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Regularity and Gröbner bases of the Rees algebra of edge ideals of bipartite graphs

Yairon Cid Ruiz University of Barcelona Journées Nationales de Calcul Formel CIRM, Luminy, January 2018

Definition

A bipartite graph G = (X, Y , E) consists of two disjoint sets of vertices X = {x1, . . . , xn} and Y = {y1, . . . , ym}, and a set of edges E ⊂

(x, y) | x ∈ X, y ∈ Y .

bipartite ⇐ ⇒ no odd cycles ⇐ ⇒ 2-colorable.

x1 x2 x3 y4 y3 y2 y1 a b c

2

Definition

A bipartite graph G = (X, Y , E) consists of two disjoint sets of vertices X = {x1, . . . , xn} and Y = {y1, . . . , ym}, and a set of edges E ⊂

(x, y) | x ∈ X, y ∈ Y .

bipartite ⇐ ⇒ no odd cycles ⇐ ⇒ 2-colorable.

x1 x2 x3 y4 y3 y2 y1 a b c

2

Definition

Let K be a field and R = K[x1, . . . , xn, y1, . . . , ym]. The edge ideal I = I(G), associated to G, is defined by I =

• xiyj | (xi, yj) ∈ E
• .

x1 x2 x3 y4 y3 y2 y1

I =

• x1y3, x2y1, x3y2, x3y3, x3y4
• ⊂ R

3

Definition

Let R(I) = ∞

i=0 Iiti ⊂ R[t] be the Rees algebra of the edge ideal I. Let

f1, . . . , fq be the square free monomials of degree two generating I. Let S = R[T1, . . . , Tq], and define the following map S = K[x1, . . . , xn, y1 . . . , ym, T1, . . . , Tq]

ψ

− → R(I) ⊂ R[t], ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Then the presentation of R(I) is given by S/K where K = Ker(ψ).

Problem

In terms of the combinatorics of the bipartite graph G, we want to: Describe the universal Gröbner basis of K. Compute the Castelnuovo-Mumford regularity of R(I). Study the regularity of the powers of the ideal I.

4

Definition

Let R(I) = ∞

i=0 Iiti ⊂ R[t] be the Rees algebra of the edge ideal I. Let

f1, . . . , fq be the square free monomials of degree two generating I. Let S = R[T1, . . . , Tq], and define the following map S = K[x1, . . . , xn, y1 . . . , ym, T1, . . . , Tq]

ψ

− → R(I) ⊂ R[t], ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Then the presentation of R(I) is given by S/K where K = Ker(ψ).

Problem

In terms of the combinatorics of the bipartite graph G, we want to: Describe the universal Gröbner basis of K. Compute the Castelnuovo-Mumford regularity of R(I). Study the regularity of the powers of the ideal I.

4

Definition

Let R(I) = ∞

i=0 Iiti ⊂ R[t] be the Rees algebra of the edge ideal I. Let

f1, . . . , fq be the square free monomials of degree two generating I. Let S = R[T1, . . . , Tq], and define the following map S = K[x1, . . . , xn, y1 . . . , ym, T1, . . . , Tq]

ψ

− → R(I) ⊂ R[t], ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Then the presentation of R(I) is given by S/K where K = Ker(ψ).

Problem

In terms of the combinatorics of the bipartite graph G, we want to: Describe the universal Gröbner basis of K. Compute the Castelnuovo-Mumford regularity of R(I). Study the regularity of the powers of the ideal I.

4

Definition

Let R(I) = ∞

i=0 Iiti ⊂ R[t] be the Rees algebra of the edge ideal I. Let

f1, . . . , fq be the square free monomials of degree two generating I. Let S = R[T1, . . . , Tq], and define the following map S = K[x1, . . . , xn, y1 . . . , ym, T1, . . . , Tq]

ψ

− → R(I) ⊂ R[t], ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Then the presentation of R(I) is given by S/K where K = Ker(ψ).

