Cubical and cosimplicial descent

TL;DR

This paper proves that key algebraic and topological invariants like K-theory, THH, and TC satisfy descent properties in cubical and cosimplicial contexts for connective structured ring spectra along 1-connected maps.

Contribution

It establishes the descent properties of algebraic K-theory, THH, and TC in cubical and cosimplicial frameworks for connective structured ring spectra.

Findings

K-theory satisfies cubical and cosimplicial descent

THH satisfies cubical and cosimplicial descent

TC satisfies cubical and cosimplicial descent

Abstract

We prove that algebraic K-theory, topological Hochschild homology and topological cyclic homology satisfy cubical and cosimplicial descent at connective structured ring spectra along 1-connected maps of such ring spectra.

Equations

$$\eta :F(A)\stackrel{\simeq }{⟶}\underset{T\in P}{\mathrm{holim}}F(X(T))$$

$$\eta :F(A)\stackrel{\simeq }{⟶}\underset{[q]\in \Delta }{\mathrm{holim}}F(Y^q)$$

$${\eta }_n:F(X(\varnothing ))⟶\underset{T\in P^n}{\mathrm{holim}}F(X(T))=\underset{P^n}{\mathrm{holim}}F(X)$$

$$\eta :F(X(\varnothing ))⟶\underset{T\in P}{\mathrm{holim}}F(X(T))=\underset{P}{\mathrm{holim}}F(X)$$

$$X^n(T)=A{\wedge }_R{X}_{1,T}{\wedge }_R\cdots {\wedge }_R{X}_{n,T},$$

$$THH(A{)}_{h{C}_{p^m}}\stackrel{N}{⟶}THH(A{)}^{C_{p^m}}\stackrel{R}{⟶}THH(A{)}^{C_{p^{m-1}}}$$

$$TC(A;p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\prod _{\text{p prime}}TC(A;p{)}_p⟶\prod _{\text{p prime}}\underset{F,m}{\mathrm{holim}}THH(A{)}_p^{h{C}_{p^m}}⟵THH(A{)}^{h\mathbb{T}},$$

$$THH(A{\wedge }_{R}B{\wedge }_R\cdots {\wedge }_{R}B)$$

$$\cong B^{cy}(A){\wedge }_{B^{cy}(R)}{B}^{cy}(B){\wedge }_{B^{cy}(R)}\cdots {\wedge }_{B^{cy}(R)}{B}^{cy}(B),$$

$$THH(X_R^n(A,B))\simeq X_{B^{cy}(R)}^n(B^{cy}(A),B^{cy}(B))$$

$$\underset{P}{\mathrm{holim}}F(X)\simeq \underset{n}{\mathrm{holim}}\underset{P^n}{\mathrm{holim}}F(X)$$

$$E_1^{s,t}={\pi }_{t-s}\mathrm{hofib}(p_s){⟹}_s{\pi }_{t-s}(\underset{n}{\mathrm{holim}}\underset{P^n}{\mathrm{holim}}F(X))$$

$$\cdots \to \underset{P^{s+1}}{\mathrm{holim}}F(X)\stackrel{p_s}{⟶}\underset{P^s}{\mathrm{holim}}F(X)\to \dots ,$$

$$d_r^{s,t}:E_r^{s,t}⟶E_r^{s+r,t+r-1}.$$

$$E_1^{s,t}={\pi }_{t-s}\mathrm{hofib}(p_s){⟹}_s{\pi }_{t-s}F(A)$$

$${\mathrm{holim}}_{P^{s+1}}K(X^{s+1})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_2^{s,t}={\underset{P}{\lim }}^s{\pi }_{t}F(X){⟹}_s{\pi }_{t-s}\underset{P}{\mathrm{holim}}F(X)$$

$$\eta :F(Y^{-1})⟶\underset{[q]\in \Delta }{\mathrm{holim}}F(Y^q)=\underset{\Delta }{\mathrm{holim}}F(Y^{\bullet })$$

$$f^{\ast }:\underset{\Delta }{\mathrm{holim}}F(Y^{\bullet })\stackrel{\simeq }{⟶}\underset{P}{\mathrm{holim}}(F(Y^{\bullet })\circ f)$$

$$f_n^{\ast }:\underset{{\Delta }^{<n}}{\mathrm{holim}}F(Y_{<n}^{\bullet })\stackrel{\simeq }{⟶}\underset{P^n}{\mathrm{holim}}(F(Y_{<n}^{\bullet })\circ f^n)$$

$$f^{\ast }:{\pi }^s{A}^{\bullet }\cong {\underset{\Delta }{\lim }}^s{A}^{\bullet }\stackrel{\cong }{⟶}{\underset{P}{\lim }}^s(A^{\bullet }\circ f)\text{and}{f}_n^{\ast }:{\underset{{\Delta }^{<n}}{\lim }}^s{A}^{\bullet }\stackrel{\cong }{⟶}{\underset{P^n}{\lim }}^s(A^{\bullet }\circ f^n)$$

$$Y^q=A{\wedge }_{R}B{\wedge }_R\cdots {\wedge }_{R}B,$$

$$d^i=i{d}_A\wedge (i{d}_B{)}^{\wedge i}\wedge \eta \wedge (i{d}_B{)}^{\wedge q-i}:Y^{q-1}⟶Y^q$$

$$s^j=i{d}_A\wedge (i{d}_B{)}^{\wedge j}\wedge \mu \wedge (i{d}_B{)}^{\wedge q-j}:Y^{q+1}⟶Y^q$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$F(A)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\eta :F(A)\stackrel{\simeq }{⟶}\underset{[q]\in \Delta }{\mathrm{holim}}F(Y^q)=\underset{\Delta }{\mathrm{holim}}F(Y^{\bullet })$$

$$A{\wedge }_{R}B{\wedge }_R\cdots {\wedge }_{R}B,$$

$$\underset{\Delta }{\mathrm{holim}}F(Y^{\bullet })\simeq \underset{n}{\mathrm{holim}}\underset{{\Delta }^{<n}}{\mathrm{holim}}F(Y^{\bullet })$$

$$E_1^{s,t}={\pi }_{t-s}\mathrm{hofib}({\delta }_s){⟹}_s{\pi }_{t-s}(\underset{n}{\mathrm{holim}}\underset{{\Delta }^{<n}}{\mathrm{holim}}F(Y^{\bullet }))$$

Full text

1. Introduction

In this paper we extend the techniques used in [Dun97] to prove that algebraic $K$-theory, topological Hochschild homology and topological cyclic homology of connective structured ring spectra all satisfy descent along $1$-connected maps of such ring spectra.

Theorem 1.1** (Cubical descent).**

Let $R$ be a connective commutative $\mathbb{S}$-algebra and let $A$ and $B$ be connective $R$-algebras. Suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Then the functors $F=K$, ${THH}$ and ${TC}$ satisfy cubical descent at $A$ along $R\to B$, in the sense that in each case the natural map

[TABLE]

is an equivalence of spectra. Here $P$ denotes the partially ordered set of nonempty finite subsets $T=\{t_{0}<\dots<t_{q}\}$ of $\mathbb{N}$, and $X(T)\cong A\wedge_{R}B\wedge_{R}\dots\wedge_{R}B$, with $(q+1)$ copies of $B$.

