The Tambara Structure of the Trace Ideal Maxine Calle Reed College, - PowerPoint PPT Presentation

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

The Big Idea The Set-up The Whole Picture The New Stuff The End The Big Idea Ring Theory � Tambara Functor Theory

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

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.

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: C N . Finite field with q elements: F q (for q a power of an odd prime). ⇒ q | p , i.e. p = q N . Then Gal( F p / F q ) = C N . F q ⊆ F p ⇐

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

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

The Big Idea The Set-up The Whole Picture The New Stuff The End The Burnside functor A on C p n [ C p / e ] + m [ C p / C p ] := nt p + m and t 2 p = pt p A ( C p / C p ) ∼ = Z [ t p ] / ( t 2 p − pt p ) N res tr A ( C p / e ) ∼ = Z n [ e / e ] := n

The Big Idea The Set-up The Whole Picture The New Stuff The End The Burnside functor A on C p n p − n nt p t p + n nt p + m p A ( C p / C p ) ∼ = Z [ t p ] / ( t 2 p − pt p ) N res tr A ( C p / e ) ∼ = Z n pn + m

The Big Idea The Set-up The Whole Picture The New Stuff The End The Grothendieck-Witt functor GW on F q ⊆ F q p n � 1 � ⊕ � α � GW ( F q ) res tr N GW ( F q p ) n � 1 � ⊕ � β �

The Big Idea The Set-up The Whole Picture The New Stuff The End The Grothendieck-Witt functor GW on F q ⊆ F q p GW ( F q ) tr N res GW ( F q p ) restriction: � 1 � �→ � 1 � � α � �→ � β � transfer: � 1 � �→ p � 1 � � β � �→ ( p − 1) � 1 � ⊕ � α � norm: n � 1 � �→ n p � 1 � ( n − 1) � 1 � ⊕ � β � �→ ( n p − 1) � 1 � ⊕ � α n �

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

The Big Idea The Set-up The Whole Picture The New Stuff The End 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 t p �− → p � 1 � 1 �− → � 1 � D Cp A ( C p / C p ) GW ( F q ) tr res tr N res N A ( C p / e ) GW ( F q p ) D e 1 �− → � 1 �

The Big Idea The Set-up The Whole Picture The New Stuff The End Example: The Whole Picture D Cp A ( C p / C p ) GW ( F q ) tr res tr N res N D e A ( C p / e ) GW ( F q p )

The Big Idea The Set-up The Whole Picture The New Stuff The End Example: The Whole Picture F e tr C p F qp qp e (1) = t p qp � 1 � = tr F q � 1 � = p � 1 � tr Cp F D Cp A ( C p / C p ) GW ( F q ) res res tr N tr N D e A ( C p / e ) GW ( F q p ) 1 � 1 �

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( D H ) } H ≤ G

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( D H ) } H ≤ G Goal Determine trace ideal (as Tambara ideal), find generators.

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( D H ) } H ≤ G Goal Determine trace ideal (as Tambara ideal), find generators. Theorem For cyclic groups, there is one generator!

The Big Idea The Set-up The Whole Picture The New Stuff The End 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 F q 2 ∞ and the algebraic closure F q 3. Prime ideals of A • In progress...

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

The Big Idea The Set-up The Whole Picture The New Stuff The End Thank You! (Questions?)