Chromatic Redshift
John Rognes
Department of Mathematics
University of Oslo, Norway
MSRI/January 2014
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
1 / 54
Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
2 / 54
Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
3 / 54
Iterated Modulation
R, +, ×
commutative ring
Mod(R), ⊕, ⊗
bipermutative category
Mod(Mod(R))
ring-like 2-category
...
Mod(n) (R)
ring-like n-category
...
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Chromatic Redshift
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Iterated K-Theory
B
commutative S-algebra
K (B)
algebraic K -theory spectrum
K (K (B))
double algebraic K -theory
...
K (n) (B)
n-fold algebraic K -theory
...
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Chromatic Redshift
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Height of Formal Group Laws
Hesselholt–Madsen ’97: Chromatic filtration of iterated topological
cyclic homology?
Chromatic filtration of iterated algebraic K -theory?
Formal coproduct on K (n) (B)∗ (CP ∞ )?
F (x1 , x2 ) = x1 + x2 + . . .
n
.
[p]F (x) = x p + . . .
John Rognes (UiO)
formal group law
height n
Chromatic Redshift
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Redshift
B
K
7−→
K (B)
ring spectrum
ring spectrum
FGL of height n
FGL of height n + 1
vn -periodic
vn+1 -periodic
|vn | = 2pn − 2
<
|vn+1 | = 2pn+1 − 2
longer wavelength, less energy
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Chromatic Redshift
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(Co-)Homological Incarnation
Nested Hopf subalgebras
0 ⊂ · · · ⊂ E (n) = E(Q0 , . . . , Qn ) ⊂ · · ·
in Steenrod algebra A = H ∗ (H).
Nested A∗ -comodule subalgebras
A∗ ⊃ · · · ⊃ (A //E (n))∗ = P(ξ¯k | k ≥ 1) ⊗ E(τ̄k | k > n) ⊃ · · ·
invariant under Dyer–Lashof operations on A∗ = H∗ (H).
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
9 / 54
K of Finite Fields
Finite field k , characteristic p > 0
Theorem (Quillen ’72)
Hi (BGL(k̄ ); Fp ) = 0 for i > 0.
K (k̄ )p ' HZp
π∗ K (k )p = π∗ K (k̄ )phGk for ∗ ≥ 0
K (k )p ' HZp
Mult. by p injective on π∗ K (k̄ )p and π∗ K (k )p
p ∈ π0 S lifts v0 ∈ π0 BP.
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Chromatic Redshift
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Lichtenbaum Conjecture
Separably closed field F̄ , characteristic 6= p
Conjecture (Lichtenbaum)
πt K (F̄ )p = Zp for t ≥ 0 even, 0 for t odd.
Proved by Suslin ’84
K (F̄ )p ' kup
L̂1 K (F̄ ) ' KUp where L̂n = LK (n)
Mult. by u ∈ π2 kup bijective on π∗ K (F̄ )p for ∗ ≥ 0.
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Chromatic Redshift
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K of Number Rings
Number field F , ring of S-integers A
/F
O
AO
Z
John Rognes (UiO)
/ Z[1/p]
Chromatic Redshift
/Q
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Quillen Conjecture
Conjecture (Quillen ’75)
−s
2
Es,t
= Hét
(Spec A; Zp (t/2)) =⇒ πs+t K (A)p
converging for s + t ≥ 1.
Zp (t/2) = πt K (F̄ )p
π∗ K (F )p = π∗ K (F̄ )phGF for ∗ ≥ 1
Mult. by β ∈ π2p−2 (S/p) bijective on π∗ (K (A); Z/p), for ∗ ≥ 1
β lifts u p−1 ∈ π∗ (ku; Z/p) and v1 ∈ π∗ BP.
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Partial and Full Verifications
Thomason ’85:
π∗ (K (F ); Z/p)[1/β] = π∗ (K (F̄ )hGF ; Z/p)
for ∗ ≥ 2.
Waldhausen ’84: Is K (A) → L1 K (A) a p-adic equivalence in high
degrees, where Ln = LE(n) ?
Bökstedt–Madsen ’94, ’95, R. ’99, Hesselholt–Madsen ’03:
Confirmed for local number fields.
Voevodsky ’03, ’11: Confirmed for (global number) fields by proof
of Milnor and Bloch–Kato conjectures.
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Chromatic Redshift
MSRI/January 2014
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K of Topological K-Theory
Adams summand L = E(1) of KU(p) , conn. cover ` = BPh1i
Theorem (Ausoni-R. ’02)
h
V (1)∗ K (`p ) = P(v2 ) ⊗ E(λ1 , λ2 , ∂)
⊕ E(λ2 ) ⊗ Fp {λ1 t d | 0 < d < p}
i
⊕ E(λ1 ) ⊗ Fp {λ2 t dp | 0 < d < p}
in degrees ≥ 2p − 2, where (. . . ).