When the cubical diagram $T\mapsto X(T)$ arises from a cosimplicial spectrum $[q]\mapsto Y^{q}$, the homotopy limit over $P$ can be replaced by a homotopy limit over the category $\Delta$ of nonempty finite totally ordered sets $[q]=\{0<\dots<q\}$. This happens, for instance, when the $R$-algebra $B$ is commutative.

Theorem 1.2** (Cosimplicial descent).**

Let $R$ be a connective commutative $\mathbb{S}$-algebra, let $A$ be a connective $R$-algebra, and let $B$ be a connective commutative $R$-algebra. Suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Then the functors $F=K$, ${THH}$ and ${TC}$ satisfy cosimplicial descent at $A$ along $R\to B$, meaning that in each case the natural map

[TABLE]

is an equivalence of spectra. Here $Y^{q}=A\wedge_{R}B\wedge_{R}\dots\wedge_{R}B$, with $(q+1)$ copies of $B$.

These results will be proved in Theorems 2.4, 2.5 and 3.7. They apply, in particular, at any connective $\mathbb{S}$-algebra $A$ along the unit map $\eta\colon\mathbb{S}\to MU$ for complex bordism. In the case of a group $\mathbb{S}$-algebra $A=\mathbb{S}[\Gamma]$, this is relevant for Waldhausen’s algebraic $K$-theory $A(X)\simeq K(\mathbb{S}[\Gamma])$ of the space $X=B\Gamma$. In the final section we discuss a program to analyze $K(\mathbb{S}[\Gamma])$ and ${TC}(\mathbb{S}[\Gamma])$ in terms of $K(Y^{\bullet})$ and ${TC}(Y^{\bullet})$ for $Y^{q}=\mathbb{S}[\Gamma]\wedge MU\wedge\dots\wedge MU$, with $(q+1)$ copies of $MU$.

2. Cubical descent

2.1. Cubical diagrams

We use terminology similar to that in [Goo91]*§1, including the notions of $k$-Cartesian and $k$-co-Cartesian cubes. For each integer $n\geq 1$, let $P^{n}_{\eta}$ be the set of subsets $T\subseteq\{1,\dots,n\}$, partially ordered by inclusion, and let $P^{n}\subset P^{n}_{\eta}$ be the partially ordered subset consisting of the nonempty such $T$. A functor $X\colon P^{n}_{\eta}\to\mathscr{C}$ from $P^{n}_{\eta}$ to any category $\mathscr{C}$ is called an $n$-dimensional cube, or an $n$-cube, in that category. The restriction of $X$ to $P^{n}$ is the subdiagram $X|P^{n}$ obtained by omitting the initial vertex $X(\varnothing)$ of the $n$-cube. Given any functor $F$ from $\mathscr{C}$ to spectra, the composite functor $F\circ X$ is an $n$-cube of spectra, which we also denote as $F(X)$. There is a natural map

[TABLE]

from the initial vertex of $F(X)$ to the homotopy limit [BK72]*Ch. XI of the remaining part of the $n$-cube. We simply write $F(X)$ in place of $F(X|P^{n})$ when it is clear that the restriction over $P^{n}\subset P^{n}_{\eta}$ is intended. When forming the homotopy limit of a diagram of spectra we implicitly assume that each vertex has been functorially replaced by a fibrant spectrum, and dually for homotopy colimits. This requires that $F$ takes values in a model category of spectra, such as that of [BF78] or one of those discussed in [MMSS01], with homotopy category equivalent to the stable homotopy category. The $n$-cube $F(X)$ is $k$-Cartesian if and only if $\eta_{n}$ is a $k$-connected map. This is equivalent to the $n$-cube being $(n+k-1)$-co-Cartesian, since the iterated homotopy cofiber of an $n$-cube of spectra is equivalent to the $n$-fold suspension of its iterated homotopy fiber. Consider also the partially ordered set $P_{\eta}$ of finite subsets $T$ of $\mathbb{N}=\{1,2,3,\dots\}$, and let $P\subset P_{\eta}$ be the partially ordered subset of nonempty such $T$. A functor $X$ from $P_{\eta}$ is an infinite-dimensional cube, or $\omega$-cube.

Definition 2.1**.**

There is a natural map

[TABLE]

and a natural equivalence $\operatorname*{holim}_{P}F(X)\simeq\operatorname*{holim}_{n}\,\operatorname*{holim}_{P^{n}}F(X)$ that connects $\eta$ to $\operatorname*{holim}_{n}\eta_{n}$. We say that $F$ satisfies cubical descent over $X$ if $\eta$ is an equivalence of spectra.

For example, if the connectivity of $\eta_{n}$ grows to infinity with $n$, then $\eta$ is an equivalence and $F$ satisfies cubical descent over $X$. Cubical descent for $F$ over $X$ ensures that the homotopy type of the spectrum $F(X(\varnothing))$ is essentially determined by the homotopy types of the spectra $F(X(T))$ for nonempty finite subsets $T\subset\mathbb{N}$.

2.2. Amitsur cubes

Let $R$ be a connective commutative $\mathbb{S}$-algebra, where $\mathbb{S}$ denotes the sphere spectrum. First, let $A$ and $B$ be connective $R$-modules, and let $\eta\colon R\to B$ be a map of $R$-modules. We can and will assume that $A$ and $B$ are flat, i.e., $R$-cofibrant as $R$-modules in the sense of [Shi04]*Thm. 2.6(1). Let $n\geq 1$, and consider the $n$-cube $X^{n}=X^{n}_{R}(A,B)\colon T\mapsto X^{n}(T)$ of spectra given by

[TABLE]

where $X_{i,T}=B$ for $i\in T$ and $X_{i,T}=R$ for $i\notin T$. For each inclusion $T^{\prime}\subseteq T$ among subsets of $\{1,\dots,n\}$, the map $X^{n}(T^{\prime})\to X^{n}(T)$ is the smash product over $R$ of $id_{A}$ with a copy of $id_{B}$ for each $i\in T^{\prime}$, a copy of $\eta\colon R\to B$ for each $i\in T\setminus T^{\prime}$, and a copy of $id_{R}$ for each $i\notin T$. Letting $n$ vary, these definitions assemble to specify an $\omega$-cube $X^{\omega}$ in spectra, whose restriction over $P^{n}_{\eta}\subset P_{\eta}$ is the $n$-cube $X^{n}$. These constructions are homotopy invariant, because of the assumption that $A$ and $B$ are flat as $R$-modules.

Lemma 2.2**.**

Suppose that $\eta\colon R\to B$ is $1$-connected. Then each $d$-dimensional subcube of the $n$-cube $X^{n}=X^{n}_{R}(A,B)$ is $d$-Cartesian and $(2d-1)$-co-Cartesian, for every $0\leq d\leq n$.

Proof.

Let $B/R$ denote the $1$-connected homotopy cofiber of $\eta\colon R\to B$. The iterated homotopy cofiber of any $d$-dimensional subcube of $X^{n}$ is equivalent to the smash product over $R$ of $A$ with $d$ copies of $B/R$ and $(n-d)$ copies of $R$ or $B$. Hence it is at least $(2d-1)$-connected. Thus the $d$-dimensional subcube is $(2d-1)$-co-Cartesian, which, as we noted earlier, is equivalent to it being $d$-Cartesian. ∎

Next, suppose that $A$ and $B$ are connective $R$-algebras, and that $\eta\colon R\to B$ is the unit map of $B$. We can assume that $A$ and $B$ are $R$-cofibrant as $R$-algebras in the sense of [Shi04]*Thm. 2.6(3). The underlying $R$-modules of $A$ and $B$ are then flat. We view $R$ as a base, $A$ as the object at which we wish to evaluate a functor, and $R\to B$ as a covering that induces a covering $A\to A\wedge_{R}B$. In this case the $n$-cube $X^{n}=X^{n}_{R}(A,B)\colon T\mapsto X^{n}(T)$, defined by the same expression as in (1), takes values in the category of connective $R$-algebras. For varying $n$, these assemble to an $\omega$-cube $X^{\omega}=X^{\omega}_{R}(A,B)$.

