideals

Know: for q ≥ q0, reg(I(G)q) = 2q + b. Problem Relate q0 and b to combinatorial data of the graph G For special classes of graphs, compute q0 and b explicitly.

Huy Tài Hà Regularity of powers of edge ideals

Asymptotic linearity of edge ideals

Know: for q ≥ q0, reg(I(G)q) = 2q + b. Problem Relate q0 and b to combinatorial data of the graph G For special classes of graphs, compute q0 and b explicitly.

Huy Tài Hà Regularity of powers of edge ideals

Known answers

Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the complement graph Gc is chordal then I(G)q has a linear resolution for all q ≥ 1; that is b = 0 and q0 = 1 Ferró-Murgia-Olteanu (2012): if I(G) is an initial or final lexsegment edge ideal then b = 0 and q0 = 1 Alilooee-Banerjee (2014): if G is a bipartite graph and reg(I(G)) = 3 then b = 1 and q0 = 1. Problem Characterize graphs for which b = 0.

Huy Tài Hà Regularity of powers of edge ideals

Known answers

Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the complement graph Gc is chordal then I(G)q has a linear resolution for all q ≥ 1; that is b = 0 and q0 = 1 Ferró-Murgia-Olteanu (2012): if I(G) is an initial or final lexsegment edge ideal then b = 0 and q0 = 1 Alilooee-Banerjee (2014): if G is a bipartite graph and reg(I(G)) = 3 then b = 1 and q0 = 1. Problem Characterize graphs for which b = 0.

Huy Tài Hà Regularity of powers of edge ideals

1

Problem and Motivation Asymptotic linearity of regularity Stabilization index and free constant

2

Literature Reviews Polynomial ideals Edge ideals Known answers

3

Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds

Huy Tài Hà Regularity of powers of edge ideals

Forests and induced matching number

Definition Let G = (V, E) be a graph. A matching in G is a collection of pairwise disjoint edges An induced matching in G is a matching {e1, . . . , es} ⊆ E such that these are also the only edges in the induced subgraph of G over the vertices s

i=1 ei

The induced matching number of G is the maximum size of an induced matching in G The graph G is a forest if it contains no cycles.

Huy Tài Hà Regularity of powers of edge ideals

Induced matching number

Example Consider the graph G as follows:

❅ ❅ ❅

• ❅

❅ ❅ ✈

x2

✈ x4 ✈ x5 ✈ x6 ✈

x3

✈

x1 {x1x2, x4x5} forms a matching, but not an induced matching in G The induced matching number of G is 1.

Huy Tài Hà Regularity of powers of edge ideals

Regularity of powers of forests

Theorem (Beyarslan, —, Trung (2014)) Let G be a forest and let ν denote its induced matching number. Then b = ν − 1 and q0 = 1. That is, for all q ≥ 1, reg(I(G)q) = 2q + ν − 1.

Huy Tài Hà Regularity of powers of edge ideals

Regularity of powers of cycles

Theorem (Beyarslan, —, Trung (2014)) Let G be an n-cycle and let ν = n 3

• denote its induced

matching number. Then b = ν − 1 and q0 = 2. In fact, reg(I(G)) = ν + 1 if n ≡ 0, 1 (mod 3) ν + 2 if n ≡ 2 (mod 3), and for all q ≥ 2, we have reg(I(G)q) = 2q + ν − 1.

Huy Tài Hà Regularity of powers of edge ideals

General lower bounds

Theorem (Katzman (2006)) Let G be a graph and let ν denote its induced matching

• number. Then

reg(I(G)) ≥ ν + 1. Theorem (Beyarslan, —, Trung (2014)) Let G be a graph and let ν denote its induced matching

• number. Then for any q ≥ 1, we have

reg(I(G)q ≥ 2q + ν − 1.

Huy Tài Hà Regularity of powers of edge ideals

General lower bounds

Theorem (Katzman (2006)) Let G be a graph and let ν denote its induced matching

• number. Then

reg(I(G)) ≥ ν + 1. Theorem (Beyarslan, —, Trung (2014)) Let G be a graph and let ν denote its induced matching

• number. Then for any q ≥ 1, we have

reg(I(G)q ≥ 2q + ν − 1.

Huy Tài Hà Regularity of powers of edge ideals

Methods

Methods: to establish the upper bound reg(I(G)q ≤ 2q + ν − 1. G is a forest: let G′ and G′′ be induced subgraphs of G with no edge in common and whose union is G, and consider ideals of the form J = I(G′) + I(G′′)q. reg(J) ≤ 2q + ν − 1. G is a cycle: make use of Banerjee’s recent work, which states that if {M1, . . . , Ms} are minimal generators of I(G)q then reg(I(G)q+1) ≤ max{reg(I(G)q), reg(I(G)q+1 : Mi) + 2q}.

Huy Tài Hà Regularity of powers of edge ideals

Methods

Methods: to establish the upper bound reg(I(G)q ≤ 2q + ν − 1. G is a forest: let G′ and G′′ be induced subgraphs of G with no edge in common and whose union is G, and consider ideals of the form J = I(G′) + I(G′′)q. reg(J) ≤ 2q + ν − 1. G is a cycle: make use of Banerjee’s recent work, which states that if {M1, . . . , Ms} are minimal generators of I(G)q then reg(I(G)q+1) ≤ max{reg(I(G)q), reg(I(G)q+1 : Mi) + 2q}.

Huy Tài Hà Regularity of powers of edge ideals

Methods

Methods: to establish the upper bound reg(I(G)q ≤ 2q + ν − 1. G is a forest: let G′ and G′′ be induced subgraphs of G with no edge in common and whose union is G, and consider ideals of the form J = I(G′) + I(G′′)q. reg(J) ≤ 2q + ν − 1. G is a cycle: make use of Banerjee’s recent work, which states that if {M1, . . . , Ms} are minimal generators of I(G)q then reg(I(G)q+1) ≤ max{reg(I(G)q), reg(I(G)q+1 : Mi) + 2q}.

Huy Tài Hà Regularity of powers of edge ideals

Hamiltonian paths and cycles

Definition Let G be a graph. A Hamiltonian path in G is a path that goes through each vertex of G exactly once A Hamiltonian cycle in G is a cycle that contains all the vertices of G (and thus, each exactly once).

Huy Tài Hà Regularity of powers of edge ideals

Hamiltonian paths and cycles

Figure: Graph with Hamiltonian path and cycle.

Huy Tài Hà Regularity of powers of edge ideals

Regularity of graphs with Hamiltonian paths and cycles

Theorem (Beyarslan, —, Trung (2014)) Let G be a graph over n vertices. If G contains a Hamiltonian path then reg(I(G)) ≤ n + 1 3

• + 1.

If G contains a Hamiltonian cycle then reg(I(G)) ≤ n 3

• + 1.

Huy Tài Hà Regularity of powers of edge ideals

Huy Tài Hà Regularity of powers of edge ideals