Problem

In terms of the combinatorics of the bipartite graph G, we want to: Describe the universal Gröbner basis of K. Compute the Castelnuovo-Mumford regularity of R(I). Study the regularity of the powers of the ideal I.

4

Matrix associated to the presentation of R(I)

Given the presentation of the Rees algebra ψ : S → R(I) ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Let A = (ai,j) ∈ Zn+m,q be the incidence matrix of G, i.e. each column corresponds to an edge fi. Then we construct the following matrix M = f1t . . . fqt x1 . . . xn y1 . . . ym           a1,1 . . . a1,q e1 . . . en en+1 . . . en+m . . . ... . . . an+m,1 . . . an+m,q 1 . . . 1

K is a toric ideal (Sturmfels 1996)

K =

• Txyα+ − Txyα− | α ∈ KerZ(M)
• 5

Matrix associated to the presentation of R(I)

Given the presentation of the Rees algebra ψ : S → R(I) ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Let A = (ai,j) ∈ Zn+m,q be the incidence matrix of G, i.e. each column corresponds to an edge fi. Then we construct the following matrix M = f1t . . . fqt x1 . . . xn y1 . . . ym           a1,1 . . . a1,q e1 . . . en en+1 . . . en+m . . . ... . . . an+m,1 . . . an+m,q 1 . . . 1

K is a toric ideal (Sturmfels 1996)

K =

• Txyα+ − Txyα− | α ∈ KerZ(M)
• 5

Matrix associated to the presentation of R(I)

Given the presentation of the Rees algebra ψ : S → R(I) ψ(xi) = xi, ψ(yi) = yi, ψ(Ti) = fit. Let A = (ai,j) ∈ Zn+m,q be the incidence matrix of G, i.e. each column corresponds to an edge fi. Then we construct the following matrix M = f1t . . . fqt x1 . . . xn y1 . . . ym           a1,1 . . . a1,q e1 . . . en en+1 . . . en+m . . . ... . . . an+m,1 . . . an+m,q 1 . . . 1

K is a toric ideal (Sturmfels 1996)

K =

• Txyα+ − Txyα− | α ∈ KerZ(M)
• 5

Example

x1 x2 y1 y2

I =

• x1y2, x2y1, x2y2
• 0 → K → S → R(I) → 0

T1 → x1y2t, T2 → x2y1t, T3 → x2y2t M = x1y2t x2y1t x2y2t x1 x2 y1 y2               x1 1 1 x2 1 1 1 y1 1 1 y2 1 1 1 t 1 1 1 K =

• T

α+

1

1 T α+

2

2 T α+

3

3 x α+

4

1 x α+

5

2 y α+

6

1 y α+

7

2

− T

α−

1

1

T

α−

2

2

T

α−

3

3

x

α−

4

1

x

α−

5

2

y

α−

6

1

y

α−

7

2

| α ∈ KerZ(M)

• 6

Example

x1 x2 y1 y2

I =

• x1y2, x2y1, x2y2
• 0 → K → S → R(I) → 0

T1 → x1y2t, T2 → x2y1t, T3 → x2y2t M = x1y2t x2y1t x2y2t x1 x2 y1 y2               x1 1 1 x2 1 1 1 y1 1 1 y2 1 1 1 t 1 1 1 K =

• T

α+

1

1 T α+

2

2 T α+

3

3 x α+

4

1 x α+

5

2 y α+

6

1 y α+

7

2

− T

α−

1

1

T

α−

2

2

T

α−

3

3

x

α−

4

1

x

α−

5

2

y

α−

6

1

y

α−

7

2

| α ∈ KerZ(M)

• 6

Example

x1 x2 y1 y2

I =

• x1y2, x2y1, x2y2
• 0 → K → S → R(I) → 0

T1 → x1y2t, T2 → x2y1t, T3 → x2y2t M = x1y2t x2y1t x2y2t x1 x2 y1 y2               x1 1 1 x2 1 1 1 y1 1 1 y2 1 1 1 t 1 1 1 K =