Definition 2.3**.**

Let $F$ be any functor from connective $R$-algebras to spectra. We call $F(X^{\omega})$ the Amitsur cube for $F$ at $A$ along $\eta\colon R\to B$, by analogy with the algebraic construction in [Ami59]. When $F$ satisfies cubical descent over $X^{\omega}$ we say that $F$ satisfies cubical descent at $A$ along $R\to B$.

Cubical descent for $F$ at $A$ along $R\to B$ ensures that $F(A)$ can be recovered from the diagram of spectra $T\mapsto F(X^{\omega}(T))$ for $T\in P$, having entries of the form $F(A\wedge_{R}B\wedge_{R}\dots\wedge_{R}B)$ with one or more copies of $B$.

2.3. Cubical descent for $K$, ${THH}$ and ${TC}$

Let $A\mapsto K(A)$ denote the algebraic $K$-theory functor from connective $\mathbb{S}$-algebras to spectra, see [BHM93]*§5 and [EKMM97]*Ch. VI.

Theorem 2.4**.**

Let $R$ be a connective commutative $\mathbb{S}$-algebra, let $A$ and $B$ be connective $R$-algebras, and suppose that the unit map $\eta\colon R\to B$ is $1$-connected.

(a) The $n$-cube $K(X^{n})=K(X^{n}_{R}(A,B))\colon T\mapsto K(X^{n}(T))$ is $(n+1)$-Cartesian, for each $n\geq 1$.

(b) Algebraic $K$-theory satisfies cubical descent at $A$ along $R\to B$.

Proof.

By Lemma 2.2 the $n$-cube $X^{n}\colon T\mapsto X^{n}(T)$ has the property that every $d$-dimensional subcube is $d$-Cartesian. Hence the assertion that the $n$-cube $K(X^{n})$ is $(n+1)$-Cartesian is the content of [Dun97]*Prop. 5.1. In other words, the natural map $\eta_{n}\colon K(A)\to\operatorname*{holim}_{P^{n}}K(X^{n})$ is $(n+1)$-connected. Thus $\eta\colon K(A)\to\operatorname*{holim}_{P}K(X^{\omega})$ is an equivalence, and $K$ satisfies cubical descent over $X^{\omega}$. ∎

Let $A\mapsto{THH}(A)$ denote the topological Hochschild homology functor from connective $\mathbb{S}$-algebras to cyclotomic spectra, see [BHM93]§3 and [HM97]§2. In particular, the circle group $\mathbb{T}$ acts naturally on the underlying spectrum of ${THH}(A)$, and for each subgroup $C=C_{p^{m}}\subset\mathbb{T}$ of order $p^{m}$, where $p$ is a prime, there is a homotopy orbit functor $A\mapsto{THH}(A)_{hC}$ and a fixed point functor $A\mapsto{THH}(A)^{C}$. These are related by a natural homotopy cofiber sequence of spectra

[TABLE]

called the norm–restriction sequence. Let ${TR}(A;p)=\operatorname*{holim}_{R,m}{THH}(A)^{C_{p^{m}}}$ be the sequential homotopy limit over the $R$-maps. Let $F\colon{THH}(A)^{C_{p^{m}}}\to{THH}(A)^{C_{p^{m-1}}}$ be the map forgetting part of the invariance, and let ${TF}(A;p)=\operatorname*{holim}_{F,m}{THH}(A)^{C_{p^{m}}}$ be the sequential homotopy limit over the $F$-maps. The $R$-maps induce a self-map of ${TF}(A;p)$, also denoted $R$, and the topological cyclic homology functor ${TC}(A;p)$ can be defined as the homotopy equalizer of $id$ and $R$:

[TABLE]

The (integral) topological cyclic homology of $A$, denoted ${TC}(A)$, is defined as the homotopy pullback of two maps

[TABLE]

see [DGM13]*Def. 6.4.3.1. The left hand map is defined in terms of $\pi\colon{TC}(A;p)\to{TF}(A;p)$ and the comparison maps $\Gamma_{m}\colon{THH}(A)^{C_{p^{m}}}\to{THH}(A)^{hC_{p^{m}}}$ from fixed points to homotopy fixed points. The right hand map is induced by the forgetful maps ${THH}(A)^{h\mathbb{T}}\to{THH}(A)^{hC_{p^{m}}}$ associated to the inclusions $C_{p^{m}}\subset\mathbb{T}$, for varying $p$ and $m$. The subscript “$p$” denotes $p$-completion. For each prime $p$, the projection ${TC}(A)\to{TC}(A;p)$ becomes an equivalence after $p$-completion.

Theorem 2.5**.**

Let $R$ be a connective commutative $\mathbb{S}$-algebra, let $A$ and $B$ be connective $R$-algebras, and suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Consider the $n$-cube $X^{n}=X^{n}_{R}(A,B)$, as above.

(a) Each $d$-dimensional subcube of ${THH}(X^{n})$ is $d$-Cartesian and $(2d-1)$-co-Cartesian.

(b) Each $d$-dimensional subcube of ${THH}(X^{n})_{hC}$, ${THH}(X^{n})^{C}$ and ${TR}(X^{n};p)$ is $d$-Cartesian and $(2d-1)$-co-Cartesian, for every $C=C_{p^{m}}\subset\mathbb{T}$.

(c) Each $d$-dimensional subcube of ${TF}(X^{n};p)$ and ${TC}(X^{n};p)$ is $(d-1)$-Cartesian and $(2d-2)$-co-Cartesian.

(d) Topological Hochschild homology, ${THH}(-)_{hC}$, ${THH}(-)^{C}$, ${TR}(-;p)$, ${TF}(-;p)$, ${TC}(-;p)$ and (integral) topological cyclic homology all satisfy cubical descent at $A$ along $R\to B$.

Proof.

(a) The underlying spectrum of ${THH}(A)$ is naturally equivalent to the realization $B^{cy}(A)$ of the cyclic bar construction $[q]\mapsto A\wedge\dots\wedge A$, with $(q+1)$ copies of $A$. Hence there is a natural equivalence

[TABLE]

where $B^{cy}(R)$ is a connective commutative $\mathbb{S}$-algebra, $B^{cy}(A)$ and $B^{cy}(B)$ are flat connective $B^{cy}(R)$-modules, and $B^{cy}(\eta)\colon B^{cy}(R)\to B^{cy}(B)$ is a $1$-connected map of $B^{cy}(R)$-modules. By Lemma 2.2 each $d$-dimensional subcube of

[TABLE]

is $(2d-1)$-co-Cartesian.

