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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences Leamer Monoids and the Huneke-Wiegand Conjecture Roberto Carlos Pelayo Christopher ONeill Brian Wissman March 23, 2019 Roberto Carlos Pelayo

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Leamer Monoids and the Huneke-Wiegand Conjecture

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman March 23, 2019

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

The Huneke-Wiegand Conjecture

In Tensor Products of Modules and the Rigidity of Tor:

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

The Huneke-Wiegand Conjecture

In Tensor Products of Modules and the Rigidity of Tor: Huneke-Wiegand Conjecture (1994) Let R be a one-dimensional Gorenstein domain. Let M = 0 be a finitely-generated R-module, which is not projective. Then the torsion submodule of M ⊗R HomR(M, R) is non-trivial.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Numerical Semigroup Rings

Let K be a field and Γ be a numerical semigroup. A numerical semigroup ring K[Γ] is the subring of K[t] given by

s∈Γ

ksts, where ks ∈ K. K[Γ] contains polynomials whose powers of t are in Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Numerical Semigroup Rings

Let K be a field and Γ be a numerical semigroup. A numerical semigroup ring K[Γ] is the subring of K[t] given by

s∈Γ

ksts, where ks ∈ K. K[Γ] contains polynomials whose powers of t are in Γ. Goal: Use semigroup structure of Γ to prove the Huneke-Wiegand conjecture for certain ideals of K[Γ].

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Arithmetic Sequences in Numerical Semigroups

Setup: Γ = numerical semigroup, s ∈ Γ is a gap element.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Arithmetic Sequences in Numerical Semigroups

Setup: Γ = numerical semigroup, s ∈ Γ is a gap element. Arithmetic Sequences in Γ: Arithmetic sequences of length m and step-size s: (n, m) := {n, n + s, n + 2s, . . . , n + ms} ⊂ Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Arithmetic Sequences in Numerical Semigroups

Setup: Γ = numerical semigroup, s ∈ Γ is a gap element. Arithmetic Sequences in Γ: Arithmetic sequences of length m and step-size s: (n, m) := {n, n + s, n + 2s, . . . , n + ms} ⊂ Γ. Arithmetic Sequence Addition: (n1, m1) + (n2, m2) = {n1, n1 + s, . . . , n1 + m1s} + {n2, n2 + s, . . . , n2 + m2s} = (n1 + n2, m1 + m2).

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Arithmetic Sequences in Numerical Semigroups

Setup: Γ = numerical semigroup, s ∈ Γ is a gap element. Arithmetic Sequences in Γ: Arithmetic sequences of length m and step-size s: (n, m) := {n, n + s, n + 2s, . . . , n + ms} ⊂ Γ. Arithmetic Sequence Addition: (n1, m1) + (n2, m2) = {n1, n1 + s, . . . , n1 + m1s} + {n2, n2 + s, . . . , n2 + m2s} = (n1 + n2, m1 + m2). ⇒ If (n1, m1), (n2, m2) ⊂ Γ, then (n1 + n2, m1 + m2) ⊂ Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Leamer Monoids

Definition Given a numerical monoid Γ and a gap element s ∈ Γ, the set Ss

Γ = {(n, m) : {n, n + s, . . . , n + ms} ⊂ Γ}

with vector addition is called the Leamer monoid of Γ with step-size s.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Leamer Monoids

Definition Given a numerical monoid Γ and a gap element s ∈ Γ, the set Ss

Γ = {(n, m) : {n, n + s, . . . , n + ms} ⊂ Γ}

with vector addition is called the Leamer monoid of Γ with step-size s. Irreducible elements: An arithmetic sequence (n, m) ∈ Ss

Γ is

irreducible if it cannot be written as the sum of two other non-trivial arithmetic sequences.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10:

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →}

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →} And s = 3 ∈ Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →} And s = 3 ∈ Γ. Examples:

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →} And s = 3 ∈ Γ. Examples: Reducible: (28, 3) = {28, 31, 34, 37} =

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →} And s = 3 ∈ Γ. Examples: Reducible: (28, 3) = {28, 31, 34, 37} = {7, 10} + {21, 24, 27} = (7, 1) + (21, 2)

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

An Arithmetic Example

Let Γ = 7, 10: 7, 10 = {0, 7, 10, 14, 17, 20, 21, 24, 27, 28, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 54, →} And s = 3 ∈ Γ. Examples: Reducible: (28, 3) = {28, 31, 34, 37} = {7, 10} + {21, 24, 27} = (7, 1) + (21, 2) Irreducible: (57, 2) = {57, 60, 63}

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

A Graphical Example

We can plot Ss

Γ in N2!

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

A Graphical Example

We can plot Ss

Γ in N2!

Γ = 7, 10, s = 3

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

A Graphical Example

We can plot Ss

Γ in N2!

