Duality on Value Semigroups Philipp Korell Technische Universitt - PowerPoint PPT Presentation

Duality on Value Semigroups Philipp Korell Technische Universität Kaiserslautern July 4, 2016 Joint work w/ Laura Tozzo & Mathias Schulze

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Example Complex algebroid curve R = C [[ x , y ]] / � x 3 y + y 6 � = C [[ x , y ]] / ( � x 3 + y 5 � ∩ � y � ) = C [[( t 5 1 , t 2 ) , ( − t 3 1 , 0 )]] ⊂ R = C [[ t 1 ]] × C [[ t 2 ]] = R / � x 3 + y 5 � × R / � y � ⊂ Q R = C [[ t 1 ]][ t − 1 1 ] × C [[ t 2 ]][ t − 1 2 ] Parametrization x �→ ( t 5 y �→ ( − t 3 1 , t 2 ) , 1 , 0 ) Discrete valuations ν i = ord t i : C [[ t i ]][ t − 1 ] → Z ∪ {∞} i

Value Semigroup Definition R complex algebroid curve � multivaluation → Z s , ν = ( ν 1 , . . . , ν s ): Q reg R � value semigroup of R Γ R = ν ( R reg ) ⊂ N s . Remark Since ν ( 1 ) = 0 and ν ( ab ) = ν ( a ) + ν ( b ) , Γ R is a monoid.

Example R = C [[ x , y ]] / � x 3 y + y 6 � ∼ = C [[( t 5 1 , t 2 ) , ( − t 3 1 , 0 )]] . Then Γ R = � ( 5 , 1 ) , ( 9 , 2 ) , ( 3 , 1 ) + N e 1 , ( 15 , 3 ) + N e 2 �

Fractional Ideals Definition ◮ A regular fractional ideal (RFI) of R is an R -submodule E ⊂ Q R such that a E ⊂ R for some a ∈ R reg and E ∩ Q reg � = ∅ . R ◮ The value semigroup ideal of E is R ) ⊂ Z s . Γ E = ν ( E ∩ Q reg Remark Applying ν to R E ⊂ E yields Γ E + Γ R ⊂ Γ E .

Definition The conductor of R is C R = R : Q R R , the largest ideal of R in R . Lemma C R = t γ R = ( t γ 1 1 , . . . , t γ s s ) R , where γ = min { α ∈ Γ R | α + N s ⊂ Γ R } is the conductor of Γ R .

Properties of Value Semigroups (E0) There is an α ∈ E such that α + N s ⊂ E . Example α ( t 11 1 , t 3 2 )( C [[ t 1 ]] × C [[ t 2 ]]) ⊂ R

Properties of Value Semigroups (E1) If α, β ∈ E , then ǫ = min { α, β } ∈ E . Example α β ǫ ( t 10 1 , t 2 2 ) + ( t 6 1 + t 25 1 , t 5 2 ) = ( t 6 1 + t 10 1 + t 25 1 , t 2 2 + t 5 2 )

Properties of Value Semigroups (E2) For any α, β ∈ E with α i = β i for some i there is ǫ in E such that ǫ i > α i = β i and ǫ j ≥ min { α j , β j } for all j � = i with equality if α j � = β j . Example i ǫ β α j ( t 6 1 + t 10 1 + t 25 1 , t 2 2 + t 5 2 ) − ( t 10 1 , t 2 2 ) = ( t 6 1 + t 25 1 , t 5 2 )

Good Semigroups and their Ideals Definition ◮ A submonoid S ⊂ N s with group of differences Z s is called a good semigroup if (E0), (E1) and (E2) hold for S . ◮ A good semigroup ideal (GSI) of S is a subset ∅ � = E ⊂ Z s such that ◮ E + S ⊂ E ( � (E0)), ◮ there is an α ∈ S such that α + E ⊂ S , ◮ E satisfies (E1) and (E2). Remark (Barucci, D’Anna, Fröberg) Not any good semigroup is a value semigroup.

General algebraic hypotheses ◮ R one-dimensional semilocal Cohen–Macaulay ring � there are finitely many valuations of Q R containing R , all are discrete ◮ R analytically reduced � (E0) ◮ R has large residue fields � (E1) ◮ R residually rational � (E2) Definition We call a one-dimensional semilocal analytically reduced and residually rational Cohen–Macaulay ring with large residue fields admissible.

