Regularity and Grbner bases of the Rees algebra of edge ideals of - PowerPoint PPT Presentation

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 = { x 1 , . . . , x n } and Y = { y 1 , . . . , y m } , and a set of edges � ( x , y ) | x ∈ X , y ∈ Y � . E ⊂ bipartite ⇐ ⇒ no odd cycles ⇐ ⇒ 2-colorable. x 1 y 1 b x 2 y 2 a x 3 y 3 c y 4 2

Definition A bipartite graph G = ( X , Y , E ) consists of two disjoint sets of vertices X = { x 1 , . . . , x n } and Y = { y 1 , . . . , y m } , and a set of edges � ( x , y ) | x ∈ X , y ∈ Y � . E ⊂ bipartite ⇐ ⇒ no odd cycles ⇐ ⇒ 2-colorable. x 1 y 1 b x 2 y 2 a x 3 y 3 c y 4 2

Definition Let K be a field and R = K [ x 1 , . . . , x n , y 1 , . . . , y m ]. The edge ideal I = I ( G ), associated to G , is defined by � � I = x i y j | ( x i , y j ) ∈ E . x 1 y 1 � � I = x 1 y 3 , x 2 y 1 , x 3 y 2 , x 3 y 3 , x 3 y 4 ⊂ R x 2 y 2 x 3 y 3 y 4 3

Definition i =0 I i t i ⊂ R [ t ] be the Rees algebra of the edge ideal I . Let Let R ( I ) = � ∞ f 1 , . . . , f q be the square free monomials of degree two generating I . Let S = R [ T 1 , . . . , T q ], and define the following map ψ S = K [ x 1 , . . . , x n , y 1 . . . , y m , T 1 , . . . , T q ] − → R ( I ) ⊂ R [ t ] , ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . 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 i =0 I i t i ⊂ R [ t ] be the Rees algebra of the edge ideal I . Let Let R ( I ) = � ∞ f 1 , . . . , f q be the square free monomials of degree two generating I . Let S = R [ T 1 , . . . , T q ], and define the following map ψ S = K [ x 1 , . . . , x n , y 1 . . . , y m , T 1 , . . . , T q ] − → R ( I ) ⊂ R [ t ] , ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . 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 i =0 I i t i ⊂ R [ t ] be the Rees algebra of the edge ideal I . Let Let R ( I ) = � ∞ f 1 , . . . , f q be the square free monomials of degree two generating I . Let S = R [ T 1 , . . . , T q ], and define the following map ψ S = K [ x 1 , . . . , x n , y 1 . . . , y m , T 1 , . . . , T q ] − → R ( I ) ⊂ R [ t ] , ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . 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 i =0 I i t i ⊂ R [ t ] be the Rees algebra of the edge ideal I . Let Let R ( I ) = � ∞ f 1 , . . . , f q be the square free monomials of degree two generating I . Let S = R [ T 1 , . . . , T q ], and define the following map ψ S = K [ x 1 , . . . , x n , y 1 . . . , y m , T 1 , . . . , T q ] − → R ( I ) ⊂ R [ t ] , ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . 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 ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . Let A = ( a i , j ) ∈ Z n + m , q be the incidence matrix of G , i.e. each column corresponds to an edge f i . Then we construct the following matrix f 1 t . . . f q t x 1 . . . x n y 1 . . . y m a 1 , 1 . . . a 1 , q e 1 . . . e n e n + 1 . . . e n + m   . . ... . .  . .  M =     a n + m , 1 . . . a n + m , q   1 . . . 1 K is a toric ideal (Sturmfels 1996) Txy α + − Txy α − | α ∈ Ker Z ( M ) � � K = 5

Matrix associated to the presentation of R ( I ) Given the presentation of the Rees algebra ψ : S → R ( I ) ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . Let A = ( a i , j ) ∈ Z n + m , q be the incidence matrix of G , i.e. each column corresponds to an edge f i . Then we construct the following matrix f 1 t . . . f q t x 1 . . . x n y 1 . . . y m a 1 , 1 . . . a 1 , q e 1 . . . e n e n + 1 . . . e n + m   . . ... . .  . .  M =     a n + m , 1 . . . a n + m , q   1 . . . 1 K is a toric ideal (Sturmfels 1996) Txy α + − Txy α − | α ∈ Ker Z ( M ) � � K = 5

