Algebraic K -Theory and Automorphisms of Manifolds Topological 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 Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Outline 1 Algebraic K -Theory and Automorphisms of Manifolds 2 Topological Cyclic Homology and p-Complete Calculations 3 Logarithmic Ring Spectra and Localization Sequences John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Outline 1 Algebraic K -Theory and Automorphisms of Manifolds 2 Topological Cyclic Homology and p-Complete Calculations 3 Logarithmic Ring Spectra and Localization Sequences John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Symmetric Spectra (Smith) A spectrum is a sequence of based spaces X0 , X1 , X2 , . . . and maps σ : Xn ∧ S 1 → Xn+1 , for n ≥ 0. A symmetric spectrum is a spectrum equipped with a Σn -action on each Xn , such that σ k : Xn ∧ S k → Xn+k is Σn × Σk -equivariant for each n, k ≥ 0. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Symmetric Ring Spectra The category SpΣ of symmetric spectra is closed symmetric monoidal, with unit the sphere spectrum S and monoidal pairing the smash product X ∧ Y . Its localization Ho(SpΣ ) with respect to the stable equivalences is Boardman’s stable homotopy category. A symmetric ring spectrum is a symmetric spectrum A with associative and unital structure maps µ : A ∧ A → A and η : S → A. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Algebraic K -Theory of Symmetric Ring Spectra Mandell defined K (A) as the algebraic K -theory of a category CA of finite cell A-modules. The algebraic K -theory spectrum K (A) exhibits a group completion |hCA | → Ω∞ K (A) of the left hand classifying space, turning cofiber sequences into sums. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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 the category of finite cell S[G]-modules, the category of retractive spaces over X . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences h-Cobordism Spaces If X is a compact smooth manifold, let H(X ) be the space of 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 stable h-cobordism space. Theorem (Igusa) H(X ) → H (X ) is about n/3-connected, for n = dim X . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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∞+1 Wh(X ) be the looped Whitehead space. Theorem (Waldhausen–Jahren–R.) There is a natural homotopy equivalence H (X ) ' ΩWh(X ). John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Diffeomorphism Groups: Rational When X is contractible, A(∗) = K (S) ' S ∨ Wh(∗). Theorem (Borel) ( Q for i = 0 or 4k + 1 6= 1, Ki (S) ⊗ Q ∼ = Ki (Z) ⊗ Q ∼ = 0 otherwise. Example (Farrell–Hsiang) ( Q for i = 4k − 1, n odd, πi Diff (D rel ∂D ) ⊗ Q ∼ = 0 otherwise, n n for i up to about n/3. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Outline 1 Algebraic K -Theory and Automorphisms of Manifolds 2 Topological Cyclic Homology and p-Complete Calculations 3 Logarithmic Ring Spectra and Localization Sequences John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Topological Cyclic Homology Bökstedt–Hsiang–Madsen constructed a natural cyclotomic trace map K (A) → TC(A; p) to the topological cyclic homology of A. It is a homotopy limit TC(A; p) = holim THH(A)Cpn n,R,F of cyclic fixed points of the topological Hochschild homology of A. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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, with π0 (A) → π0 (B) surjective with nilpotent kernel. The square K (A) / K (B) / TC(B) TC(A) is homotopy Cartesian. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences The Sphere Spectrum and the Integers Example Homotopy Cartesian square K (S)∧ p / K (Z)∧ p / TC(Z; p)∧ . TC(S; p)∧ p p R. used this to calculate H∗ and π∗ of ∧ ∧ K (S)∧ p ' Sp ∨ Wh(∗)p for regular primes p. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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 Sq I 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−2 C/A (Sq 1 , Sq 3 ) → H ∗ Wh(∗) → Σ3 A /A (Sq 1 , Sq 2 ) of A -modules. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences K -Theory of the Sphere Spectrum: Homotopy Example (R.) The homotopy groups of Wh(∗), modulo p-torsion for irregular primes p, begin: i πi Wh(∗) i πi Wh(∗) i πi Wh(∗) 0 0 1 0 2 0 3 Z/2 10 Z/8 ⊕ (Z/2)2 15 (Z/2)2 4 0 11 Z/6 16 Z/24 ⊕ Z/2 John Rognes 5 Z 6 0 12 Z/4 7 Z/2 13 Z 17 Z ⊕ (Z/2)2 8 0 9 Z ⊕ Z/2 14 Z/36 ⊕ Z/3 18 Z/480 ⊕ (Z/2)3 Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Localization and Descent for Algebraic K -Theory Seek a conceptual understanding of these calculational results on K (A)p for A = S. Can we recover K (A)p from K (B)p for suitably local symmetric ring spectra B? Can we descend to K (B)p from K (C)p for appropriate extensions B → C? Is there a simple description of K (Ω)p for sufficiently large such extensions B → Ω? John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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 [v1±1 ]. / KUp O Lp O Sp / `p φ / kup / HZp Theorem (Blumberg-Mandell) Homotopy cofiber sequence K (`p ) → K (Lp ) → ΣK (Zp ) . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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 −2 V (1). Theorem (Ausoni–R.) V (1)∗ K (`p ) and V (1)∗ K (Lp ) are finitely generated free Fp [v2 ]-modules, each on 4p + 4 generators, up to small error terms. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Lichtenbaum–Quillen Conjecture Suggests that K (Ω)p is a connective form of the Lubin–Tate spectrum E2 , with π∗ E2 = WFp2 [[u1 ]][u ±1 ] and V (1)∗ E2 = Fp2 [u ±1 ]. Conjecture (R.) For purely v1 -periodic commutative symmetric ring spectra B there is a spectral sequence −s 2 Es,t = Hmot (B; Fp2 (t/2)) =⇒ V (1)s+t K (B) . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences E 2 -Term for V (1)∗ K (Lp ), p = 5 −3 • −2 • • • −1 0 • s/t 0 • • • • • • • • • • • • • • • • 2p | | | John Rognes Algebraic K -Theory of Strict Ring Spectra • • • 2p2 Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Beilinson–Lichtenbaum Conjecture Set r r (Lp ; Fp2 (∗)) . Het (Lp ; Fp2 (∗)) = v2−1 Hmot Observe motivic truncation: ( r (L ; F (m)) for 0 ≤ r ≤ m, Het p p2 r Hmot (Lp ; Fp2 (m)) ∼ = 0 otherwise. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Tate–Poitou Duality Symmetry about (s, t) = (−3/2, p + 1) similar to arithmetic duality. Conjecture (R.) For finite extensions B of Lp there is a perfect pairing 3−r r Het (B; Fp2 (m)) ⊗ Het (B; Fp2 (p+1−m)) ∪ 3 → Het (B; Fp2 (p+1)) ∼ = Z/p for each r and m. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Outline 1 Algebraic K -Theory and Automorphisms of Manifolds 2 Topological Cyclic Homology and p-Complete Calculations 3 Logarithmic Ring Spectra and Localization Sequences John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Logarithmic Geometry Seek to realize more of motivic cohomology as Galois cohomology. Difficult to classify/construct ramified extensions B → C by obstruction theory. Tamely ramified extensions behave as unramified when rigidified by logarithmic structures. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Logarithmic Rings (Fontaine–Illusie, Kato) A pre-log ring consists of a commutative ring R; a commutative monoid M; a monoid homomorphism α : M → (R, ·). Log ring if α−1 GL1 (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 ] . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences J -spaces (R., Sagave, Schlichtkrull) The “underlying graded space” of a symmetric spectrum A is a J -shaped diagram of spaces ΩJ (A) : (n1 , n2 ) 7→ Ωn2 An1 Indexing category J is isomorphic to Quillen’s construction Σ−1 Σ, with BJ ' QS 0 . Homotopy type of a J -space X : J → S is detected by XhJ = hocolimJ X . Positive projective model structure. Convolution product X Y maps to smash product under S J [−] : S J → SpΣ , Quillen adjoint to ΩJ (−) : SpΣ → S J . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Topological logarithmic structures Definition A 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). J Log ring spectrum if α−1 GLJ 1 (A) → GL1 (A) is J -equivalence. J Trivial log structure on A has M = GLJ 1 (A) ⊂ Ω (A). Localization A → A[M −1 ] = A ∧S J [M] S J [M gp ] factors as A → (A, M) → A[M −1 ] . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences The replete bar construction The group completion η : M → M gp makes (M gp )hJ a group completion of the E∞ space MhJ . The cyclic bar construction B cy (M) is the usual simplicial object [q] 7→ M M · · · M. The replete bar construction is a homotopy pullback B cy (M) M ρ / B rep (M) / B cy (M gp ) /M / M gp = Repletion in topology plays the role of working with fine and saturated log structures in algebra. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Logarithmic Topological Hochschild Homology Definition Log THH of a pre-log ring spectrum (A, M, α) is the pushout S J [B cy (M)] THH(A) ρ ρ / S J [B rep (M)] / THH(A, M) of cyclic commutative symmetric ring spectra. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Log Étale Extensions (A, M) → (B, N) is formally log étale if B ∧A THH(A, M) ' THH(B, N). The direct image log structure of (B, N) along j : A → B is j∗ N = ΩJ (A) ×ΩJ (B) N. Theorem (R.–Sagave–Schlichtkrull) J φ : (`p , j∗ GLJ 1 (Lp )) → (kup , j∗ GL1 (KUp )) is log étale. John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences 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 ρ ∂ THH(e) → THH(e, j∗ GLJ 1 (E)) → ΣTHH(e[0, d)) where e[0, d) is the (d − 1)-th Postnikov section of e. Example Homotopy cofiber sequence THH(`p ) → THH(`p , j∗ GLJ 1 (Lp )) → ΣTHH(Zp ) . John Rognes Algebraic K -Theory of Strict Ring Spectra Algebraic K -Theory and Automorphisms of Manifolds Topological Cyclic Homology and p-Complete Calculations Logarithmic Ring Spectra and Localization Sequences Future Work Develop log TC, with a cyclotomic trace map from log K -theory, related to K (A[M −1 ]). Develop log obstruction theory to realize tamely ramified extensions A → B as part of log étale extensions (A, M) → (B, N). John Rognes Algebraic K -Theory of Strict Ring Spectra
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