I = ( M 1 , . . . , M q ) monomial ideal in polynomial ring. - PowerPoint PPT Presentation

O N THE R ESOLUTIONS OF ( SOME ) S IMPLICIAL F ORESTS S ARA F ARIDI D ALHOUSIE U NIVERSITY

I = ( M 1 , . . . , M q ) monomial ideal in polynomial ring. Question. What are the Betti numbers β i,j ( I ) ? Eliahou-Kervaire Splittings: When I = J + K where G ( J ) ∩ G ( K ) = ∅ , and there is a “splitting function” with certain properties, one has a recursive formula: β i,j ( I ) = β i,j ( J ) + β i,j ( K ) + β i − 1 ,j ( J ∩ K )

u w v s x z I = ( xyv, vw, ws, yzv, zuv ) y Question. Can one give an order to the facets of ∆ so that induces a splitting on the generators of I ?

Trees and Good Leafs Definition. A leaf is a facet that intersects the complex in a face . u w v s x z y

Trees and Good Leafs Definition. A leaf is a facet that intersects the complex in a face . u w v s x z y Definition. A forest is a complex whose every subset (of facets) has a leaf. A tree is a connected forest. has no leaf

Trees and Good Leafs Definition. A leaf is a facet that intersects the complex in a face . u w v s x z y Definition. A forest is a complex whose every subset (of facets) has a leaf. A tree is a connected forest. has no leaf Definition. A good leaf is a facet that is a leaf of every subset.

Definition. A good leaf is a facet that is a leaf of every subset. Fact. Every tree has a good leaf [Herzog-Hibi-Trung-Zheng 2008]

Definition. A good leaf is a facet that is a leaf of every subset. Fact. Every tree has a good leaf [Herzog-Hibi-Trung-Zheng 2008] Orders induced by good leafs – F 0 , . . . , F q where each F i is the leaf of � F 1 , . . . , F i �

Definition. A good leaf is a facet that is a leaf of every subset. Fact. Every tree has a good leaf [Herzog-Hibi-Trung-Zheng 2008] Orders induced by good leafs – F 0 , . . . , F q where each F i is the leaf of � F 1 , . . . , F i � – F 0 , F 1 , . . . , F q where F 0 is a good leaf of ∆ and F 0 ∩ F 1 ⊇ F 0 ∩ F 2 ⊇ · · · ⊇ F 0 ∩ F q

Theorem. If ∆ is a forest, then its facets can be ordered as F 0 , F 1 , . . . , F q such that 1. F 0 is a good leaf of ∆ 2. F 0 ∩ F 1 ⊇ F 0 ∩ F 2 ⊇ · · · ⊇ F 0 ∩ F q 3. each F i is a leaf of � F 0 , F 1 , . . . , F i � for 0 ≤ i ≤ q

Theorem. If ∆ is a forest, then its facets can be ordered as F 0 , F 1 , . . . , F q such that 1. F 0 is a good leaf of ∆ 2. F 0 ∩ F 1 ⊇ F 0 ∩ F 2 ⊇ · · · ⊇ F 0 ∩ F q 3. each F i is a leaf of � F 0 , F 1 , . . . , F i � for 0 ≤ i ≤ q 4 3 u w v s 2 x 0 1 z y vy ⊇ v ⊇ v ⊇ ∅

Theorem. (H` a - Van Tuyl 2007) If F is a leaf, then there is an Eliahou-Kervaire type splitting for ∆ described as follows: j −| F | i � � β ij (∆) = β ij (∆ \ F ) + β ℓ 1 − 1 ,ℓ 2 ( C ( F )) β i − ℓ 1 − 1 ,j −| F |− ℓ 2 (∆ / C ( F )) ℓ 1 =0 ℓ 2 =0 where C ( F ) = ( F ′ ∈ ∆ | F ′ ∩ F � = ∅ ) C ( F ) = ( F ′ \ F | F ′ ∈ C ( F ))

Theorem. (H` a - Van Tuyl 2007) If F is a leaf, then there is an Eliahou-Kervaire type splitting for ∆ described as follows: j −| F | i � � β ij (∆) = β ij (∆ \ F ) + β ℓ 1 − 1 ,ℓ 2 ( C ( F )) β i − ℓ 1 − 1 ,j −| F |− ℓ 2 (∆ / C ( F )) ℓ 1 =0 ℓ 2 =0 where C ( F ) = ( F ′ ∈ ∆ | F ′ ∩ F � = ∅ ) C ( F ) = ( F ′ \ F | F ′ ∈ C ( F )) Note. This formula is recursive if ∆ is a forest as C ( F ) = subset of a forest= also a forest C ( F ) = localization of a forest = also a forest

