Presentations of rings with a chain of semidualizing modules . - PowerPoint PPT Presentation

. Presentations of rings with a chain of semidualizing modules . Ensiyeh Amanzadeh with Mohammad T. Dibaei IPM 12th seminar on commutative algebra and related topics School of Mathemaics, IPM, 2015 . . . . . .

Presentations of rings with a chain of semidualizing modules Semidualizing modules Throughout R is a commutative Noetherian local ring. . Definition . An R –module C is called semidualizing , if • C is finite (i.e. finitely generated) • The natural homothety map χ R C : R − → Hom R ( C , C ) is an isomorphism • For all i > 0, Ext i R ( C , C ) = 0 . . Example . Examples of semidualizing modules include • R • The dualizing module of R if it exists (dualizing module is a semidualizing module with finite injective dimension). . . . . . . .

Presentations of rings with a chain of semidualizing modules Semidualizing modules Throughout C assumed to be a semidualizing R –module. . Basic properties . • Ann R ( C ) = 0 and Supp R ( C ) = Spec ( R ). • dim R ( C ) = dim ( R ) and Ass R ( C ) = Ass R ( R ). • If R is local, then depth R ( C ) = depth ( R ). . If R is Gorenstein and local, then R is the only semidualizing R –module. Conversely, if the dualizing R –module is just the only semidualizing R –module, then R is Gorenstein. . . . . . .

Presentations of rings with a chain of semidualizing modules Totally C –reflexive modules . Definition . A finite R –module M is totally C – reflexive when it satisfies the following conditions. • The natural homomorphism δ C M : M − → Hom R ( Hom R ( M , C ) , C ) is an isomorphism. • For all i > 0, Ext i R ( M , C ) = 0 = Ext i R ( Hom R ( M , C ) , C ). . • Every finite projective R –module is totally C –reflexive. • The G C -dimension of a finite R –module M , denoted G C - dim R ( M ), is defined as { } � � there is an exact sequence of R − modules � G C − dim R ( M ) = inf n ⩾ 0 0 → G n → · · · → G 1 → G 0 → M → 0 � such that each G i is totally C − reflexive . . . . . .

Presentations of rings with a chain of semidualizing modules The set G 0 ( R ) The set of all isomorphism classes of semidualizing R –modules is denoted by G 0 ( R ), and the isomorphism class of a semidualizing R –module C is denoted [ C ]. • Write [ C ] ⊴ [ B ] when B is totally C –reflexive. • Write [ C ] ◁ [ B ] when [ C ] ⊴ [ B ] and [ C ] ̸ = [ B ]. • For each [ C ] ∈ G 0 ( R ) set � { } � [ C ] ⊴ [ B ] G C ( R ) = [ B ] ∈ G 0 ( R ) . • If [ C ] ⊴ [ B ], then (1) Hom R ( B , C ) is a semidualizing, and (2) [ C ] ⊴ [ Hom R ( B , C )]. . . . . . .

Presentations of rings with a chain of semidualizing modules Chain in G 0 ( R ) A chain in G 0 ( R ) is a sequence [ C n ] ⊴ · · · ⊴ [ C 1 ] ⊴ [ C 0 ], and such a chain has length n if [ C i ] ̸ = [ C j ] whenever i ̸ = j . . Theorem (Gerko) . If [ C n ] ⊴ · · · ⊴ [ C 1 ] ⊴ [ C 0 ] is a chain in G 0 ( R ), then one gets C n ∼ = C 0 ⊗ R Hom R ( C 0 , C 1 ) ⊗ R · · · ⊗ R Hom R ( C n − 1 , C n ) . . . . . . . .

Presentations of rings with a chain of semidualizing modules Chain in G 0 ( R ) Assume that [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] is a chain in G 0 ( R ). • For each i ∈ [ n ] set B i = Hom R ( C i − 1 , C i ). • For each sequence of integers i = { i 1 , · · · , i j } with j ⩾ 1 and 1 ⩽ i 1 < · · · < i j ⩽ n , set B i = B i 1 ⊗ R · · · ⊗ R B i j . ( B { i 1 } = B i 1 and set B ∅ = C 0 .) . . . . . .

Presentations of rings with a chain of semidualizing modules Chain in G 0 ( R ) . Proposition (Sather-Wagstaff) . Assume that [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] is a chain in G 0 ( R ) such that G C 1 ( R ) ⊆ G C 2 ( R ) ⊆ · · · ⊆ G C n ( R ). (1) For each sequence i = { i 1 , · · · , i j } ⊆ [ n ], the R –module B i is a semidualizing. (2) If i = { i 1 , · · · , i j } ⊆ [ n ] and s = { s 1 , · · · , s t } ⊆ [ n ] are two sequences with s ⊆ i , then [ B i ] ⊴ [ B s ] and Hom R ( B s , B i ) ∼ = B i \ s . (3) If i = { i 1 , · · · , i j } ⊆ [ n ] and s = { s 1 , · · · , s t } ⊆ [ n ] are two sequences, then the following conditions are equivalent. (i) The R –module B i ⊗ R B s is semidualizing. (ii) i ∩ s = ∅ . . . . . . . .

Presentations of rings with a chain of semidualizing modules Chain in G 0 ( R ) For a semidualizing R –module C , set ( − ) † C = Hom R ( − , C ). . Definition . Let [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] be a chain in G 0 ( R ) of length n . For each sequence of integers i = { i 1 , · · · , i j } such that j ⩾ 0 and † Ci 1 † Ci 2 ···† Cij 1 ⩽ i 1 < · · · < i j ⩽ n , set C i = C . 0 (When j = 0, set C i = C ∅ = C 0 ). We say that the above chain is suitable if C 0 = R and C i is totally C t –reflexive, for all i and t with i j ⩽ t ⩽ n . . • If [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ R ] is a suitable chain, then C i is a semidualizing R –module for each i ⊆ [ n ]. • For each sequence of integers { x 1 , · · · , x m } with 1 ⩽ x 1 < · · · < x m ⩽ n , the sequence [ C x m ] ◁ · · · ◁ [ C x 1 ] ◁ [ R ] is a suitable chain in G 0 ( R ). . . . . . .