(b) Each $d$-dimensional subcube of ${THH}(X^{n})_{hC}$ is at least as co-Cartesian as the corresponding $d$-dimensional subcube of ${THH}(X^{n})$, because homotopy orbits preserve connectivity. The analogous claim for the subcubes of ${THH}(X^{n})^{C}$, with $C=C_{p^{m}}$, follows by induction on $m$ from the norm–restriction homotopy cofiber sequence. The equivalent Cartesian claim follows. The Cartesian claim for the subcubes of the sequential homotopy limit ${TR}(X^{n};p)$ follows from the Milnor $\lim$-$\lim^{1}$ sequence. In this case no connectivity is lost, because $\lim^{1}$ vanishes on sequences of surjections, see [Dun97]*Lem. 4.3.

(c) The Cartesian claims for ${TF}(X^{n};p)$ and ${TC}(X^{n};p)$ follow from those for the cubes ${THH}(X^{n})^{C}$, since sequential homotopy limits and homotopy equalizers reduce connectivity by at most one, see [Dun97]*Prop. 4.4.

(d) For each of the functors $F={THH}$, ${THH}(-)_{hC}$, ${THH}(-)^{C}$, ${TR}(-;p)$, ${TF}(-;p)$ and ${TC}(-;p)$, the natural map $\eta_{n}\colon F(A)\to\operatorname*{holim}_{P^{n}}F(X^{n})$ is $n$- or $(n-1)$-connected. Thus the natural map $\eta\colon F(A)\to\operatorname*{holim}_{P}F(X^{\omega})$ is an equivalence, and each of these functors satisfies cubical descent over $X^{\omega}$. Finally, for integral topological cyclic homology we use that each vertex in the natural diagram defining it satisfies cubical descent, since $\operatorname*{holim}_{P}$ commutes up to a natural chain of equivalences with other homotopy limits. ∎

2.4. Two spectral sequences

For each $\omega$-cube $F(X)$ of spectra, the equivalent homotopy limits

[TABLE]

give rise to two spectral sequences. On the one hand, we have the homotopy spectral sequence

[TABLE]

associated to the tower of fibrations

[TABLE]

see [BK72]*IX.4.2. Here $\operatorname{hofib}(p_{s})$ denotes the homotopy fiber of the map $p_{s}$, which is equivalent to the iterated homotopy fiber of the $s$-dimensional subcube of $F(X|P^{s+1})$ with vertices indexed by the $T\in P^{s+1}$ with $s+1\in T$. This spectral sequence is conditionally convergent [Boa99]*Def. 5.10 to the sequential limit $\lim_{s}\pi_{*}(\operatorname*{holim}_{P^{s}}F(X))$ of the homotopy groups of that tower. For each $r\geq 1$, the bigrading of the $d_{r}$-differential is given by

[TABLE]

Following the usual conventions for Adams spectral sequences, $d_{r}$ maps bidegree $(x,y)$ in the $(t-s,s)$-plane to bidegree $(x-1,y+r)$. If the derived limit $RE_{\infty}=\lim_{r}^{1}E_{r}$ of $E_{r}$-terms vanishes in each bidegree, then the spectral sequence converges strongly to $\pi_{*}(\operatorname*{holim}_{n}\,\operatorname*{holim}_{P^{n}}F(X))$, see [Boa99]*Thm. 7.4, which is isomorphic to $\pi_{*}F(X(\varnothing))$ when $F$ satisfies cubical descent over $X$. We call this the cubical descent spectral sequence.

Corollary 2.6**.**

Let $R$, $A$, $B$ and $\eta$ be as in Theorem 2.4 and Theorem 2.5. When $F$ is one of the functors $K$, ${THH}$, $\dots$, ${TC}(-;p)$ or ${TC}$ of those theorems, and $X=X^{\omega}_{R}(A,B)$, the cubical descent spectral sequence

[TABLE]

vanishes above a line of slope $+1$ in the $(t-s,s)$-plane, starting at the $E_{1}$-term, hence collapses at a finite stage in each bidegree. Thus $RE_{\infty}=0$ and the spectral sequence is strongly convergent.

Proof.

Let $c=1$ when $F$ is $K$, let $c=0$ when $F$ is one of the functors ${THH}$, ${THH}(-)_{hC}$, ${THH}(-)^{C}$ or ${TR}(-;p)$, and let $c=-1$ when $F$ is ${TF}(-;p)$ or ${TC}(-;p)$. Then $\eta_{n}\colon F(A)\to\operatorname*{holim}_{P^{n}}F(X^{n})$ is $(n+c)$-connected for each $n\geq 1$, so $p_{s}\colon\operatorname*{holim}_{P^{s+1}}F(X^{s+1})\to\operatorname*{holim}_{P^{s}}F(X^{s})$ is $(s+c)$-connected for each $s\geq 1$. Thus $E_{1}^{s,t}=0$ for $t-s<s+c$ and $s\geq 1$.

The case $F={TC}$ remains. For this we appeal to [DGM13]*Thm. 7.0.0.2 (where $K(B)$ in the lower left-hand corner of the displayed square should be replaced with $K(A)$), to see that for each $s\geq 1$ the square

[TABLE]

is homotopy Cartesian. This uses that $\pi_{0}X^{s+1}(T)$ is constant as a functor of $T$, by our assumptions on $R$, $A$, $B$ and $\eta$. Hence the spectral sequences for $K$ and ${TC}$ have the same groups $E_{1}^{s,t}$ for all $s\geq 1$, and from the case $F=K$ we know that these groups vanish for $t-s<s+1$. ∎

On the other hand, we have the homotopy limit spectral sequence

[TABLE]

of [BK72]*XI.7.1, associated to $P$-shaped diagrams of spectra. We call this the cubical homotopy limit spectral sequence. The abutment is isomorphic to $\pi_{*}F(X(\varnothing))$ when $F$ satisfies cubical descent over $X$. We do not know whether the cubical descent spectral sequence and the cubical homotopy limit spectral sequence are isomorphic for all $\omega$-cubical diagrams $F(X)$, but in Subsection 3.4 we will see that this is the case whenever the diagram $F(X|P)$ arises from a cosimplicial object $F(Y^{\bullet})$ by composition with a specific functor $f\colon P\to\Delta$, which we will now introduce.

3. Cosimplicial descent

3.1. Cosimplicial objects

Let $\Delta_{\eta}$ be the category of finite totally ordered sets $[q]=\{0<\dots<q\}$ and order-preserving functions, where $q\geq-1$ is an integer. The object $[-1]=\varnothing$ is initial in this category. Let $\Delta\subset\Delta_{\eta}$ be the full subcategory generated by the objects $[q]$ with $q\geq 0$. For $n\geq 1$, let $\Delta^{<n}_{\eta}\subset\Delta_{\eta}$ and $\Delta^{<n}\subset\Delta$ be the respective full subcategories generated by the objects $[q]$ with $q<n$.

A coaugmented cosimplicial object in $\mathscr{C}$ is a functor $Y\colon\Delta_{\eta}\to\mathscr{C}$. We let $Y^{q}=Y([q])$, for each $q\geq-1$. We can also write $Y$ as $\eta\colon Y^{-1}\to Y^{\bullet}$, where $Y^{\bullet}$ is the cosimplicial object given by the restriction of $Y$ over $\Delta\subset\Delta_{\eta}$. For $n\geq 1$, a functor $Y_{<n}\colon\Delta^{<n}_{\eta}\to\mathscr{C}$ is called a coaugmented $(n-1)$-truncated cosimplicial object, also written as $\eta\colon Y^{-1}\to Y_{<n}^{\bullet}$, where $Y_{<n}^{\bullet}=Y_{<n}|\Delta^{<n}$.

