Stabilization of multigraded Betti numbers Huy Ti H Tulane - PowerPoint PPT Presentation

Stabilization of multigraded Betti numbers Huy Tài Hà Tulane University Joint with Amir Bagheri and Marc Chardin

Outlines Motivation - asymptotic linearity of regularity 1 Multigraded (or G -graded) situation 2 Problem and approach 3 Equi-generated case 4 General case 5

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 | M l � = 0 } , The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999), Kodiyalam (2000), 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 ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b .

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 | M l � = 0 } , The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999), Kodiyalam (2000), 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 ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b .

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 | M l � = 0 } , The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999), Kodiyalam (2000), 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 ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b .

G -graded Betti numbers G a finitely generated abelian group, k a field. R = k [ x 1 , . . . , x n ] a G -graded polynomial ring. M a finitely generated G -graded module over R . The minimal G -graded free resolution of M : R ( − η ) β p ,η ( M ) → · · · → R ( − η ) β 0 ,η ( M ) → M → 0 . � � 0 → η ∈ G η ∈ G The numbers β i ,η ( M ) are called the G -graded Betti numbers of M . β i ,η ( M ) = dim k Tor R i ( M , k ) η � study the support Supp G ( Tor R i ( M , k )) .

G -graded Betti numbers G a finitely generated abelian group, k a field. R = k [ x 1 , . . . , x n ] a G -graded polynomial ring. M a finitely generated G -graded module over R . The minimal G -graded free resolution of M : R ( − η ) β p ,η ( M ) → · · · → R ( − η ) β 0 ,η ( M ) → M → 0 . � � 0 → η ∈ G η ∈ G The numbers β i ,η ( M ) are called the G -graded Betti numbers of M . β i ,η ( M ) = dim k Tor R i ( M , k ) η � study the support Supp G ( Tor R i ( M , k )) .

G -graded Betti numbers G a finitely generated abelian group, k a field. R = k [ x 1 , . . . , x n ] a G -graded polynomial ring. M a finitely generated G -graded module over R . The minimal G -graded free resolution of M : R ( − η ) β p ,η ( M ) → · · · → R ( − η ) β 0 ,η ( M ) → M → 0 . � � 0 → η ∈ G η ∈ G The numbers β i ,η ( M ) are called the G -graded Betti numbers of M . β i ,η ( M ) = dim k Tor R i ( M , k ) η � study the support Supp G ( Tor R i ( M , k )) .

G -graded Betti numbers G a finitely generated abelian group, k a field. R = k [ x 1 , . . . , x n ] a G -graded polynomial ring. M a finitely generated G -graded module over R . The minimal G -graded free resolution of M : R ( − η ) β p ,η ( M ) → · · · → R ( − η ) β 0 ,η ( M ) → M → 0 . � � 0 → η ∈ G η ∈ G The numbers β i ,η ( M ) are called the G -graded Betti numbers of M . β i ,η ( M ) = dim k Tor R i ( M , k ) η � study the support Supp G ( Tor R i ( M , k )) .

Problem Problem Let I 1 , . . . , I s be G-graded homogeneous ideal in R, and let M be a finitely generated G-graded R-module. Investigate the asymptotic behavior of Supp G ( Tor R i ( I t 1 1 . . . I t s s M , k )) as t = ( t 1 , . . . , t s ) ∈ N s gets large.

Approach to the problem t ∈ N s I t 1 1 . . . I t s t ∈ N s I t 1 1 . . . I t s R = � s , M R = � s M . I i = ( F i , 1 , . . . , F i , r i ) . S = R [ T i , j | 1 ≤ i ≤ s , 1 ≤ j ≤ r i ] is G × Z s -graded polynomial extension of R , where deg G × Z s ( a ) = ( deg G ( a ) , 0 ) ∀ a ∈ R , deg G × Z s ( T i , j ) = ( deg G ( F i , j , e i ) . M R is a finitely generated G × Z s -graded module over S , and I t 1 1 . . . I t s s M = ( M R ) ( ∗ , t ) = ( M R ) G × t . For a finitely generated G × Z s -graded module M over S , study Tor R i ( M G × t , k ) .

Approach to the problem t ∈ N s I t 1 1 . . . I t s t ∈ N s I t 1 1 . . . I t s R = � s , M R = � s M . I i = ( F i , 1 , . . . , F i , r i ) . S = R [ T i , j | 1 ≤ i ≤ s , 1 ≤ j ≤ r i ] is G × Z s -graded polynomial extension of R , where deg G × Z s ( a ) = ( deg G ( a ) , 0 ) ∀ a ∈ R , deg G × Z s ( T i , j ) = ( deg G ( F i , j , e i ) . M R is a finitely generated G × Z s -graded module over S , and I t 1 1 . . . I t s s M = ( M R ) ( ∗ , t ) = ( M R ) G × t . For a finitely generated G × Z s -graded module M over S , study Tor R i ( M G × t , k ) .