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Chromatic Redshift
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K of Topological K-Theory, II
Adams summand L = E(1) of KU(p) , conn. cover ` = BPh1i
Theorem (Ausoni-R. ’02, Ausoni ’10)
V (1)∗ K (`p ) = (. . . ) and V (1)∗ K (kup ) = (. . . ).
V (1) = S/(p, v1 ) = S ∪p e1 ∪α1 e2p−1 ∪p e2p
Blumberg–Mandell ’08: Also V (1)∗ K (Lp ) and V (1)∗ K (KUp )
Mult. by v2 ∈ π2p2 −2 V (1) bijective on each answer, for ∗ ≥ 2p − 2
v2 lifts v2 ∈ π∗ BP.
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
17 / 54
Quillen Conjecture, II
`p -algebra of S-integers B
Conjecture (à la Quillen/Voevodsky)
−s
2
Es,t
= Hmot
(Spec B; Fp2 (t/2)) =⇒ V (1)s+t K (B)
converging for s + t 0.
∗ (−) motivic cohomology for commutative S-algebras?
Hmot
Fp2 (t/2) = V (1)t E2 , where π∗ E2 = WFp2 [[u1 ]][u ±1 ]
Need Fp2 due to sign in Ausoni’s relation bp−1 = −v2 .
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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`p -Algebra of Integers
Definition (`p -algebra of integers B)
A connected commutative `p -algebra over a Galois extension of
Lp [1/p], semi-finite as `p -module.
/M
O
BO
G
`p
/ Lp
/ Lp [1/p]
Examples!? For S-integers, allow localizations.
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Chromatic Redshift
MSRI/January 2014
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`p -Algebraic Integers
Definition (`p -algebraic integers Ω1 )
p-completed homotopy colimit of all such B.
ΩO 1
/M
O
BO
G
`p
/ Lp
O
/ Lp [1/p]
Jp
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Chromatic Redshift
MSRI/January 2014
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Schloß Ringberg ’99 Conjecture
Conjecture (à la Lichtenbaum/Suslin)
Mult. by v2 bijective on V (1)∗ K (B) for ∗ 0.
V (1)∗ K (Ω1 ) = V (1)∗ E2 for ∗ 0.
L̂2 K (Ω1 ) ' E2 .
For G-Galois B → Ω1 , expect V (1)∗ K (B) = V (1)∗ K (Ω1 )hG for
∗0
Ring spectrum map K (ku) → E2 ?
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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en -Algebra of Integers
Lubin–Tate spectrum En , connective cover en
Definition (en -algebra of integers B)
A connected commutative en -algebra over a Galois extension of
En [1/p], semi-finite as en -module.
/M
O
BO
G
en
John Rognes (UiO)
/ En
/ En [1/p]
Chromatic Redshift
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en -Algebraic Integers
Definition (en -algebraic integers Ωn )
p-completed homotopy colimit of all such B.
ΩO n
/M
O
BO
G
en
/ En
O
/ En [1/p]
L̂n S
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Finite Localizations
Hopkins–Smith ’98: Finite p-local spectrum F , with vn+1 self map
v : Σd F → F .
Miller ’92: Finite localization Lfn+1 annihilates finite E(n+1)-acyclic
spectra.
F∗ Lfn+1 X = F∗ X [1/v ] for any X .
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Schloß Ringberg ’99 Conjecture, II
en → B → Ωn and (F , v ) as above
Conjecture
Mult. by v bijective on F∗ K (B) for ∗ 0.
K (B) → Lfn+1 K (B) a p-adic equivalence in high degrees.
F∗ K (Ωn ) = F∗ En+1 for ∗ 0.
L̂n+1 K (Ωn ) ' En+1 .
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Topological Hochschild Homology
B commutative S-algebra
∼ functions on X
THH(B) = B ⊗ S 1 topological Hochschild homology
∼ functions on free loop space L X
1
1
1 , THH(B))S homotopy fixed points
THH(B)hS = F (ES+
1 ∧
∼ functions on Borel construction ES+
S1 L X
1
1
1
f ∧ F (ES 1 , THH(B))]S Tate construction
THH(B)tS = [ES
+
∼ functions on periodicized Borel construction.
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Chromatic Redshift
MSRI/January 2014
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Trace Maps
THH(B)hCpn
8
K (B)
O
/ THH(B)Cpn
&
/ THH(B)
tCpn+1
THH(B)
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Chromatic Redshift
MSRI/January 2014
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Trace Maps, II
THH(B)hS
p
9
K (B)
O
/ THH(B)
/ TF (B; p)
%
THH(B)tS
p
John Rognes (UiO)
1
1
Chromatic Redshift
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Traces of Redshift
No redshift in THH(B).