Definition 3.1**.**

Given any functor $F$ from $\mathscr{C}$ to spectra, the composite functor $F\circ Y$ is a coaugmented cosimplicial spectrum $F(Y)$, which we can write as $\eta\colon F(Y^{-1})\to F(Y^{\bullet})$. We say that $F$ satisfies cosimplicial descent over $Y$ if the natural map

[TABLE]

is an equivalence of spectra. This ensures that the homotopy type of $F(Y^{-1})$ is essentially determined by the homotopy types of the spectra $F(Y^{q})$ for $q\geq 0$.

3.2. Comparison of cubical and cosimplicial objects

There is a well-defined functor $f_{\eta}\colon P_{\eta}\longrightarrow\Delta_{\eta}$ that maps the element $T=\{t_{0}<\dots<t_{q}\}\subset\mathbb{N}$ to the object $[q]$, and maps the inclusion $T\setminus\{t_{i}\}\subset T$ to the $i$-th face operator $\delta^{i}\colon[q-1]\to[q]$, for each $0\leq i\leq q$. By restricting $f_{\eta}$, one gets functors $f\colon P\to\Delta$, $f^{n}_{\eta}\colon P^{n}_{\eta}\to\Delta^{<n}_{\eta}$ and $f^{n}\colon P^{n}\to\Delta^{<n}$.

Composition with $f_{\eta}\colon P_{\eta}\to\Delta_{\eta}$ takes each coaugmented cosimplicial spectrum $F(Y)$ to an $\omega$-cube $F(X)=F(Y)\circ f_{\eta}$ in the category of spectra. Likewise, composition with $f^{n}_{\eta}\colon P^{n}_{\eta}\to\Delta^{<n}_{\eta}$ takes each coaugmented $(n-1)$-truncated cosimplicial spectrum $F(Y_{<n})$ to an $n$-cube $F(X^{n})=F(Y_{<n})\circ f^{n}_{\eta}$ of spectra. If $F(Y_{<n})$ is given by restricting $F(Y)$ over $\Delta^{<n}_{\eta}\subset\Delta_{\eta}$ then $F(X^{n})$ is given by restricting $F(X)$ over $P^{n}_{\eta}\subset P_{\eta}$.

Proposition 3.2**.**

The functors $f\colon P\to\Delta$ and $f^{n}\colon P^{n}\to\Delta^{<n}$ are left cofinal. Hence, for any cosimplicial spectrum $F(Y^{\bullet})$ the canonical map

[TABLE]

is an equivalence, and for any $(n-1)$-truncated cosimplicial spectrum $F(Y_{<n}^{\bullet})$ the canonical map

[TABLE]

is an equivalence.

Proof.

The assertion that $f^{n}$ is left cofinal, i.e., that the left fiber $f_{n}/[q]$ has contractible nerve for each object $[q]$ of $\Delta^{<n}$, is proved in [Car08]*§6. Note that the argument offered at this point in [DGM13]*A8.1.1 is flawed. The left fiber $f/[q]$ is the increasing union of the left fibers $f_{n}/[q]$, hence its nerve is also contractible. The equivalences of homotopy limits then follow from the cofinality theorem of [BK72]*XI.9.2. ∎

Corollary 3.3**.**

Let $Y$ be a coaugmented cosimplicial object in $\mathscr{C}$, and let $F$ be a functor from $\mathscr{C}$ to spectra. Then $F$ satisfies cosimplicial descent over $Y$ if and only if $F$ satisfies cubical descent over $X=Y\circ f_{\eta}$.

Corollary 3.4**.**

For any cosimplicial abelian group $A^{\bullet}$, the canonical homomorphisms

[TABLE]

are isomorphisms, for each $s\geq 0$.

Proof.

See [BK72]*XI.7.2 and XI.7.3. ∎

3.3. Amitsur resolutions

We now suppose that $R$ is a connective commutative $\mathbb{S}$-algebra, that $A$ is a connective $R$-algebra, and that $B$ is a connective commutative $R$-algebra with unit map $\eta\colon R\to B$ and multiplication map $\mu\colon B\wedge_{R}B\to B$. The condition that $B$ is commutative ensures that $\mu$ is a morphism of connective $R$-algebras. We can assume that $A$ is $R$-cofibrant as an $R$-algebra, and that $B$ is $R$-com-alg-cofibrant as a commutative $R$-algebra in the sense of [Shi04]*Thm. 3.2. (This follows if $B$ is positive stable cofibrant in the sense of [MMSS01]*Thm. 15.2(i).) The underlying $R$-modules of $A$ and $B$ are then flat, by [Shi04]*Cor. 4.3, so that the results of Section 2 carry over to this situation.

Consider the cosimplicial connective $R$-algebra $Y^{\bullet}=Y^{\bullet}_{R}(A,B)\colon[q]\mapsto Y^{q}$ given by

[TABLE]

with $(q+1)$ copies of $B$. The coface maps

[TABLE]

for $0\leq i\leq q$, and the codegeneracy maps

[TABLE]

for $0\leq j\leq q$, are induced by the unit map $\eta$ and the multiplication map $\mu$, respectively. The unit map $\eta$ also induces a coaugmentation $\eta\colon A\to Y^{0}=A\wedge_{R}B$ that makes $A\to Y^{\bullet}$ a coaugmented cosimplicial connective $R$-algebra, i.e., a functor $Y=Y_{R}(A,B)$ from $\Delta_{\eta}$ to the category of connective $R$-algebras:

[TABLE]

Definition 3.5**.**

Given a functor $F$ from connective $R$-algebras to spectra, we call the coaugmented cosimplicial spectrum $\eta\colon F(A)\to F(Y^{\bullet})=F(Y^{\bullet}_{R}(A,B))$ the Amitsur resolution for $F$ at $A$ along $\eta\colon R\to B$. When $F$ satisfies cosimplicial descent over $A\to Y^{\bullet}$ we say that $F$ satisfies cosimplicial descent at $A$ along $R\to B$.

Cosimplicial descent for $F$ at $A$ along $R\to B$ ensures that the coaugmented cosimplicial spectrum

[TABLE]

induces an equivalence

[TABLE]

from $F(A)$ to the homotopy limit of the remainder of the diagram, having entries of the form $F(A\wedge_{R}B\wedge_{R}\dots\wedge_{R}B)$ with one or more copies of $B$.

Lemma 3.6**.**

There is a natural isomorphism $X^{\omega}_{R}(A,B)\cong Y_{R}(A,B)\circ f_{\eta}$.

Proof.

For $T=\{t_{0}<\dots<t_{q}\}\subset\mathbb{N}$, with $f_{\eta}(T)=[q]$, both $X^{\omega}_{R}(A,B)(T)$ and $Y_{R}(A,B)([q])$ are identified with

[TABLE]

where there are $(q+1)$ copies of $B$. For each $T^{\prime}\subset T$, the induced morphisms in $X^{\omega}_{R}(A,B)$ and $Y_{R}(A,B)\circ f_{\eta}$ are evidently compatible with these identifications. ∎

Theorem 3.7**.**

*Let $R$ be a connective commutative $\mathbb{S}$-algebra, $A$ a connective $R$-algebra and $B$ a connective commutative $R$-algebra, and suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Then algebraic $K$-theory, topological Hochschild homology, ${THH}(-)_{hC}$, ${THH}(-)^{C}$, ${TR}(-;p)$, ${TF}(-;p)$, ${TC}(-;p)$, and (integral) topological cyclic homology all satisfy cosimplicial descent at $A$ along $R\to B$. *

Proof.