Theorem Let R be an admissible ring, E a RFI of R. Then: ◮ Γ R is a good semigroup. ◮ Γ E is a good semigroup ideal. ◮ Γ E = � m ∈ Max ( R ) Γ E m . ◮ Γ E = Γ � E .

Remark In general, ◮ Γ E : F � Γ E − Γ F , ◮ Γ E + Γ F � Γ EF , ◮ Γ E − Γ F not GSI, ◮ Γ E + Γ F not GSI.

Example R = C [[( − t 4 1 , t 2 ) , ( − t 3 1 , 0 ) , ( 0 , t 2 ) , ( t 5 1 , 0 )]] E = � ( t 3 1 , t 2 ) , ( t 2 1 , 0 ) � R F = � ( t 3 1 , t 2 ) , ( t 4 1 , 0 ) , ( t 5 1 , 0 ) � R Γ R Γ E Γ F Γ E + Γ F not GSI

Definition (Delgado) For α ∈ Z s , consider set s � { β ∈ Z s | α i = β i , α j < β j for all j � = i } . ∆( α ) = i = 1 β α i

Definition For E ⊂ Z s , set ∆ E ( α ) = ∆( α ) ∩ E . β ∆ E ( α ) α i

Definition The conductor of a good semigroup ideal E is γ E = min { α ∈ E | α + N s ⊂ E } , and we set τ = γ S − 1 . Theorem (Delgado / Campillo, Delgado, Kiyek) Let R be a local admissible ring. Then R is Gorenstein if and only if Γ R = { α ∈ Z s | ∆ S ( τ − α ) = ∅} ( Γ R symmetric).

Example Irreducible plane curve R = C [[ x , y ]] / � x 7 − y 4 � ∼ = C [[ t 4 , t 7 ]] τ 0 0 1 2 3 4 4 5 6 7 7 8 8 9 10 11 11 12 12 13 14 14 15 15 16 16 17 18 18

Example ∆ S ( τ − α ) = ∅ τ τ − α α

Example ∆ S ( τ − α ) � = ∅ α τ τ − α

Definition ◮ A RFI K is canonical if K : ( K : E ) = E for all RFI E . ◮ R is Gorenstein if R is a canonical ideal. Definition (D’Anna) The canonical semigroup ideal of a good semigroup S is S = { α ∈ Z s | ∆ S ( τ − α ) = ∅} . K 0 Remark R is Gorenstein if and only if Γ R = K 0 Γ R .

Theorem (D’Anna) Let R be local and K a RFI such that R ⊂ K ⊂ R. Then ⇒ Γ K = K 0 K canonical ⇐ Γ R . Theorem (Pol) Let R be a Gorenstein algebroid curve and E a RFI. Then Γ R : E = { α ∈ Z s | ∆ Γ E ( τ − α ) = ∅} = Γ R − Γ E .

Definition (KTS) Let S be a good semigroup. We call K a canonical semigroup ideal if ◮ K GSI ◮ If E GSI with K ⊂ E and γ K = γ E , then K = E . Theorem (KTS) The following are equivalent: ◮ K is a canonical semigroup ideal. ◮ α + K = K 0 S for some α ∈ Z s . ◮ K − ( K − E ) = E for all GSI E. If these are satisfied, then K − E = { β ∈ Z s | ∆ E ( τ − β ) = ∅} + α is a GSI.

Example E is (E1) but not (E2), K 0 S − E not GSI, E � K 0 S − ( K 0 S − E ) . K 0 S E S K 0 K 0 S − ( K 0 S − E S − E )

� � � Main Result Theorem (KTS) Let R be an admissible ring, K a RFI. ◮ K is canonical if and only if Γ K is canonical. ◮ If K is canonical, then E�→K : E { RFI of R } { RFI of R } � E�→ Γ E E�→ Γ E E �→ Γ K − E � { GSI of Γ R } { GSI of Γ R }

Reference [KTS] Philipp Korell, Laura Tozzo, and Mathias Schulze: “Duality on value semigroups”, arXiv 1510.04072 (2015).