algebraic k-theory of strict ring spectrafolk.uio.no/rognes/papers/seoul.pdf · the algebraic...

Algebraic K -Theory and Automorphisms of ManifoldsTopological Cyclic Homology and p-Complete Calculations

Logarithmic Ring Spectra and Localization Sequences

Algebraic K -Theory of Strict Ring Spectra

John Rognes

University of Oslo, Norway

Seoul ICM 2014

John Rognes Algebraic K -Theory of Strict Ring Spectra



Outline

1 Algebraic K -Theory and Automorphisms of Manifolds

2 Topological Cyclic Homology and p-Complete Calculations

3 Logarithmic Ring Spectra and Localization Sequences




Symmetric Spectra (Smith)

A spectrum is a sequence of based spaces

X0,X1,X2, . . .

and maps σ : Xn ∧ S1 → Xn+1, for n ≥ 0.A symmetric spectrum is a spectrum equipped with aΣn-action on each Xn, such that

σk : Xn ∧ Sk → Xn+k

is Σn × Σk -equivariant for each n, k ≥ 0.




Symmetric Ring Spectra

The category SpΣ of symmetric spectra is closedsymmetric monoidal, with unit the sphere spectrum S andmonoidal pairing the smash product X ∧ Y .Its localization Ho(SpΣ) with respect to the stableequivalences is Boardman’s stable homotopy category.A symmetric ring spectrum is a symmetric spectrum A withassociative and unital structure maps µ : A ∧ A→ A andη : S → A.




Algebraic K -Theory of Symmetric Ring Spectra

Mandell defined K (A) as the algebraic K -theory of acategory CA of finite cell A-modules.The algebraic K -theory spectrum K (A) exhibits a groupcompletion

|hCA| → Ω∞K (A)

of the left hand classifying space, turning cofibersequences into sums.




Algebraic K -Theory of Spaces

Let X ' BG be a space, with loop group G ' ΩX .Let S[G] be the spherical group ring spectrum.Waldhausen first defined

A(X ) = K (S[G])

as the algebraic K -theory of an unstable model for thecategory of finite cell S[G]-modules, the category ofretractive spaces over X .




h-Cobordism Spaces

If X is a compact smooth manifold, let H(X ) be the spaceof h-cobordisms (W ; X ,Y ) with X at one end:

∂W = X ∪ Y , X '→W '← Y

Let H (X ) = colimk H(X × [0,1]k ) be the stableh-cobordism space.

Theorem (Igusa)

H(X )→H (X ) is about n/3-connected, for n = dim X.




The Stable Parametrized h-Cobordism Theorem

A(X ) = K (S[G]) splits as

A(X ) ' S[X ] ∨Wh(X ) ,

defining the Whitehead spectrum.Let ΩWh(X ) = Ω∞+1Wh(X ) be the looped Whiteheadspace.

Theorem (Waldhausen–Jahren–R.)

There is a natural homotopy equivalence H (X ) ' ΩWh(X ).




Diffeomorphism Groups: Rational

When X is contractible, A(∗) = K (S) ' S ∨Wh(∗).

Theorem (Borel)

Ki(S)⊗Q ∼= Ki(Z)⊗Q ∼=

Q for i = 0 or 4k + 1 6= 1,0 otherwise.

Example (Farrell–Hsiang)

πiDiff (Dn rel ∂Dn)⊗Q ∼=

Q for i = 4k − 1, n odd,0 otherwise,

for i up to about n/3.




Outline







Topological Cyclic Homology

Bökstedt–Hsiang–Madsen constructed a naturalcyclotomic trace map

K (A)→ TC(A; p)

to the topological cyclic homology of A.It is a homotopy limit

TC(A; p) = holimn,R,F

THH(A)Cpn

of cyclic fixed points of the topological Hochschildhomology of A.




Nilpotent extensions

An integral version satisfies TC(A)∧p ' TC(A; p)∧p .