Combine Theorems 2.4 and 2.5, Corollary 3.3 and Lemma 3.6. ∎

3.4. More spectral sequences

For each coaugmented cosimplicial spectrum $\eta\colon F(Y^{-1})\to F(Y^{\bullet})$, the equivalent homotopy limits

[TABLE]

give rise to two spectral sequences. These turn out to be isomorphic to one another, as well as to the two spectral sequences of Subsection 2.4, when we consider cubical diagrams that arise from cosimplicial diagrams by composition with the left cofinal functor $f$.

On the one hand, we have the homotopy spectral sequence

[TABLE]

associated to the tower of fibrations

[TABLE]

which we call the cosimplicial descent spectral sequence.

By Proposition 3.2, the tower of fibrations (5) is equivalent to the tower (3) when $X=Y\circ f_{\eta}$, $F(X)=F(Y)\circ f_{\eta}$ and $F(X|P)=F(Y^{\bullet})\circ f$. Hence in these cases the cosimplicial descent spectral sequence for $\eta\colon F(Y^{-1})\to F(Y^{\bullet})$ is isomorphic to the cubical descent spectral sequence for $F(X)$.

There is also the Bousfield–Kan homotopy spectral sequence

[TABLE]

of the cosimplicial spectrum $F(Y^{\bullet})$, associated to the tower of maps

[TABLE]

see [BK72]*X.6.1. To ensure that the maps $\tau_{s}$ are fibrations, we first implicitly replace $F(Y^{\bullet})$ with an equivalent fibrant cosimplicial spectrum $F(Y^{\bullet})^{\prime}$ [BK72]*X.4.6. This fibrant replacement does not change the homotopy type of $F(X|P)=F(Y^{\bullet})\circ f$ or any of the associated homotopy limits that we consider. The $E_{2}$-term of this homotopy spectral sequence can then be expressed as

[TABLE]

see [BK72]*X.7.2. We prove in Proposition 3.10 that there are natural equivalences

[TABLE]

(after the implicit fibrant replacement of $F(Y^{\bullet})$), compatible for varying $n$ with the equivalence

[TABLE]

of [BK72]*XI.4.4. Hence the tower (6) is equivalent to the tower (5), and the cosimplicial descent spectral sequence is isomorphic to the Bousfield–Kan homotopy spectral sequence of $F(Y^{\bullet})$, starting with the $E_{1}$-term.

On the other hand, we have the cosimplicial homotopy limit spectral sequence

[TABLE]

By [BK72]*X.7.5 there is a natural map from the Bousfield–Kan homotopy spectral sequence of $F(Y^{\bullet})$ to the cosimplicial homotopy limit spectral sequence, and this map is an isomorphism at the $E_{2}$-term, hence also at all later terms.

Finally, the functor $f\colon P\to\Delta$ induces a map $f^{*}$ of homotopy limit spectral sequences from

[TABLE]

to

[TABLE]

where $F(X|P)=F(Y^{\bullet})\circ f$. This is the map of Bousfield–Kan spectral sequences associated to a canonical map $f^{*}\colon\Pi^{\bullet}_{\Delta}F(Y^{\bullet})\to\Pi^{\bullet}_{P}F(X|P)$ of cosimplicial spectra, see [BK72]*XI.5.1, so the map of $E_{2}$-terms is the isomorphism $f^{*}$ of Corollary 3.4. Hence in these cases the cosimplicial homotopy limit spectral sequence for $F(Y^{\bullet})$ is isomorphic to the cubical homotopy limit spectral sequence for $F(X|P)$.

Proposition 3.8**.**

(a) Let $F(Y^{\bullet})$ be any cosimplicial spectrum, with fibrant replacement $F(Y^{\bullet})^{\prime}$, and let $F(X|P)=F(Y^{\bullet})\circ f$. The first three of the spectral sequences

[TABLE]

are isomorphic at the $E_{1}$-term, and all five are isomorphic at the $E_{2}$-term, hence also at all later terms. If $RE_{\infty}=0$ these spectral sequences all converge strongly to the indicated abutments.

(b) Let $F(Y)$ be any coaugmented cosimplicial spectrum, and let $F(X)=F(Y)\circ f_{\eta}$. If $F$ satisfies cubical descent over $X$, or equivalently, if $F$ satisfies cosimplicial descent over $Y$, then each abutment in (a) is isomorphic to $\pi_{*}F(Y^{-1})=\pi_{*}F(X(\varnothing))$. If $F(Y)$ is the Amitsur resolution $\eta\colon F(A)\to F(Y^{\bullet}_{R}(A,B))$, so that $F(X)$ is the Amitsur cube $F(X^{\omega}_{R}(A,B))$, then this common abutment is $\pi_{*}F(A)$.

Proof.

(a) The stated isomorphisms were all discussed before the statement of the proposition. Each spectral sequence is derived from a tower of fibrations, hence is conditionally convergent to the sequential limit of the homotopy groups of the terms in this tower. When $RE_{\infty}$ vanishes, each homotopy group of the homotopy limit of the tower is isomorphic to that sequential limit, and the spectral sequence is strongly convergent, by [BK72]*IX.5.4 or [Boa99]*Thm. 7.4.

(b) Cosimplicial descent for $F$ over $Y$ ensures that $F(Y^{-1})\simeq\operatorname*{holim}_{\Delta}F(Y^{\bullet})$. ∎

Theorem 3.9**.**

Let $R$ be a connective commutative $\mathbb{S}$-algebra, $A$ a connective $R$-algebra, $B$ a connective commutative $R$-algebra and suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Let $F$ be one of the functors

• •

$K$, ${TC}$ (with $c=+1$),

• •

${THH}$, ${THH}(-)_{hC}$, ${THH}(-)^{C}$, ${TR}(-;p)$ (with $c=0$),

• •

${TF}(-;p)$, ${TC}(-;p)$ (with $c=-1$).

Then the $E_{1}$-term of the (cubical/cosimplicial) descent spectral sequence for $F$ at $A$ along $\eta\colon R\to B$ vanishes in all bidegrees $(s,t)$ with $t-s<s+c$ and $s\geq 1$. It is strongly convergent to $\pi_{*}F(A)$, with $E_{2}$-term given by

[TABLE]

Proof.

The description of the $E_{2}$-term is that of the Bousfield–Kan spectral sequence. The vanishing line for the $E_{1}$-term of the cubical descent spectral sequence is that of Corollary 2.6. It follows that in each bidegree $(s,t)$ there is a finite $r$ such that $E_{r}^{s,t}=E_{\infty}^{s,t}$, which implies that $RE_{\infty}^{s,t}=0$. Hence each version of the spectral sequence converges strongly to the homotopy groups of $F(Y^{-1}_{R}(A,B))=F(A)$. ∎

In the discussion above we used the following result, for which a proof does not seem to have appeared in the literature.

Proposition 3.10**.**

Let $Z^{\bullet}$ be any fibrant cosimplicial space. There are compatible natural equivalences

[TABLE]

for all $0\leq n\leq\infty$.

Proof.