Approach to the problem t ∈ N s I t 1 1 . . . I t s t ∈ N s I t 1 1 . . . I t s R = � s , M R = � s M . I i = ( F i , 1 , . . . , F i , r i ) . S = R [ T i , j | 1 ≤ i ≤ s , 1 ≤ j ≤ r i ] is G × Z s -graded polynomial extension of R , where deg G × Z s ( a ) = ( deg G ( a ) , 0 ) ∀ a ∈ R , deg G × Z s ( T i , j ) = ( deg G ( F i , j , e i ) . M R is a finitely generated G × Z s -graded module over S , and I t 1 1 . . . I t s s M = ( M R ) ( ∗ , t ) = ( M R ) G × t . For a finitely generated G × Z s -graded module M over S , study Tor R i ( M G × t , k ) .

Approach to the problem t ∈ N s I t 1 1 . . . I t s t ∈ N s I t 1 1 . . . I t s R = � s , M R = � s M . I i = ( F i , 1 , . . . , F i , r i ) . S = R [ T i , j | 1 ≤ i ≤ s , 1 ≤ j ≤ r i ] is G × Z s -graded polynomial extension of R , where deg G × Z s ( a ) = ( deg G ( a ) , 0 ) ∀ a ∈ R , deg G × Z s ( T i , j ) = ( deg G ( F i , j , e i ) . M R is a finitely generated G × Z s -graded module over S , and I t 1 1 . . . I t s s M = ( M R ) ( ∗ , t ) = ( M R ) G × t . For a finitely generated G × Z s -graded module M over S , study Tor R i ( M G × t , k ) .

Approach to the problem If F • is a G × Z s -graded complex of free S -modules, then for δ ∈ Z s , H i (( F • ) G × δ ⊗ R k ) = H i ( F • ⊗ R k ) G × δ . If F • is a G × Z s -graded free resolution of M , then ( F • ) G × δ is a G -graded free resolution of M G × δ . Hence Tor R i ( M G × δ , k ) = H i ( F • ⊗ R k ) G × δ . where F • ⊗ R k is viewed as a G × Z s -graded complex of free modules over B = k [ T i , j ] .

Approach to the problem If F • is a G × Z s -graded complex of free S -modules, then for δ ∈ Z s , H i (( F • ) G × δ ⊗ R k ) = H i ( F • ⊗ R k ) G × δ . If F • is a G × Z s -graded free resolution of M , then ( F • ) G × δ is a G -graded free resolution of M G × δ . Hence Tor R i ( M G × δ , k ) = H i ( F • ⊗ R k ) G × δ . where F • ⊗ R k is viewed as a G × Z s -graded complex of free modules over B = k [ T i , j ] .

Equi-generated case I i = ( F i , 1 , . . . , F i , r i ) is generated in degree γ i ∈ G . t ∈ N s I t 1 1 ( t 1 γ 1 ) . . . I t s There is a natural map S → R ≃ � s ( t s γ s ) θ,ℓ S ( − θ, − ℓ ) β i θ,ℓ be the i th module of F • Let F i = � H i (( F • ) G × δ ⊗ R k ) η = H i ( F [ η ] • ⊗ R k ) δ , where S ( − η, − ℓ ) β i [ R ( − η ) ⊗ k B ( − ℓ )] β i F [ η ] � η,ℓ = � = η,ℓ . i ℓ ℓ

Equi-generated case I i = ( F i , 1 , . . . , F i , r i ) is generated in degree γ i ∈ G . t ∈ N s I t 1 1 ( t 1 γ 1 ) . . . I t s There is a natural map S → R ≃ � s ( t s γ s ) θ,ℓ S ( − θ, − ℓ ) β i θ,ℓ be the i th module of F • Let F i = � H i (( F • ) G × δ ⊗ R k ) η = H i ( F [ η ] • ⊗ R k ) δ , where S ( − η, − ℓ ) β i [ R ( − η ) ⊗ k B ( − ℓ )] β i F [ η ] � η,ℓ = � = η,ℓ . i ℓ ℓ

Equi-generated case I i = ( F i , 1 , . . . , F i , r i ) is generated in degree γ i ∈ G . t ∈ N s I t 1 1 ( t 1 γ 1 ) . . . I t s There is a natural map S → R ≃ � s ( t s γ s ) θ,ℓ S ( − θ, − ℓ ) β i θ,ℓ be the i th module of F • Let F i = � H i (( F • ) G × δ ⊗ R k ) η = H i ( F [ η ] • ⊗ R k ) δ , where S ( − η, − ℓ ) β i [ R ( − η ) ⊗ k B ( − ℓ )] β i F [ η ] � η,ℓ = � = η,ℓ . i ℓ ℓ

Equi-generated case I i = ( F i , 1 , . . . , F i , r i ) is generated in degree γ i ∈ G . t ∈ N s I t 1 1 ( t 1 γ 1 ) . . . I t s There is a natural map S → R ≃ � s ( t s γ s ) θ,ℓ S ( − θ, − ℓ ) β i θ,ℓ be the i th module of F • Let F i = � H i (( F • ) G × δ ⊗ R k ) η = H i ( F [ η ] • ⊗ R k ) δ , where S ( − η, − ℓ ) β i [ R ( − η ) ⊗ k B ( − ℓ )] β i F [ η ] � η,ℓ = � = η,ℓ . i ℓ ℓ