Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Huneke–Wiegand conjecture of rank one with the change of rings Naoki Taniguchi Meiji University Joint work with S. Goto, R. Takahashi, and H. L. Truong RIMS Workshop The 35th Symposium on Commutative Algebra December 5, 2013 Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 1 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Introduction R an integral domain 1 M , N finitely generated torsionfree R -modules 2 Question When is the tensor product M ⊗ R N torsionfree? Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 2 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Conjecture 1.1 ( Huneke–Wiegand conjecture [4]) Let R be a Gorenstein local domain. Let M be a maximal C–M R -module. If M ⊗ R Hom R ( M , R ) is torsionfree, then M is free. Conjecture 1.2 Let R be a Gorenstein local domain with dim R = 1 and I an ideal of R . If I ⊗ R Hom R ( I , R ) is torsionfree, then I is principal. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 3 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Conjecture 1.1 ( Huneke–Wiegand conjecture [4]) Let R be a Gorenstein local domain. Let M be a maximal C–M R -module. If M ⊗ R Hom R ( M , R ) is torsionfree, then M is free. Conjecture 1.2 Let R be a Gorenstein local domain with dim R = 1 and I an ideal of R . If I ⊗ R Hom R ( I , R ) is torsionfree, then I is principal. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 3 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References In my lecture we are interested in the question of what happens if we replace Hom R ( I , R ) by Hom R ( I , K R ). Conjecture 1.3 Let R be a C–M local ring with dim R = 1 and assume ∃ K R . Let I be a faithful ideal of R . If I ⊗ R Hom R ( I , K R ) is torsionfree, then I ∼ = R or K R as an R -module. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 4 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References In my lecture we are interested in the question of what happens if we replace Hom R ( I , R ) by Hom R ( I , K R ). Conjecture 1.3 Let R be a C–M local ring with dim R = 1 and assume ∃ K R . Let I be a faithful ideal of R . If I ⊗ R Hom R ( I , K R ) is torsionfree, then I ∼ = R or K R as an R -module. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 4 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Theorem 1.4 (Main Theorem) Let R be a C–M local ring with dim R = 1 and assume ∃ K R . Let I be a faithful ideal of R . (1) Assume that the canonical map t : I ⊗ R Hom R ( I , K R ) → K R , x ⊗ f �→ f ( x ) is an isomorphism. If r , s ≥ 2, then e ( R ) > ( r + 1) s ≥ 6, where r = µ R ( I ) and s = µ R (Hom R ( I , K R )). (2) Suppose that I ⊗ R Hom R ( I , K R ) is torsionfree. If e ( R ) ≤ 6, then I ∼ = R or K R . Corollary 1.5 Let R be a C–M local ring with dim R ≥ 1. Assume that R p is Gorenstein and e ( R p ) ≤ 6 for every height one prime p . Let I be a faithful ideal of R . If I ⊗ R Hom R ( I , R ) is reflexive, then I is principal. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 5 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Theorem 1.4 (Main Theorem) Let R be a C–M local ring with dim R = 1 and assume ∃ K R . Let I be a faithful ideal of R . (1) Assume that the canonical map t : I ⊗ R Hom R ( I , K R ) → K R , x ⊗ f �→ f ( x ) is an isomorphism. If r , s ≥ 2, then e ( R ) > ( r + 1) s ≥ 6, where r = µ R ( I ) and s = µ R (Hom R ( I , K R )). (2) Suppose that I ⊗ R Hom R ( I , K R ) is torsionfree. If e ( R ) ≤ 6, then I ∼ = R or K R . Corollary 1.5 Let R be a C–M local ring with dim R ≥ 1. Assume that R p is Gorenstein and e ( R p ) ≤ 6 for every height one prime p . Let I be a faithful ideal of R . If I ⊗ R Hom R ( I , R ) is reflexive, then I is principal. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 5 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Contents Introduction 1 Change of rings 2 Proof of Theorem 1.4 3 Numerical semigroup rings and monomial ideals 4 The case where e ( R ) = 7 5 Examples 6 Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 6 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Notation In what follows, unless other specified, we assume ( R , m ) a C–M local ring, dim R = 1 1 F = Q( R ) the total ring of fractions of R . 2 F = { I | I is a fractional ideal such that FI = F } 3 ∃ a canonical module K R of R 4 M ∨ = Hom R ( M , K R ) for each R -module M 5 µ R ( M ) = ℓ R ( M / m M ) for each R -module M 6 Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 7 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Change of rings Let I ∈ F . Denote by t : I ⊗ R I ∨ → K R , x ⊗ f �→ f ( x ) . Then the diagram ∼ = F ⊗ R ( I ⊗ R I ∨ ) − − − → F ⊗ R K R � � α t I ⊗ R I ∨ − − − → K R is commutative. Hence T := T ( I ⊗ R I ∨ ) = Ker t . Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 8 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Lemma 2.1 I ⊗ R I ∨ is torsionfree ⇐ ⇒ t : I ⊗ R I ∨ − → K R is injective. t We set L = Im( I ⊗ R I ∨ → K R ). − Consider t 0 → T → I ⊗ R I ∨ − → L → 0 . Hence L ∨ ∼ = ( I ⊗ R I ∨ ) ∨ = Hom R ( I , I ∨∨ ) ∼ = I : I =: B ⊆ F . Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 9 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Lemma 2.1 I ⊗ R I ∨ is torsionfree ⇐ ⇒ t : I ⊗ R I ∨ − → K R is injective. t We set L = Im( I ⊗ R I ∨ → K R ). − Consider t 0 → T → I ⊗ R I ∨ − → L → 0 . Hence L ∨ ∼ = ( I ⊗ R I ∨ ) ∨ = Hom R ( I , I ∨∨ ) ∼ = I : I =: B ⊆ F . Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 9 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Let R ⊆ S ⊆ B . Then I is also a fractional ideal of S . L = L ∨∨ = B ∨ = K B ⊆ S ∨ = K S and Hom S ( I , K S ) = Hom S ( I , Hom R ( S , K R )) ∼ Hom R ( I ⊗ S S , K R ) = Hom R ( I , K R ) = I ∨ . = Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 10 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References t S I ⊗ S Hom S ( I , K S ) − − → K S � � ρ ι t I ⊗ R I ∨ − − → L where ρ ( x ⊗ f ) = x ⊗ f . Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 11 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Lemma 2.2 Let I ∈ F and R ⊆ S ⊆ B = I : I . If I ⊗ R I ∨ is torsionfree, then I ⊗ S Hom S ( I , K S ) is a torsionfree S -module and ρ : I ⊗ R I ∨ → I ⊗ S Hom S ( I , K S ) is bijective. In particular, if S = B , then t B : I ⊗ B Hom B ( I , K B ) → K B , x ⊗ f �→ f ( x ) is an isomorphism of B -modules. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 12 / 33
Introduction Change of rings Proof of Thm 1.4 Numerical semigroup rings The case where e ( R ) = 7 Examples References Proposition 2.3 ( Change of rings ) Let I ∈ F and assume that I ⊗ R I ∨ is torsionfree. If there exists R ⊆ S ⊆ B such that I ∼ = S or K S as an S -module, then I ∼ = R or K R as an R -module. Proof. Suppose I ∼ = S and consider I ⊗ R I ∨ ρ ∼ = I ⊗ S Hom S ( I , K S ) ∼ = Hom S ( I , K S ) ∼ = I ∨ . Then µ R ( I ) · µ R ( I ∨ ) = µ R ( I ∨ ) , so that I ∼ = R , since µ R ( I ) = 1. Naoki Taniguchi (Meiji University) Huneke–Wiegand conjecture December 5, 2013 13 / 33
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