Tree ∆ = ( F 0 , F 1 , . . . , F q − 1 , F q ) ❄ ❄ good leaf leaf β ij (∆) = β ij ( F 0 , . . . , F q − 1 ) + j −| F q | i � � β ℓ 1 − 1 ,ℓ 2 ( C ( F q )) β i − ℓ 1 − 1 ,j −| F q |− ℓ 2 (∆ / C ( F q )) ℓ 1 =0 ℓ 2 =0

Tree ∆ = ( F 0 , F 1 , . . . , F q − 1 , F q ) ❄ ❄ good leaf leaf β ij (∆) = β ij ( F 0 , . . . , F q − 1 ) + 0 j −| F q | i � �� � � � β ℓ 1 − 1 ,ℓ 2 ( C ( F q )) β i − ℓ 1 − 1 ,j −| F q |− ℓ 2 ( ∆ / C ( F q )) ℓ 1 =0 ℓ 2 =0

Tree ∆ = ( F 0 , F 1 , . . . , F q − 1 , F q ) ❄ ❄ good leaf leaf β ij (∆) = β ij ( F 0 , . . . , F q − 1 ) + 0 j −| F q | i � �� � � � β ℓ 1 − 1 ,ℓ 2 ( C ( F q )) β i − ℓ 1 − 1 ,j −| F q |− ℓ 2 ( ∆ / C ( F q )) ℓ 1 =0 ℓ 2 =0 � �� � β i − 1 ,j −| Fq | ( C ( F q ))

Tree ∆ = ( F 0 , F 1 , . . . , F q − 1 , F q ) ❄ ❄ good leaf leaf β ij ( F 0 , . . . , F q ) = β ij ( F 0 , . . . , F q − 1 ) + β i − 1 ,j −| F q | ( C ( F q )) = β ij ( F 0 , . . . , F q − 2 ) + β i − 1 ,j −| F q − 1 | ( C ( F q − 1 )) + β i − 1 ,j −| F q | ( C ( F q )) . . . q � = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 This formula is inductive but not recursive !

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 This formula is inductive but not recursive ! Compute β 0 j (∆) : q � β 0 ,j ( F 0 , . . . , F q ) = δ j, | F u | u =0 where δ a,b is the Kronecker delta function.

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 This formula is inductive but not recursive ! Compute β 0 j (∆) : q � β 0 ,j ( F 0 , . . . , F q ) = δ j, | F u | u =0 where δ a,b is the Kronecker delta function. Compute β 1 j (∆) : q � β 1 j ( F 0 , . . . , F q ) = β 0 ,j −| F u | ( C ( F u )) u =1 We need to know the generators of C ( F u ) !

Theorem. Given a good-leaf-ordering F 0 ∩ F 1 ⊇ · · · ⊇ F 0 ∩ F q – C ( F u ) = ( F i 1 \ F u , . . . , F i s \ F u ) 0 ≤ i 1 < i 2 < · · · < i s < u is a forest – F i s \ F u has a free vertex and is therefore a “splitting facet” of C ( F u )

Theorem. Given a good-leaf-ordering F 0 ∩ F 1 ⊇ · · · ⊇ F 0 ∩ F q – C ( F u ) = ( F i 1 \ F u , . . . , F i s \ F u ) 0 ≤ i 1 < i 2 < · · · < i s < u is a forest – F i s \ F u has a free vertex and is therefore a “splitting facet” of C ( F u ) Moreover if F 0 ∩ F 1 � · · · � F 0 ∩ F q then – i s = u − 1 – C ( F u ) is connected.

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 Compute β 1 j (∆) : q � β 1 j (∆) = β 0 ,j −| F u | ( C ( F u )) u =1 q u − 1 � � = γ j, | F u ∪ F v | , { F s ∪ F u | s<u } u =1 v =0 � j = | N | , N ′ � | N for all N ′ ∈ A 1 where γ j,N,A = 0 otherwise

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 Compute β 2 j (∆) : q � β 2 j (∆) = β 1 ,j −| F u | ( C ( F u )) u =1 = � · · ·

q � β ij ( F 0 , . . . , F q ) = β ij ( F 0 ) + β i − 1 ,j −| F u | ( C ( F u )) u =1 More Generally u 1 − 1 u i − 1 q � � � β ij ( F 0 , . . . , F q ) = · · · γ j, | F u 1 ∪···∪ F ui +1 | , { F u 1 ∪···∪ F ui ∪ F s | s<u i +1 } u 1 =1 u 2 =0 u i +1 =0 where � 1 j = | N | , some division properties related to elements of A γ j,N,A = 0 otherwise