Matrix associated to the presentation of R ( I ) Given the presentation of the Rees algebra ψ : S → R ( I ) ψ ( x i ) = x i , ψ ( y i ) = y i , ψ ( T i ) = f i t . Let A = ( a i , j ) ∈ Z n + m , q be the incidence matrix of G , i.e. each column corresponds to an edge f i . Then we construct the following matrix f 1 t . . . f q t x 1 . . . x n y 1 . . . y m a 1 , 1 . . . a 1 , q e 1 . . . e n e n + 1 . . . e n + m   . . ... . .  . .  M =     a n + m , 1 . . . a n + m , q   1 . . . 1 K is a toric ideal (Sturmfels 1996) Txy α + − Txy α − | α ∈ Ker Z ( M ) � � K = 5

Example x 1 y 1 � � I = x 1 y 2 , x 2 y 1 , x 2 y 2 x 2 0 → K → S → R ( I ) → 0 y 2 T 1 �→ x 1 y 2 t , T 2 �→ x 2 y 1 t , T 3 �→ x 2 y 2 t x 1 y 2 t x 2 y 1 t x 2 y 2 t x 1 x 2 y 1 y 2 x 1  1 0 0 1 0 0 0  x 2 0 1 1 0 1 0 0     M = y 1 0 1 0 0 0 1 0       y 2 1 0 1 0 0 0 1   t 1 1 1 0 0 0 0 � α + α + α + α + α + α + α + K = T 1 T 1 2 T 2 3 x 3 1 x 4 2 y 5 1 y 6 7 2 � α − α − α − α − α − α − α − − T T T x x y y | α ∈ Ker Z ( M ) 1 2 3 4 5 6 7 1 2 3 1 2 1 2 6

Example x 1 y 1 � � I = x 1 y 2 , x 2 y 1 , x 2 y 2 x 2 0 → K → S → R ( I ) → 0 y 2 T 1 �→ x 1 y 2 t , T 2 �→ x 2 y 1 t , T 3 �→ x 2 y 2 t x 1 y 2 t x 2 y 1 t x 2 y 2 t x 1 x 2 y 1 y 2 x 1  1 0 0 1 0 0 0  x 2 0 1 1 0 1 0 0     M = y 1 0 1 0 0 0 1 0       y 2 1 0 1 0 0 0 1   t 1 1 1 0 0 0 0 � α + α + α + α + α + α + α + K = T 1 T 1 2 T 2 3 x 3 1 x 4 2 y 5 1 y 6 7 2 � α − α − α − α − α − α − α − − T T T x x y y | α ∈ Ker Z ( M ) 1 2 3 4 5 6 7 1 2 3 1 2 1 2 6

Example x 1 y 1 � � I = x 1 y 2 , x 2 y 1 , x 2 y 2 x 2 0 → K → S → R ( I ) → 0 y 2 T 1 �→ x 1 y 2 t , T 2 �→ x 2 y 1 t , T 3 �→ x 2 y 2 t x 1 y 2 t x 2 y 1 t x 2 y 2 t x 1 x 2 y 1 y 2 x 1  1 0 0 1 0 0 0  x 2 0 1 1 0 1 0 0     M = y 1 0 1 0 0 0 1 0       y 2 1 0 1 0 0 0 1   t 1 1 1 0 0 0 0 � α + α + α + α + α + α + α + K = T 1 T 1 2 T 2 3 x 3 1 x 4 2 y 5 1 y 6 7 2 � α − α − α − α − α − α − α − − T T T x x y y | α ∈ Ker Z ( M ) 1 2 3 4 5 6 7 1 2 3 1 2 1 2 6

Universal Gröbner basis of K � U = G < ( K ) < runs over all possible term orders ( G < ( K ) denotes reduced Gröbner basis with respect to < ) Circuit α ∈ Ker Z ( 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 Txy α + − Txy α − | α is a circuit of M � � If G is a bipartite graph then U = . 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 = G < ( K ) < runs over all possible term orders ( G < ( K ) denotes reduced Gröbner basis with respect to < ) Circuit α ∈ Ker Z ( 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 Txy α + − Txy α − | α is a circuit of M � � If G is a bipartite graph then U = . 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 = G < ( K ) < runs over all possible term orders ( G < ( K ) denotes reduced Gröbner basis with respect to < ) Circuit α ∈ Ker Z ( 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 Txy α + − Txy α − | α is a circuit of M � � If G is a bipartite graph then U = . Proof. From Gitler, Valencia, and Villarreal 2005, then M is totally unimodular. Hence, by Sturmfels 1996 we get the equality. 7