• T

α+

1

1 T α+

2

2 T α+

3

3 x α+

4

1 x α+

5

2 y α+

6

1 y α+

7

2

− T

α−

1

1

T

α−

2

2

T

α−

3

3

x

α−

4

1

x

α−

5

2

y

α−

6

1

y

α−

7

2

| α ∈ KerZ(M)

• 6

Universal Gröbner basis of K

U =

• < runs over all possible term orders

G<(K) (G<(K) denotes reduced Gröbner basis with respect to <)

Circuit

α ∈ KerZ(M) is called a circuit if it has minimal support supp(α) with respect to inclusion and its coordinates are relatively prime. In general we have that the set of circuits is contained in U.

Lemma

If G is a bipartite graph then U =

• Txyα+ − Txyα− | α is a circuit of M
• .

Proof.

From Gitler, Valencia, and Villarreal 2005, then M is totally unimodular. Hence, by Sturmfels 1996 we get the equality.

7

Universal Gröbner basis of K

U =

• < runs over all possible term orders

G<(K) (G<(K) denotes reduced Gröbner basis with respect to <)

Circuit

α ∈ KerZ(M) is called a circuit if it has minimal support supp(α) with respect to inclusion and its coordinates are relatively prime. In general we have that the set of circuits is contained in U.

Lemma

If G is a bipartite graph then U =

• Txyα+ − Txyα− | α is a circuit of M
• .

Proof.

From Gitler, Valencia, and Villarreal 2005, then M is totally unimodular. Hence, by Sturmfels 1996 we get the equality.

7

Universal Gröbner basis of K

U =

• < runs over all possible term orders

G<(K) (G<(K) denotes reduced Gröbner basis with respect to <)

Circuit

α ∈ KerZ(M) is called a circuit if it has minimal support supp(α) with respect to inclusion and its coordinates are relatively prime. In general we have that the set of circuits is contained in U.

Lemma

If G is a bipartite graph then U =

• Txyα+ − Txyα− | α is a circuit of M
• .

Proof.

From Gitler, Valencia, and Villarreal 2005, then M is totally unimodular. Hence, by Sturmfels 1996 we get the equality.

7

Theorem

Let G be bipartite graph, then U is given by U = {Tw+ − Tw− | w is an even cycle} ∪ {v0Tw+ − vaTw− | w = (v0, . . . , va) is an even path} ∪ {u0uaTw+

1 Tw− 2 − v0vbTw− 1 Tw+ 2 | w1 = (u0, . . . , ua) and

w2 = (v0, . . . , vb) are disjoint odd paths}. T2T4 − T1T3 x1T2 − x2T1 x1y1T2 − x2y2T1

x1 y1 x2 y2 T1 T2 T3 T4 x1 y1 x2 T1 T2 x1 y1 x2 y2 T1 T2

8

• Proof. (sketch).

We construct the cone graph C(G) of G (add a new vertex z and connect it to all vertices of G).

x1 y1 x2 x1 x2 y1 z

Let K[C(G)] = K[e | e ∈ E(C(G))] ⊂ R[z]. Then we have a canonical map π : S − → K[C(G)] ⊂ R[z], π(xi) = xiz, π(yi) = yiz, π(Ti) = fi. We have that R(I) ∼ = K[C(G)] (Vasconcelos 1998), and so K = Ker(π). From Villarreal 1995 we can determine the circuits of the incidence matrix of C(G). Finally, we translate them into the circuits of M.

9

• Proof. (sketch).

We construct the cone graph C(G) of G (add a new vertex z and connect it to all vertices of G).

x1 y1 x2 x1 x2 y1 z

Let K[C(G)] = K[e | e ∈ E(C(G))] ⊂ R[z]. Then we have a canonical map π : S − → K[C(G)] ⊂ R[z], π(xi) = xiz, π(yi) = yiz, π(Ti) = fi. We have that R(I) ∼ = K[C(G)] (Vasconcelos 1998), and so K = Ker(π). From Villarreal 1995 we can determine the circuits of the incidence matrix of C(G). Finally, we translate them into the circuits of M.