Theorem (Dundas–Goodwillie–McCarthy)Let A→ B be a map of connective symmetric ring spectra, withπ0(A)→ π0(B) surjective with nilpotent kernel. The square

K (A) //

K (B)

TC(A) // TC(B)

is homotopy Cartesian.




The Sphere Spectrum and the Integers

ExampleHomotopy Cartesian square

K (S)∧p //

K (Z)∧p

TC(S; p)∧p // TC(Z; p)∧p .

R. used this to calculate H∗ and π∗ of

K (S)∧p ' S∧p ∨Wh(∗)∧p

for regular primes p.




K -Theory of the Sphere Spectrum: Cohomology

Let A be the mod p Steenrod algebra.For p = 2 let C ⊂ A be generated by admissible SqI

where I = (i1, . . . , in) with n ≥ 2 or I = (i) with i odd.

Theorem (R.)

The mod 2 cohomology of Wh(∗) is the nontrivial extension

Σ−2C/A (Sq1,Sq3)→ H∗Wh(∗)→ Σ3A /A (Sq1,Sq2)

of A -modules.




K -Theory of the Sphere Spectrum: Homotopy

Example (R.)

The homotopy groups of Wh(∗), modulo p-torsion for irregularprimes p, begin:

i 0 1 2 3 4 5 6 7 8 9πiWh(∗) 0 0 0 Z/2 0 Z 0 Z/2 0 Z⊕ Z/2

i 10 11 12 13 14πiWh(∗) Z/8⊕ (Z/2)2 Z/6 Z/4 Z Z/36⊕ Z/3

i 15 16 17 18πiWh(∗) (Z/2)2 Z/24⊕ Z/2 Z⊕ (Z/2)2 Z/480⊕ (Z/2)3




Localization and Descent for Algebraic K -Theory

Seek a conceptual understanding of these calculationalresults on K (A)p for A = S.Can we recover K (A)p from K (B)p for suitably localsymmetric ring spectra B?Can we descend to K (B)p from K (C)p for appropriateextensions B → C?Is there a simple description of K (Ω)p for sufficiently largesuch extensions B → Ω?




Algebraic K -Theory of Topological K -Theory

Adams summand A = `p of kup, with π∗`p = Zp[v1].

Localization B = Lp, with π∗Lp = Zp[v±11 ].

Lp // KUp

Sp // `p

OO

φ // kup

OO

// HZp

Theorem (Blumberg-Mandell)

Homotopy cofiber sequence

K (`p)→ K (Lp)→ ΣK (Zp) .




Chromatic Redshift

For p ≥ 5, the type 2 Smith–Toda complex

V (1) = S ∪p e1 ∪α1 e2p−1 ∪p e2p

is a ring spectrum up to homotopy, with v2 ∈ π2p2−2V (1).

Theorem (Ausoni–R.)

V (1)∗K (`p) and V (1)∗K (Lp)

are finitely generated free Fp[v2]-modules, each on 4p + 4generators, up to small error terms.




Lichtenbaum–Quillen Conjecture

Suggests that K (Ω)p is a connective form of the Lubin–Tatespectrum E2, with π∗E2 = WFp2 [[u1]][u±1] andV (1)∗E2 = Fp2 [u±1].

Conjecture (R.)For purely v1-periodic commutative symmetric ring spectra Bthere is a spectral sequence

E2s,t = H−s

mot(B;Fp2(t/2)) =⇒ V (1)s+tK (B) .




E2-Term for V (1)∗K (Lp), p = 5

••

2p2

•

•

•

|

•

•

|

•

•

|

•

•••

••

2p

•

•

•

•

•

•••

•

0

−3

−2

−1

0

s/t




Beilinson–Lichtenbaum Conjecture

SetH r

et(Lp;Fp2(∗)) = v−12 H r

mot(Lp;Fp2(∗)) .

Observe motivic truncation:

H rmot(Lp;Fp2(m)) ∼=

H r

et(Lp;Fp2(m)) for 0 ≤ r ≤ m,0 otherwise.




Tate–Poitou Duality

Symmetry about (s, t) = (−3/2,p + 1) similar to arithmeticduality.