1
All redshift yet seen in K (B) also visible in THH(B)tS .
Detectable in A∗ -coaction on homology.
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Homological Approach
en -algebra of integers B
Ho
E(n)
O
/ En
O
BPhni
/ en
/B
H∗ (en ) awkward for n ≥ 2
H∗ (BPhni) = P(ξ¯k | k ≥ 1) ⊗ E(τ̄k | k > n)
Subalgebra of A∗ = P(ξ¯k | k ≥ 1) ⊗ E(τ̄k | k ≥ 0)
H∗ (B) commutative H∗ (BPhni)-algebra
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vn -Periodic Input
Adams spectral sequence
E2s,t (B) = Exts,t
A∗ (Fp , H∗ (B)) =⇒ πt−s (B)
algebra over
E2∗,∗ = P(v0 , . . . , vn )
converging to
π∗ BPhnip = Zp [v1 , . . . , vn ]
generating vn -periodicity in π∗ (B).
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Chromatic Redshift
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Add Circle Action
Circle action gives suspension operator σ.
Bökstedt spectral sequence
2
Es,t
(B) = HHs (H∗ (B))t =⇒ Hs+t (THH(B))
algebra over
2
E∗,∗
= H∗ (BPhni) ⊗ E(σ ξ¯k | k ≥ 1) ⊗ Γ(στ̄k | k > n)
converging to H∗ (THH(BPhni)).
John Rognes (UiO)
Chromatic Redshift
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Differentials and Extensions
Dyer–Lashof operations
k
Q p (τ̄k ) = τ̄k +1
imply multiplicative extensions (στ̄k )p = στ̄k +1 and differentials
.
d p−1 (γj στ̄k ) = σ ξ¯k +1 · γj−p στ̄k
p
∞ converging to
for k > n, j ≥ p, leaving E∗,∗
= E∗,∗
H∗ (THH(BPhni)) = H∗ (BPhni) ⊗ E(σ ξ¯1 , . . . , σ ξ¯n+1 ) ⊗ P(στ̄n+1 ) .
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Remove Circle Action
Homological Tate spectral sequence
1
2
c
Es,t
(B) = Ĥ −s (S 1 ; Ht (THH(B))) =⇒ Hs+t
(THH(B)tS )
algebra over
2
E∗,∗
= P(t ±1 ) ⊗ H∗ (THH(BPhni))
1
converging to H∗c (THH(BPhni)tS ).
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Differentials, II
Circle invariance gives differentials
d 2 (t i · x) = t i+1 · σx
leaving
p
3
= P(t ±1 ) ⊗ P(ξ¯1p , . . . , ξ¯n+1
, ξ¯k | k > n+1)
E∗,∗
p−1 ¯
σ ξn+1 )
⊗ E(τk0 | k > n+1) ⊗ E(ξ¯1p−1 σ ξ¯1 , . . . , ξ¯n+1
where τk0 = τ̄k − τ̄k −1 (στ̄k −1 )p−1 for k > n+1.
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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(E 2 , d 2 )-Charts
O
t · ξ¯kp−1 σ ξ¯k
O
t · (στ̄n+1 )p
ck
ξ¯kp
t · σ ξ¯k g
τ̄n+1 (στ̄n+1 )p−1 , τ̄n+2
t · στ̄n+1
g
ξ¯
k
1
τ̄n+1
1
1≤k ≤n+1
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Chromatic Redshift
MSRI/January 2014
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Extensions, II
Bruner–R. ’05: Often collapses at E 3 = E ∞ .
Lunøe-Nielsen–R. ’11, ’12, Knut Berg: Often A∗ -comodule
extensions combining copies of
p
P(ξ¯1p , . . . , ξ¯n+1
, ξ¯k | k > n+1) ⊗ E(τk0 | k > n+1)
to
P(ξ¯k | k ≥ 1) ⊗ E(τk0 | k > n+1) ∼
= H∗ (BPhn+1i) .
Lose τ̄n+1 to kill στ̄n+1 .
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Chromatic Redshift
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vn+1 -Periodic Output
1
1
H∗c (THH(B)tS ) algebra over H∗c (THH(BPhni)tS ), typically with
associated graded
P(t ±p
n+1
) ⊗ H∗ (BPhn+1i) ⊗ E(ν1 , . . . , νn+1 ) .