9

• Proof. (sketch).

We construct the cone graph C(G) of G (add a new vertex z and connect it to all vertices of G).

x1 y1 x2 x1 x2 y1 z

Let K[C(G)] = K[e | e ∈ E(C(G))] ⊂ R[z]. Then we have a canonical map π : S − → K[C(G)] ⊂ R[z], π(xi) = xiz, π(yi) = yiz, π(Ti) = fi. We have that R(I) ∼ = K[C(G)] (Vasconcelos 1998), and so K = Ker(π). From Villarreal 1995 we can determine the circuits of the incidence matrix of C(G). Finally, we translate them into the circuits of M.

9

S is bigraded with bigrad(xi) = bigrad(yi) = (1, 0) and bigrad(Ti) = (0, 1). R(I) as a bigraded S-module has a minimal bigraded free resolution 0 − → Fp − → · · · − → F1 − → F0 − → R(I) − → 0, where Fi = ⊕jS(−aij, −bij). As in Römer 2001, we can define regxy(R(I)) = max

i,j {aij − i},

regT(R(I)) = max

i,j {bij − i},

reg(R(I)) = max

i,j {aij + bij − i}.

Theorem (Römer 2001, Chardin 2013)

reg(Is) ≤ 2s + regxy(R(I)) for all s ≥ 1.

Theorem

Let < be any term order in S, then we have regxy(R(I)) ≤ regxy(S/in<(K)).

10

S is bigraded with bigrad(xi) = bigrad(yi) = (1, 0) and bigrad(Ti) = (0, 1). R(I) as a bigraded S-module has a minimal bigraded free resolution 0 − → Fp − → · · · − → F1 − → F0 − → R(I) − → 0, where Fi = ⊕jS(−aij, −bij). As in Römer 2001, we can define regxy(R(I)) = max

i,j {aij − i},

regT(R(I)) = max

i,j {bij − i},

reg(R(I)) = max

i,j {aij + bij − i}.

Theorem (Römer 2001, Chardin 2013)

reg(Is) ≤ 2s + regxy(R(I)) for all s ≥ 1.

Theorem

Let < be any term order in S, then we have regxy(R(I)) ≤ regxy(S/in<(K)).

10

S is bigraded with bigrad(xi) = bigrad(yi) = (1, 0) and bigrad(Ti) = (0, 1). R(I) as a bigraded S-module has a minimal bigraded free resolution 0 − → Fp − → · · · − → F1 − → F0 − → R(I) − → 0, where Fi = ⊕jS(−aij, −bij). As in Römer 2001, we can define regxy(R(I)) = max

i,j {aij − i},

regT(R(I)) = max

i,j {bij − i},

reg(R(I)) = max

i,j {aij + bij − i}.

Theorem (Römer 2001, Chardin 2013)

reg(Is) ≤ 2s + regxy(R(I)) for all s ≥ 1.

Theorem

Let < be any term order in S, then we have regxy(R(I)) ≤ regxy(S/in<(K)).

10

S is bigraded with bigrad(xi) = bigrad(yi) = (1, 0) and bigrad(Ti) = (0, 1). R(I) as a bigraded S-module has a minimal bigraded free resolution 0 − → Fp − → · · · − → F1 − → F0 − → R(I) − → 0, where Fi = ⊕jS(−aij, −bij). As in Römer 2001, we can define regxy(R(I)) = max

i,j {aij − i},

regT(R(I)) = max

i,j {bij − i},

reg(R(I)) = max

i,j {aij + bij − i}.

Theorem (Römer 2001, Chardin 2013)

reg(Is) ≤ 2s + regxy(R(I)) for all s ≥ 1.

Theorem

Let < be any term order in S, then we have regxy(R(I)) ≤ regxy(S/in<(K)).

