The COE workshop on algebraic K-theory, topological Hochschild homology and de Rham-Witt complex

Jan. 19 (Wed.)

11:00-12:00 Hesselholt Introduction
13:00-13:50 Hagihara Cyclic sets and Hochschild homology
14:00-15:00 Hagihara Stable homotopy
15:30-16:20 Hagihara Equivariant stable homotopy

Jan. 20 (Thu.)

13:00-15:00 Hagihara Topological Hochschild homology
15:30-16:30 Hesselholt Topological Hochschild homology of a finite field
16:40-17:40 Hagihara The Tate spectrum and its application

Jan. 21 (Fri.)

10:30-12:00 Hagihara The de Rham-Witt complex in mixed characteristic
13:00-15:00 Hesselholt The structure of T R for a complete discrete

valuation ring of mixed characteristic

Jan. 19: Room 122 Jan. 20, 21: Room 128

Introduction

This series of talks is devoted to topological Hochschild homology and its relation to algebraic K-theory. One associates with every ring A an object T (A) of the triangulated category of 𝕋-spectra and defines

TR_{q}^{n}(A; p) = Hom(𝕋/C_{p^{n−1}+}, T (A)[q]).

These groups form, as n and q vary, an algebraic structure called a Witt complex of which the de Rham-Witt complex, by definition, is the initial example. We will show that the canonical map W_{n}Ω_{A}^{q} → TR_{q}^{n}(A; p) behaves similarly to the canonical map from Milnor K-theory to algebraic K-theory. The TR-groups and algebraic K-groups are related as follows: There is a long-exact sequence

⋯ → TC_{q}^{n}(A; p) → TR_{q}^{n}(A; p) ⟶^{R−F} TR_{q}^{n−1}(A; p) ⟶^{∂} TC_{q−1}^{n}(A; p) → ⋯

and a map of pro-abelian groups called the cyclotomic trace

tr: K_{q}(A) → {TC_{q}^{n}(A; p)}_{n≥1}.

In many cases, including local number rings, this map induces an isomorphism of the corresponding pro-abelian groups with ℤ/p^{v}ℤ-coefficients.

Cyclic sets and Hochschild homology

First we define a cyclic set and investigate its relation to 𝕋-topological spaces, where 𝕋 denotes the circle group. As an important example, we define a Hochschild space H(A) for a ring A, whose homotopy groups HH_{∗}(A) are Hochschild homology groups.

(Equivariant) stable homotopy

In this talk, using the concept of (G-)spectra, we construct the stable homotopy category and its variant, the G-equivariant stable homotopy category, for a compact Lie group G (The case of G = 𝕋 is actually used). They are necessary for the definition and the calculation of a topological Hochschild homology spectrum T (A) defined in the next talk.

Topological Hochschild homology

Now we define a 𝕋-spectrum T (A) for a ring A, a topological variant of H(A), and examine its equivariant homotopy groups T R_{∗}^{·}(A; p), in particular, the topological Hochschild homology groups T HH_{∗}(A). In addition, we define the operations V, F, R and d on them.

Topological Hochschild homology of a finite field

We evaluate the topological Hochschild groups THH_{q}(𝔽_{p}) = TR_{q}^{1}(𝔽_{p}; p).

The Tate spectrum and its application

We introduce a Tate spectrum and observe its fundamental properties. And, by using this, we calculate T R_{∗}^{·}(𝔽_{p}; p).

The de Rham-Witt complex in mixed characteristic

Here we give the definition and the properties of de Rham-Witt complex W_{·}Ω_{A}^{∗} for a ℤ_{(p)}-algebra A. And we prove the comparison theorem between W_{·}Ω_{A}^{∗} and T R_{∗}^{·}(A; p) for a regular 𝔽_{p}-algebra A.

The structure of TR for a complete discrete valuation ring of mixed characteristic

Let V be a complete discrete valuation ring with quotient field K of characteristic 0 and residue field k of odd characteristic p. We construct a map

\xymatrix{⋯ → K_{q}(k) \ar[r]^{i_{∗}} \ar[d] & K_{q}(V) \ar[r]^{j^{∗}} \ar[d] & K_{q}(K) \ar[r]^{∂} \ar[d] & K_{q−1}(k) → ⋯ \ar[d] \\ ⋯ → TR_{q}^{n}(k; p) \ar[r]^{i_{∗}} & TR_{q}^{n}(V; p) \ar[r]^{j^{∗}} & TR_{q}^{n}(V |K; p) \ar[r]^{δ} & TR_{q−1}^{n}(k; p) → ⋯}

from the localization sequence in algebraic K-theory to a similar sequence of TR-groups. The groups TR_{q}^{n}(V |K; p) form a Witt complex over the log-ring (V, M), where α: M = V ∩ K^{∗} → V is the canonical log-structure, and the canonical map

W_{n}Ω_{(V,M)}^{q} → TR_{q}^{n}(V |K; p)

is an isomorphism, for q ≤ 2. If K contains the p^{v}th roots of unity, then the map

W_{n}Ω_{(V,M)}^{∗} ⊗ S_{ℤ/p^{v}ℤ}(μ_{p^{v}}) → TR_{∗}^{n}(V |K; p, ℤ/p^{v}ℤ)

that to ζ associates the image by the cyclotomic trace map of the associated Bott element b_{ζ} ∈ K_{2}(K, ℤ/p^{v}ℤ).