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The Big Idea The Set-up The Whole Picture The New Stuff The End The Tambara Structure of the Trace Ideal Maxine Calle Reed College, Portland OR with S. Ginnett Collaborative Mathematics Research Group, 2019 supervised by K. Ormsby and A.

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Tambara Structure of the Trace Ideal

Maxine Calle

Reed College, Portland OR with S. Ginnett Collaborative Mathematics Research Group, 2019 supervised by K. Ormsby and A. Osorno callem@reed.edu

February 1, 2020

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Big Idea Ring Theory Tambara Functor Theory

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Big Idea Ring Theory Tambara Functor Theory

Trace homomorphism: Dress map: A → GW A → GW Kernel Trace ideal T I

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Big Idea Ring Theory Tambara Functor Theory

Trace homomorphism: Dress map: A → GW A → GW Kernel Trace ideal T I

Goal: Determine T I

Then GW ∼ = A / T I when the Dress map is surjective.

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Basic Ingredients

Our Main Example

Cyclic group with N elements: CN. Finite field with q elements: Fq (for q a power of an odd prime). Fq ⊆ Fp ⇐ ⇒ q | p, i.e. p = qN. Then Gal(Fp/Fq) = CN.

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The Less Basic Ingredients

Tambara functors (D. Tambara [4], 1993)

Specified by data:

A commutative ring for each subgroup of G
Tambara structure maps restriction, transfer, norm, and

conjugation satisfying various compatibility conditions and commutative diagrams

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Less Basic Ingredients

Tambara functors (D. Tambara [4], 1993)

Specified by data:

A commutative ring for each subgroup of G
Tambara structure maps restriction, transfer, norm, and

conjugation satisfying various compatibility conditions and commutative diagrams

Examples we care about:

Burnside Tambara functor A
Grothendieck-Witt (Galois) Tambara functor GW

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The Burnside functor A on Cp

n[Cp/e] + m[Cp/Cp] := ntp + m and t2

p = ptp

A(Cp/Cp) ∼ = Z[tp]/(t2

p − ptp)

A(Cp/e) ∼ = Z n[e/e] := n

res tr N

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The Burnside functor A on Cp

ntp

np−n p

tp + n ntp + m A(Cp/Cp) ∼ = Z[tp]/(t2

p − ptp)

A(Cp/e) ∼ = Z n pn + m

res tr N

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Grothendieck-Witt functor GW on Fq ⊆ Fqp

n1 ⊕ α GW (Fq) GW (Fqp) n1 ⊕ β

res tr N

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The Grothendieck-Witt functor GW on Fq ⊆ Fqp

GW (Fq) GW (Fqp)

res tr N

restriction: 1 → 1 α → β transfer: 1 → p1 β → (p − 1)1 ⊕ α norm: n1 → np1 (n − 1)1 ⊕ β → (np − 1)1 ⊕ αn

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The Dress Map

Definition

For rings, trace homomorphism (A. Dress [2], 1971)
For Tambara functors, Dress map D is given by trace

homomorphism at each level

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The Dress Map

Definition

For rings, trace homomorphism (A. Dress [2], 1971)
For Tambara functors, Dress map D is given by trace

homomorphism at each level tp − → p1 1 − → 1 A(Cp/Cp) GW (Fq) A(Cp/e) GW (Fqp)

DCp res res De N tr tr N

1 − → 1

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Example: The Whole Picture

A(Cp/Cp) GW (Fq) A(Cp/e) GW (Fqp)

DCp res res De N tr tr N

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The Big Idea The Set-up The Whole Picture The New Stuff The End

Example: The Whole Picture

trCp

e (1) = tp

tr

Fe

F

Cp qp 1 = tr

Fqp Fq 1 = p1

A(Cp/Cp) GW (Fq) A(Cp/e) GW (Fqp) 1 1

DCp res res De N tr tr N

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Trace Ideal

Definition

The trace ideal is the kernel of the Dress map, T I = {ker(DH)}H≤G

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The Big Idea The Set-up The Whole Picture The New Stuff The End

The Trace Ideal

Definition

The trace ideal is the kernel of the Dress map, T I = {ker(DH)}H≤G

Goal

Determine trace ideal (as Tambara ideal), find generators.

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The Trace Ideal

Definition

The trace ideal is the kernel of the Dress map, T I = {ker(DH)}H≤G

Goal

Determine trace ideal (as Tambara ideal), find generators.

Theorem

For cyclic groups, there is one generator!

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Other Results and Future Work

1. Arbitrary cyclic extensions (non-finite fields)
For C/R, e.g., the trace ideal is 0, implying GW ∼

= A

2. Profinite extensions of finite fields
Quadratic closure Fq2∞ and the algebraic closure Fq
3. Prime ideals of A
In progress...

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References

M. Calle and S. Ginnett (2019)

The Tambara Structure of the Trace Ideal for Cyclic Extensions. Submitted for publication. Available: arxiv:1910.03029

A. Dress (1971)

Notes on the theory of representations of finite groups. Universit¨ at Bielefeld, Fakult¨ at f¨ ur Mathematik, Bielefeld.

H. Nakaoka (2011)

Ideals of Tambara functors. Advances in Mathematics, 230:2295–2331.

D. Tambara (1993)

On multiplicative transfer. Communications in Algebra, 28: 1393–1420.

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