10

Regularity of the powers of I

A celebrated result of Cutkosky, Herzog, and Trung 1999 and Kodiyalam 2000 says that (for a general ideal in a polynomial ring) reg(Is) = as + b for s ≫ 0. But the exact form of this linear function and when reg(Is) starts to be linear is still wide open even for monomial ideals.

Corollary

G bipartite graph with bipartition V (G) = X ∪ Y . Then, for all s ≥ 1 we have reg(Is) ≤ 2s + min{|X|, |Y |} − 1.

Proof.

Using our characterization of U, a “suitable” term order and the Taylor resolution, then we can bound regxy(S/in<(K)).

11

Regularity of the powers of I

A celebrated result of Cutkosky, Herzog, and Trung 1999 and Kodiyalam 2000 says that (for a general ideal in a polynomial ring) reg(Is) = as + b for s ≫ 0. But the exact form of this linear function and when reg(Is) starts to be linear is still wide open even for monomial ideals.

Corollary

G bipartite graph with bipartition V (G) = X ∪ Y . Then, for all s ≥ 1 we have reg(Is) ≤ 2s + min{|X|, |Y |} − 1.

Proof.

Using our characterization of U, a “suitable” term order and the Taylor resolution, then we can bound regxy(S/in<(K)).

11

Theorem

Let G be a bipartite graph and I = I(G) be its edge ideal. The total regularity of R(I) is given by reg(R(I)) = match(G).

Proof (sketch).

Since M is totally unimodular, then by Gitler, Valencia, and Villarreal 2005 we have that R(I) is a normal domain. From Hochster 1972, then R(I) is Cohen-Macaulay and so it has a canonical module ωR(I). The minimal free resolutions of R(I) and ωR(I) are dual. ωR(I) can be computed using a formula of Danilov and Stanley (Gitler, Valencia, and Villarreal 2005).

12

Theorem

Let G be a bipartite graph and I = I(G) be its edge ideal. The total regularity of R(I) is given by reg(R(I)) = match(G).

Proof (sketch).

Since M is totally unimodular, then by Gitler, Valencia, and Villarreal 2005 we have that R(I) is a normal domain. From Hochster 1972, then R(I) is Cohen-Macaulay and so it has a canonical module ωR(I). The minimal free resolutions of R(I) and ωR(I) are dual. ωR(I) can be computed using a formula of Danilov and Stanley (Gitler, Valencia, and Villarreal 2005).

12

Theorem

Let G be a bipartite graph and I = I(G) be its edge ideal. The total regularity of R(I) is given by reg(R(I)) = match(G).

Proof (sketch).

Since M is totally unimodular, then by Gitler, Valencia, and Villarreal 2005 we have that R(I) is a normal domain. From Hochster 1972, then R(I) is Cohen-Macaulay and so it has a canonical module ωR(I). The minimal free resolutions of R(I) and ωR(I) are dual. ωR(I) can be computed using a formula of Danilov and Stanley (Gitler, Valencia, and Villarreal 2005).

12

Theorem

Let G be a bipartite graph and I = I(G) be its edge ideal. The total regularity of R(I) is given by reg(R(I)) = match(G).

Proof (sketch).

Since M is totally unimodular, then by Gitler, Valencia, and Villarreal 2005 we have that R(I) is a normal domain. From Hochster 1972, then R(I) is Cohen-Macaulay and so it has a canonical module ωR(I). The minimal free resolutions of R(I) and ωR(I) are dual. ωR(I) can be computed using a formula of Danilov and Stanley (Gitler, Valencia, and Villarreal 2005).

12

Theorem

Let G be a bipartite graph and I = I(G) be its edge ideal. The total regularity of R(I) is given by reg(R(I)) = match(G).

Proof (sketch).

Since M is totally unimodular, then by Gitler, Valencia, and Villarreal 2005 we have that R(I) is a normal domain. From Hochster 1972, then R(I) is Cohen-Macaulay and so it has a canonical module ωR(I). The minimal free resolutions of R(I) and ωR(I) are dual. ωR(I) can be computed using a formula of Danilov and Stanley (Gitler, Valencia, and Villarreal 2005).