Presentations of rings with a chain of semidualizing modules Chain in G 0 ( R ) . Theorem (Sather-Wagstaff) . Let G 0 ( R ) admit a chain [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] such that G C 1 ( R ) ⊆ G C 2 ( R ) ⊆ · · · ⊆ G C n ( R ). { } | = 2 n . • | G 0 ( R ) | ⩾ | [ C i ] | i ⊆ [ n ] { } { } • If C 0 = R , then [ B u ] | u ⊆ [ n ] [ C i ] | i ⊆ [ n ] = . . . . . . . .

Presentations of rings with a chain of semidualizing modules Suitable chains . Lemma (Dibaei and me) . Assume that R admits a suitable chain [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] = [ R ] in G 0 ( R ). Then for any k ∈ [ n ], there exists a suitable chain † Ck † Ck † Ck [ C n ] ◁ · · · ◁ [ C k +1 ] ◁ [ C k ] ◁ [ C ] ◁ · · · ◁ [ C k − 2 ] ◁ [ C k − 1 ] ◁ [ R ] 1 in G 0 ( R ) of length n . . . . . . . .

Presentations of rings with a chain of semidualizing modules Proposition (Suitable chains in G 0 ( R k ) )(Dibaei and me) Let R be Cohen-Macaulay and [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ C 0 ] be a † Ck suitable chain in G 0 ( R ). For any k ∈ [ n ], set R k = R ⋉ C k − 1 the † Ck trivial extension of R by C k − 1 . Set  † Ck   Hom R ( R k , C k − 1 − l ) if 0 ⩽ l < k − 1 C ( k ) = l   Hom R ( R k , C l +1 ) if k − 1 ⩽ l ⩽ n − 1 . • For all l , 0 ⩽ l ⩽ n − 1, C ( k ) is a semidualizing R k –module. l • For any k ∈ [ n ], [ C ( k ) n − 1 ] ◁ · · · ◁ [ C ( k ) ] ◁ [ R k ] 1 is a suitable chain in G 0 ( R k ) of length n − 1. 21 . . . . . .

Presentations of rings with a chain of semidualizing modules Main result . Theorem (Dibaei and me) . Let R be a Cohen–Macaulay ring with a dualizing module D . Assume that R admits a suitable chain [ C n ] ◁ · · · ◁ [ C 1 ] ◁ [ R ] in G 0 ( R ) and that C n ∼ = D . Then there exist a Gorenstein local ring Q and ideals I 1 , · · · , I n of Q , which satisfy the following conditions. In this situation, for each Λ ⊆ [ n ], set R Λ = Q / (Σ l ∈ Λ I l ), in particular R ∅ = Q . (1) There is a ring isomorphism R ∼ = Q / ( I 1 + · · · + I n ). (2) For each Λ ⊆ [ n ] with Λ ̸ = ∅ , the ring R Λ is non-Gorenstein Cohen–Macaulay with a dualizing module. (3) For each Λ ⊆ [ n ] with Λ ̸ = ∅ , we have ∩ l ∈ Λ I l = ∏ l ∈ Λ I l . (4) For subsets Λ, Γ of [ n ] with Γ ⊊ Λ, we have G − dim R Γ R Λ = 0, and Hom R Γ ( R Λ , R Γ ) is a non-free semidualizing R Λ –module. . . . . . . .

Presentations of rings with a chain of semidualizing modules Main result . Theorem (Dibaei and me) . (5) For subsets Λ, Γ of [ n ] with Λ ̸ = Γ, the module Hom R Λ ∩ Γ ( R Λ , R Γ ) is not cyclic and R Λ ∩ Γ Ext ⩾ 1 R Λ ∩ Γ ( R Λ , R Γ ) = 0 = Tor ( R Λ , R Γ ) . ⩾ 1 (6) For subsets Λ, Γ of [ n ] with | Λ \ Γ | = 1, we have i R Λ ∩ Γ � R Λ ∩ Γ ( R Λ , R Γ ) = 0 = � Ext Tor ( R Λ , R Γ ) i for all i ∈ Z . . 22 . . . . . .

Presentations of rings with a chain of semidualizing modules . Construction . We construct the ring Q by induction on n . We claim that the ring Q , as an R –module, has the form Q = ⊕ i ⊆ [ n ] B i and the ring structure on it is as follows. ( ) ( ) For two elements α i i ⊆ [ n ] and θ i i ⊆ [ n ] of Q ∑ ( ) ( ) ( ) α i θ i i ⊆ [ n ] = σ i i ⊆ [ n ] , where σ i = α v · θ w . i ⊆ [ n ] v ⊆ i w = i \ v . • n = 1: set Q = R ⋉ C 1 and I 1 = 0 ⊕ C 1 . (Proved by Foxby and Reiten) • n = 2: The extension ring Q has the form † C 2 Q = R ⊕ C 1 ⊕ C ⊕ C 2 as an R –module. The ring structure 1 ( r , c , f , d )( r ′ , c ′ , f ′ , d ′ ) = on Q is given by ( rr ′ , rc ′ + r ′ c , rf ′ + r ′ f , f ′ ( c ) + f ( c ′ ) + rd ′ + r ′ d ). (Proved by Jorgensen, Leuschke and Sather-Wagstaff) . . . . . .