Conjecture (R.)For finite extensions B of Lp there is a perfect pairing

H ret(B;Fp2(m))⊗ H3−r

et (B;Fp2(p+1−m))

∪→ H3et(B;Fp2(p+1)) ∼= Z/p

for each r and m.




Outline







Logarithmic Geometry

Seek to realize more of motivic cohomology as Galoiscohomology.Difficult to classify/construct ramified extensions B → C byobstruction theory.Tamely ramified extensions behave as unramified whenrigidified by logarithmic structures.




Logarithmic Rings (Fontaine–Illusie, Kato)

A pre-log ring consists ofa commutative ring R;a commutative monoid M;a monoid homomorphism α : M → (R, ·).

Log ring if α−1GL1(R)→ GL1(R) is an isomorphism.Trivial log structure on R has M = GL1(R).Localization R → R[M−1] factors in log rings as

R → (R,M)→ R[M−1] .




J -spaces (R., Sagave, Schlichtkrull)

The “underlying graded space” of a symmetric spectrum Ais a J -shaped diagram of spaces

ΩJ (A) : (n1,n2) 7→ Ωn2An1

Indexing category J is isomorphic to Quillen’s constructionΣ−1Σ, with BJ ' QS0.Homotopy type of a J -space X : J → S is detected byXhJ = hocolimJ X . Positive projective model structure.Convolution product X Y maps to smash product underSJ [−] : SJ → SpΣ, Quillen adjoint to ΩJ (−) : SpΣ → SJ .




Topological logarithmic structures

DefinitionA pre-log ring spectrum consists of

a commutative symmetric ring spectrum A;a commutative J -space monoid M;a commutative J -space monoid map α : M → ΩJ (A).

Log ring spectrum if α−1GLJ1 (A)→ GLJ1 (A) isJ -equivalence.Trivial log structure on A has M = GLJ1 (A) ⊂ ΩJ (A).Localization A→ A[M−1] = A ∧SJ [M] SJ [Mgp] factors as

A→ (A,M)→ A[M−1] .




The replete bar construction

The group completion η : M → Mgp makes (Mgp)hJ agroup completion of the E∞ space MhJ .The cyclic bar construction Bcy(M) is the usual simplicialobject [q] 7→ M M · · · M.The replete bar construction is a homotopy pullback

Bcy(M)ρ //

Brep(M) //

Bcy(Mgp)

M = // M // Mgp

Repletion in topology plays the role of working with fineand saturated log structures in algebra.




Logarithmic Topological Hochschild Homology

DefinitionLog THH of a pre-log ring spectrum (A,M, α) is the pushout

SJ [Bcy(M)]ρ //

SJ [Brep(M)]

THH(A)

ρ // THH(A,M)

of cyclic commutative symmetric ring spectra.




Log Étale Extensions

(A,M)→ (B,N) is formally log étale ifB ∧A THH(A,M) ' THH(B,N).The direct image log structure of (B,N) along j : A→ B isj∗N = ΩJ (A)×ΩJ (B) N.

Theorem (R.–Sagave–Schlichtkrull)

φ : (`p, j∗GLJ1 (Lp))→ (kup, j∗GLJ1 (KUp))

is log étale.




Localization Sequences

Theorem (R.–Sagave–Schlichtkrull)Let E be a d-periodic commutative symmetric ring spectrum,with connective cover j : e→ E. Homotopy cofiber sequence

THH(e)ρ→ THH(e, j∗GLJ1 (E))

∂→ ΣTHH(e[0,d))

where e[0,d) is the (d − 1)-th Postnikov section of e.

ExampleHomotopy cofiber sequence

THH(`p)→ THH(`p, j∗GLJ1 (Lp))→ ΣTHH(Zp) .




Future Work

Develop log TC, with a cyclotomic trace map from logK -theory, related to K (A[M−1]).Develop log obstruction theory to realize tamely ramifiedextensions A→ B as part of log étale extensions(A,M)→ (B,N).


algebraic k-theory of strict ring spectrafolk.uio.no/rognes/papers/seoul.pdf · the algebraic...

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