When $n=\infty$, this result is [BK72]*XI.4.4. We indicate how to adapt their proof to the case of finite $n$.

Let $D=\Delta^{<n+1}$, let $i\colon D\to\Delta$ be the inclusion of the full subcategory, let $\mathscr{S}$ be the category of spaces (= simplicial sets), and let $\mathscr{S}^{\Delta}=c\mathscr{S}$ and $\mathscr{S}^{D}$ be the functor categories of cosimplicial spaces and $n$-truncated cosimplicial spaces, respectively. Composition with $i$ defines the restriction functor $i^{*}\colon c\mathscr{S}\to\mathscr{S}^{D}$. Let $i_{*}=\operatorname{LKan}_{i}\colon\mathscr{S}^{D}\to c\mathscr{S}$ be the left Kan extension, left adjoint to $i^{*}$. For each $[k]\in D$, with $0\leq k\leq n$, let $D/[k]$ be the over category and let $N(D/[k])$ denote its nerve (also known as its underlying or classifying space). For $[k]\in D$, let $z\colon N(D/[k])\to\Delta[k]$ be the “zeroth vertex map” sending each vertex $\alpha\colon[p]\to[k]$ in $N(D/[k])$ to the vertex $\alpha(0)$ in $\Delta[k]$, see [BK72]*XI.2.6(ii). It is a weak equivalence for each $[k]$ in $D$, since both $N(D/[k])$ and $\Delta[k]$ are contractible. Composition with $z$ defines a morphism of mapping spaces

[TABLE]

Here $(i_{*}i^{*}\Delta)[q]=\operatorname*{colim}_{[k]\to[q]}\Delta[k]\cong\operatorname{sk}_{n}\Delta[q]$ for $[q]\in\Delta$, where $[k]\to[q]$ ranges over the left fiber category $i/[q]$ of $i$ at $[q]$. Hence we can rewrite $z^{*}$ as

[TABLE]

To prove that $z^{*}$ is a weak equivalence, we will use that $\mathscr{S}^{D}$ with the Reedy model structure is a simplicial model category [Hir03]*Thm. 15.3.4. Here $D$ has the evident Reedy category structure inherited from $\Delta$. It suffices to prove that $z\colon N(D/-)\to i^{*}\Delta$ is a weak equivalence of cofibrant objects and that $i^{*}Z^{\bullet}$ is a fibrant object, in the Reedy model structure on $\mathscr{S}^{D}$. We have already observed that $z$ is a Reedy weak equivalence. The cosimplicial space $\Delta$ is unaugmentable, hence cofibrant in the model structure on $c\mathscr{S}$, see [BK72]*X.4.2. The latching map of $i^{*}\Delta$ at each object $[k]$ in $D$ is equal to the latching map of $\Delta$ at $[k]\in\Delta$, and therefore $i^{*}\Delta$ is Reedy cofibrant. By assumption, $Z^{\bullet}$ is fibrant in the model structure on $c\mathscr{S}$, see [BK72]*X.4.6. The matching map of $i^{*}Z^{\bullet}$ at each object $[k]$ in $D$ is equal to the matching map of $Z^{\bullet}$ at $[k]\in\Delta$, hence $i^{*}Z^{\bullet}$ is Reedy fibrant. It remains to check that $N(D/-)$ is Reedy cofibrant. This follows immediately from the fact that it is cofibrant in the projective model structure on $\mathscr{S}^{D}$, see [Hir03]*Prop. 14.8.9, Thm. 11.6.1. ∎

3.5. Tensored structure and topological André–Quillen homology

In this subsection we specialize to the case when both $A$ and $B$ are commutative $R$-algebras. This category is tensored over spaces, taking any simplicial set $S\colon[p]\mapsto S_{p}$ and any commutative $R$-algebra $A$ to the realization $S\otimes_{R}A$ of $[p]\mapsto S_{p}\otimes_{R}A=A\wedge_{R}\dots\wedge_{R}A$, with one copy of $A$ for each element of $S_{p}$. In the case when $S=S^{1}=\Delta[1]/\partial\Delta[1]$ is the simplicial circle and $R=\mathbb{S}$ is the sphere spectrum, $S^{1}\otimes_{\mathbb{S}}A=B^{cy}(A)$ is the cyclic bar construction considered in the proof of Theorem 2.5, which is equivalent to ${THH}(A)$ when $A$ is flat. Suppose now that $A$ and $B$ are $R$-com-alg-cofibrant, hence flat as $R$-modules. Our results about descent for ${THH}$ generalize as follows.

Proposition 3.11**.**

If the unit map $R\to B$ is $1$-connected, then tensoring with any simplicial set $S$ satisfies cosimplicial descent at $A$ along $R\to B$, meaning that

[TABLE]

is an equivalence.

Proof.

Tensors commute with coproducts, so $S\otimes_{R}Y^{\bullet}_{R}(A,B)$ is isomorphic to $Y^{\bullet}_{R}(S\otimes_{R}A,S\otimes_{R}B)$. Lemma 2.2 shows that $X^{n}_{R}(S_{p}\otimes_{R}A,S_{p}\otimes_{R}B)$ is $(2n-1)$-co-Cartesian for each $p\geq 0$, which implies that $X^{n}_{R}(S\otimes_{R}A,S\otimes_{R}B)$ is $(2n-1)$-co-Cartesian and $n$-Cartesian, for each $n\geq 1$. Hence

[TABLE]

is $n$-connected, and therefore $\eta$ is an equivalence. ∎

Using either the Bökstedt type model as in [BCD10] (extended mutatis mutandis to connective orthogonal or symmetric spectra) or a model with more categorical control as in [BDS], the fundamental homotopy cofiber sequence (2) leads to the equivariant extension given below. This generalizes our descent results for ${TC}(-;p)$ to the various forms of covering homology considered in [BCD10]*Sec. 7. In particular, the higher topological cyclic homology of [CDD11] satisfies cosimplicial descent at connective commutative $\mathbb{S}$-algebras along $1$-connected unit maps.

Corollary 3.12**.**

If $G$ is a finite group acting freely on $S$, and $R\to B$ is $1$-connected, then tensoring with $S$ satisfies equivariant cosimplicial descent at $A$ along $R\to B$, meaning that

[TABLE]

is a $G$-equivariant equivalence.

Proof.

We have to show that for every subgroup $H\subseteq G$, the map $\eta^{H}$ of $H$-fixed points is an equivalence. This follows by essentially the same argument as for ${THH}$, e.g. by [BCD10]*Lem. 5.1.3, using induction over the closed families of subgroups and the fact that homotopy orbits preserve connectivity. ∎

Coming back to the non-equivariant situation, cosimplicial descent is satisfied by topological André–Quillen homology, which we will denote by $TAQ^{R}(A)$.

Proposition 3.13**.**

If the unit map $R\to B$ is $1$-connected, then topological André–Quillen homology satisfies cosimplicial descent at $A$ along $R\to B$, in the sense that

[TABLE]

is an equivalence.

Proof.