Γ = 7, 10, s = 3 Examples: (28, 3) = (7, 1) + (21, 2) is reducible (57, 2) is irreducible

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

What Does This Have to Do with the HW Conjecture?

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

What Does This Have to Do with the HW Conjecture?

Theorem [Garc´ ıa-Sanchez, Leamer] Let Γ be a symmetric numerical semigroup and K a field. The numerical semigroup ring K[Γ] satisfies the Huneke-Wiegand conjecture for 2-generated monomial ideals if and only if, for each gap element s ∈ N \ Γ, there exists an irreducible arithmetic sequence (x, 2) ∈ Ss

Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

What Does This Have to Do with the HW Conjecture?

Theorem [Garc´ ıa-Sanchez, Leamer] Let Γ be a symmetric numerical semigroup and K a field. The numerical semigroup ring K[Γ] satisfies the Huneke-Wiegand conjecture for 2-generated monomial ideals if and only if, for each gap element s ∈ N \ Γ, there exists an irreducible arithmetic sequence (x, 2) ∈ Ss

Γ.

There exists an irreducible (x, 2) ∈ Ss

Γ iff I = ta, ta+s

satisfies the HW conjecture for any a ∈ Γ. For R = K[Γ], the length of the torsion submodule of I ⊗R HomR(I, R) is the number of height 2 irreducible elements in Ss

Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Finding Irreducible Elements of Height 2

Goal: For every gap element s ∈ Γ, find a height 2 irreducible.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Finding Irreducible Elements of Height 2

Goal: For every gap element s ∈ Γ, find a height 2 irreducible. Problem: This graph only corresponds to one gap element s.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Finding Irreducible Elements of Height 2 for every s

Solution: For every gap element s ∈ Γ, plot a point at (s, n) if there exists a height 2 irreducible (n, 2) ∈ Ss

Γ...

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Finding Irreducible Elements of Height 2 for every s

Solution: For every gap element s ∈ Γ, plot a point at (s, n) if there exists a height 2 irreducible (n, 2) ∈ Ss

Γ...

Γ = 4, 9 ...and use this plot to find a relationship between s and n.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Finding Irreducible Elements of Height 2 for every s

Solution: For every gap element s ∈ Γ, plot a point at (s, n) if there exists a height 2 irreducible (n, 2) ∈ Ss

Γ...

Γ = 4, 9 ...and use this plot to find a relationship between s and n. Example: 4, 9 will have a height 2 irreducible at n = 27 − s.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Embedding Dimension 3

Consider Γ = 6, 10, 15.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Embedding Dimension 3

Consider Γ = 6, 10, 15. We need to use multiple lines!

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Embedding Dimension 3

Theorem Given a symmetric numerical semigroup Γ = n1, n2, n3 with embedding dimension 3, Ss

Γ has a height 2 irreducible at

F(Γ) − s + nj for some minimal generators nj. F(Γ) = Frobenius number of Γ.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

When the numerical semigroup is generated by a (generalized) arithmetic sequence, finding arithmetic sequences becomes tractable!

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

When the numerical semigroup is generated by a (generalized) arithmetic sequence, finding arithmetic sequences becomes tractable! Definition Let n, d, h, k be positive integers. A numerical monoid generated by a generalized arithmetic sequence is of the form n, nh + d, nh + 2d, . . . , nh + kd, where gcd(n, d) = 1 and k < n.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

When the numerical semigroup is generated by a (generalized) arithmetic sequence, finding arithmetic sequences becomes tractable! Definition Let n, d, h, k be positive integers. A numerical monoid generated by a generalized arithmetic sequence is of the form n, nh + d, nh + 2d, . . . , nh + kd, where gcd(n, d) = 1 and k < n. For example, when n = 6, d = 1, h = 2, k = 4, we obtain 6, 13, 14, 15, 16.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

Consider Γ = 6, 13, 14, 15, 16.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

Consider Γ = 6, 13, 14, 15, 16. Here, we need one non-constant linear function and one constant function.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Generated by Generalized Arithmetic Sequence

Consider Γ = 6, 13, 14, 15, 16. Theorem For a numerical semigroup Γ = n, nh + d, nh + 2d, . . . , nh + kd generated by a generalized arithmetic sequence, there exists a height 2 irreducible at F(Γ) − s + d or nh + d.

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture

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The Huneke-Wiegand Conjecture and Leamer Monoids Finding Irreducible Arithmetic Sequences

Thank you!

Mahalo to the following: The NSF-funded PURE Math program Team Zoboomafoo: Miguel Landeros, Karina Pe˜ na, and Jimmy Ren Scott Chapman for suggesting these problems Chris O’Neill, Scott Chapman, and Jim Coykendall for

rganizing this awesome session!

Roberto Carlos Pelayo Christopher O’Neill Brian Wissman Leamer Monoids and the Huneke-Wiegand Conjecture