Limit of Adams spectral sequences
1
s,t
(Fp , H∗c (THH(B)tS )) =⇒ πt−s THH(B)tS
E2s,t (B) = ExtA
∗
1
algebra over E2∗,∗ (BPhni), containing factors like
∗,∗
ExtA
(Fp , H∗ (BPhn+1i)) = P(v0 , . . . , vn , vn+1 ) .
∗
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Chromatic Redshift
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State of Affairs
Room for differentials due to exterior factor E(ν1 , . . . , νn+1 ).
Might truncate the periodic vn+1 -action.
Empirically this does not happen.
A general explanation is currently lacking.
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Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
42 / 54
Hyperelliptic Cohomology
Topological modular forms tmf .
π∗ (tmf ) is v2 -periodic.
1
Do π∗ K (tmf ) and π∗ THH(tmf )tS detect v3 -periodic families in
π∗ S?
Work in progress for p = 2 with Bruner ’08.
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
44 / 54
Chromatic and Telescopic Localizations
EO n
KUp
O
Z×
p
Gn
Jp
L̂nO S
O
HQ
O
'
S(p)
/ ...
/ Lf S
n
/ Lf
n−1 S
/ ...
/ Lf S
1
/ Lf S
0
'
Ravenel’s Telescope Conjecture ’84: Lfn S −→ Ln S ?
True for n ≤ 1.
Mahowald–Ravenel–Shick ’01: Probably false for n ≥ 2.
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Chromatic Redshift
MSRI/January 2014
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K-Theory Localization Tower
Theorem (Waldhausen ’84, trading Ln for Lfn )
Tower of fiber sequences for n ≥ 1
K (S(p) )
/ ...
K (Cn , wn )
...
/ K (Lf S)
n
/ K (Lf S)
n−1
/ ...
/ K (Q)
Cn category of finite spectra of type ≥ n
wn subcategory of E(n)-equivalences
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Chromatic Redshift
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Monochromatic K-Theory
Knsm category of small K (n)-local spectra
Endf category of En -module spectra with each πi finite
Proposition (Hovey–Strickland ’99)
(Cn , wn ) → (Knsm , h) is an idempotent completion.
πi K (Cn , wn ) ∼
= πi K (Knsm )
for i > 0.
Base change along L̂n S → En takes Knsm to Endf .
Conjecture (à la wishful thinking)
K (Knsm ) −→ K (Endf )hGn
close to an equivalence.
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Broken Dévissage
Express K (Endf ) using K of En and localizations.
n = 1 for simplicity.
Barwick: Fiber sequence
K (E1df ) −→ K (KUp ) −→ K (KUp [1/p]) .
Ausoni–R. ’12: Transfer map
K (KU/p) −→ K (E1df )
is far from an equivalence.
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Chromatic Redshift
MSRI/January 2014
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Congregation
K (Endf )
O
/ K (En )
Gn
K (Cn , wn )
/ K (K sm )
n
...
&
/ K (Lf S)
n
/ K (Lf S)
n−1
/ ...
K (En ) governs change in K -theory along Lfn S → Lfn−1 S.
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Chromatic Redshift
MSRI/January 2014
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
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K of the Sphere Spectrum
Waldhausen et al.: K (S) geometrically important.
R. ’02: Compute A -module H ∗ (K (S)) for p = 2.
R. ’03: Compute A -module H ∗ (K (S)) for p regular, up to an
extension.
Gives π∗ K (S)p in a range.
As complicated as π∗ S.
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K of Complex Bordism
S → MU → H halfway house.
S is totalization of cosimplicial spectrum
[q] 7→ MU ∧ MU ∧q .
Is K (S) close to totalization of
[q] 7→ K (MU ∧ MU ∧q ) ?
Seek conceptual understanding of K (S) by Hopf–Galois descent
from K (MU).
Pursued by Bruner–R. ’05, R. ’08, R. ’09 and
Lunøe-Nielsen–R. ’11.
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Outline
1
K-Theoretical Redshift
Iteration, Height and Nesting
Redshift in Algebraic K-Theory
Spectral Lichtenbaum–Quillen Conjectures
2
Topological Cyclic Redshift
The Cyclotomic Trace Map
Circle-Equivariant Redshift
Beyond Elliptic Cohomology
3
Variations
Waldhausen’s Localization Tower
Infinite Complexity
Higher Redshift
John Rognes (UiO)
Chromatic Redshift
MSRI/January 2014
53 / 54
Higher Redshift
B commutative S-algebra, G Lie group of rank k .
Study
B 7−→ (B ⊗ G)hG
or Tate-like construction.
Expect shift from vn - to vn+k -periodicity.
Carlsson–Dundas et al. ’10, ’11: G = T k .
R. ’11: General G.
R. ’08, Torleif Veen ’13: Partial verifications.
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