12

Corollary

For all s ≥ match(G) + |E(G)| + 1 we have reg(I(G)s+1) = reg(I(G)s) + 2. For all s ≥ 1 we have reg(I(G)s) ≤ 2s + match(G) − 1.

Proof.

Using the upper bound for the total regularity we get regT(R(I)) ≤ match(G), regxy(R(I)) ≤ match(G) − 1. Then the results follow from Cutkosky, Herzog, and Trung 1999 and Römer 2001, respectively.

13

A sharper upper bound and a Conjecture

For bipartite graphs, we have the following inequalities reg(Is) ≤ 2s + co-chord(G) − 1 ≤ 2s + match(G) − 1 ≤ 2s + min{|X|, |Y |} − 1. The upper bound reg(Is) ≤ 2s + co-chord(G) − 1 was obtained in Jayanthan, Narayanan, and Selvaraja 2016 using a combinatorial argument called “even connection”.

Conjecture (Alilooee, Banerjee, Beyarslan and Hà)

Let G be an arbitrary graph then reg(I(G)s) ≤ 2s + reg(I(G)) − 2 for all s ≥ 1. (We always have 2s + co-chord(G) − 1 ≤ 2s + reg(I(G)) − 2.)

14

A sharper upper bound and a Conjecture

For bipartite graphs, we have the following inequalities reg(Is) ≤ 2s + co-chord(G) − 1 ≤ 2s + match(G) − 1 ≤ 2s + min{|X|, |Y |} − 1. The upper bound reg(Is) ≤ 2s + co-chord(G) − 1 was obtained in Jayanthan, Narayanan, and Selvaraja 2016 using a combinatorial argument called “even connection”.

Conjecture (Alilooee, Banerjee, Beyarslan and Hà)

Let G be an arbitrary graph then reg(I(G)s) ≤ 2s + reg(I(G)) − 2 for all s ≥ 1. (We always have 2s + co-chord(G) − 1 ≤ 2s + reg(I(G)) − 2.)

14

References I

Chardin, Marc (2013). “Powers of ideals: Betti numbers, cohomology and regularity”. In: Commutative algebra. Springer, New York, pp. 317–333. url: https://doi.org/10.1007/978-1-4614-5292-8_9. Cutkosky, S. Dale, Jürgen Herzog, and Ngô Viêt Trung (1999). “Asymptotic behaviour of the Castelnuovo-Mumford regularity”. In: Compositio Math. 118.3, pp. 243–261. Gitler, Isidoro, Carlos Valencia, and Rafael H. Villarreal (2005). “A note on the Rees algebra of a bipartite graph”. In: J. Pure Appl. Algebra 201.1-3,

• pp. 17–24.

Hochster, M. (1972). “Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes”. In: Ann. of Math. (2) 96, pp. 318–337. Jayanthan, AV, N Narayanan, and S Selvaraja (2016). “Regularity of powers of bipartite graphs”. In: Journal of Algebraic Combinatorics, pp. 1–22.

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References II

Kodiyalam, Vijay (2000). “Asymptotic behaviour of Castelnuovo-Mumford regularity”. In: Proc. Amer. Math. Soc. 128.2, pp. 407–411. Römer, Tim (2001). “Homological properties of bigraded algebras”. In: Illinois J.

• Math. 45.4, pp. 1361–1376.

Sturmfels, Bernd (1996). Gröbner bases and convex polytopes. Vol. 8. University Lecture Series. American Mathematical Society, Providence, RI, pp. xii+162. isbn: 0-8218-0487-1. Vasconcelos, Wolmer V. (1998). Computational methods in commutative algebra and algebraic geometry. Vol. 2. Algorithms and Computation in Mathematics. Springer-Verlag, Berlin. Villarreal, Rafael H. (1995). “Rees algebras of edge ideals”. In: Comm. Algebra 23.9, pp. 3513–3524.

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Merci beaucoup!