By [BM05]*Thm. 4, there is an equivalence

[TABLE]

of $A$-module spectra, where $(S^{m}\otimes_{R}A)/A$ denotes the homotopy cofiber of the map $A\to S^{m}\otimes_{R}A$ associated to the base point in $S^{m}=(S^{1})^{\wedge m}$. The map is at least $(m-1)$-connected, for each $m\geq 0$, and likewise for $B$ in place of $A$. It follows that the map

[TABLE]

(with $n\geq 1$ copies of $B/R$ and of $(S^{m}\otimes_{R}B)/R$) is at least $(m+2n-3)$-connected. Hence the $n$-cube

[TABLE]

(with $X^{n}=X^{n}_{R}(A,B)$) is $(2n-3)$-co-Cartesian, for each $m\geq 0$. Thus the $n$-cube $T\mapsto TAQ^{R}(X^{n}(T))$ is $(2n-3)$-co-Cartesian and $(n-2)$-Cartesian, so that

[TABLE]

is at least $(n-2)$-connected. Passing to the homotopy limit over $n$, it follows that $\eta$ is an equivalence. ∎

3.6. Less commutative examples

Let us return to the situation where $A$ is not necessarily commutative. In Subsection 3.3 we took $B$ to be commutative to ensure that $\mu\colon B\wedge_{R}B\to B$ and the codegeneracy maps $s^{j}\colon Y^{q+1}\to Y^{q}$ are morphisms of $R$-algebras. For non-commutative $B$, we can still define what we might call a precosimplicial $R$-algebra $[q]\mapsto Y^{q}$, where now $[q]$ ranges over the subcategory $M\subset\Delta$ with morphisms the injective, order-preserving functions. The functor $f\colon P\to\Delta$ defined at the beginning of Subsection 3.2 factors as the composite of a functor $e\colon P\to M$ and the inclusion $i\colon M\to\Delta$. Here $e$ is not left cofinal, so the analogue of Proposition 3.2 does not hold for general precosimplicial spectra $Z^{\bullet}$. On the other hand, $i\colon M\to\Delta$ is left cofinal, see [DD77]*3.17, so for cosimplicial $Y^{\bullet}$ the canonical map

[TABLE]

is an equivalence. Hence

[TABLE]

is an equivalence for each precosimplicial spectrum $Z^{\bullet}=Y^{\bullet}\circ i$ that admits an extension to a cosimplicial spectrum.

By an $\mathscr{O}$ $R$-ring spectrum, for an operad $\mathscr{O}$, we mean an $\mathscr{O}$-algebra in the category of $R$-modules. We now relax the commutativity condition on $B$ to only ask that it is an $E_{2}$ $R$-ring spectrum, i.e., an $\mathscr{O}$ $R$-ring spectrum for some $E_{2}$ operad $\mathscr{O}$. Then $B$ is equivalent to a monoid in a category of $A_{\infty}$ $R$-ring spectra, by [BFV07]*Thm. C, since the tensor product of the associative operad and the little intervals operad $\mathscr{C}_{1}$ is an $E_{2}$ operad, and $\mathscr{C}_{1}$ is an $A_{\infty}$ operad. In other words, we may assume that $\eta\colon R\to B$ and $\mu\colon B\wedge_{R}B\to B$ are morphisms of $\mathscr{C}_{1}$ $R$-ring spectra, so that the Amitsur resolution $A\to Y^{\bullet}_{R}(A,B)$ is a coaugmented cosimplicial object in the category of connective $\mathscr{C}_{1}$ $R$-ring spectra.

Using a monadic bar construction [May72]*§9 to functorially turn $\mathscr{C}_{1}$-algebras into monoids, we may replace this Amitsur resolution with an equivalent coaugmented cosimplicial connective $R$-algebra $A\to\bar{Y}^{\bullet}_{R}(A,B)$, which we might denote by $\bar{Y}_{R}(A,B)$. Applying a homotopy functor $F$ from connective $R$-algebras to spectra, we thus obtain a coaugmented cosimplicial spectrum $F(A)\to F(\bar{Y}^{\bullet}_{R}(A,B))$.

The $\omega$-cubical diagram $\bar{Y}_{R}(A,B)\circ f_{\eta}$ remains equivalent to the Amitsur $\omega$-cube associated to the $R$-module $A$ and the unit map $\eta\colon R\to B$ of $\mathscr{C}_{1}$ $R$-ring spectra. Replacing $B$ with an equivalent $R$-algebra $\bar{B}$, we obtain an equivalent $\omega$-cube $X^{\omega}_{R}(A,\bar{B})$. Hence there is a chain of equivalences

[TABLE]

Substituting this for Lemma 3.6 in the proof of Theorem 3.7, we obtain the following generalization of that theorem.

Theorem 3.14**.**

Let $R$ be a connective commutative $\mathbb{S}$-algebra, let $A$ be a connective $R$-algebra, and let $B$ be a connective $E_{2}$ $R$-ring spectrum. Suppose that the unit map $\eta\colon R\to B$ is $1$-connected. Then the functors $F=K$, ${THH}$ and ${TC}$, as well as their intermediate variants, satisfy cosimplicial descent at $A$ along $R\to B$, in the sense that

[TABLE]

is an equivalence. Here $\bar{Y}^{q}_{R}(A,B)$ and $Y^{q}_{R}(A,B)$ are equivalent as $A_{\infty}$ $R$-ring spectra, for each $q\geq 0$.

4. Applications

4.1. Algebraic $K$-theory of spaces

For each topological group $\Gamma$, the spherical group ring $\mathbb{S}[\Gamma]=\Sigma^{\infty}\Gamma_{+}$ is a connective $\mathbb{S}$-algebra. When $X\simeq B\Gamma$, the algebraic $K$-theory of $\mathbb{S}[\Gamma]$ is a model for Waldhausen’s algebraic $K$-theory $A(X)$ of the space $X$, see [Wal85]. When $X$ is a high-dimensional compact (topological, piecewise-linear or differentiable) manifold, $A(X)$ is closely related to the space of $h$-cobordisms on $X$ and the group of automorphisms of $X$ [WJR13]. This motivates the interest in the algebraic $K$-theory of $\mathbb{S}$ and the associated spherical group rings.

Base change along the Hurewicz map $\mathbb{S}\to H\mathbb{Z}$ induces rational equivalences $\mathbb{S}[\Gamma]\to H\mathbb{Z}[\Gamma]$ and $K(\mathbb{S}[\Gamma])\to K(H\mathbb{Z}[\Gamma])$. In particular, $A(*)=K(\mathbb{S})\to K(H\mathbb{Z})=K(\mathbb{Z})$ is a rational equivalence, and Borel’s rational computation of the algebraic $K$-theory of the integers [Bor74] gives strong rational information about the $h$-cobordism spaces and automorphism groups of high-dimensional highly-connected manifolds [Igu88]*p. 7.

4.2. Descent along $\mathbb{S}\to H\mathbb{Z}$

To obtain torsion information about $A(X)=K(\mathbb{S}[\Gamma])$ one can instead consider the cosimplicial resolution

[TABLE]

in the category of connective $\mathbb{S}$-algebras, and the induced coaugmented cosimplicial spectrum

[TABLE]

By Theorem 3.7 the natural map from $K(\mathbb{S}[\Gamma])$ to the homotopy limit of this cosimplicial spectrum is an equivalence, and similarly for ${THH}$ and ${TC}$. These cases of descent, along $\mathbb{S}\to H\mathbb{Z}$, are essentially those studied in [Dun97]. See also [Tsa00] for descent results in the context of commutative rings.

A computational drawback with this approach is the structure of the smash product

[TABLE]

with $(q+1)$ copies of $H\mathbb{Z}$. It is a connective $H\mathbb{Z}$-algebra, hence equivalent to a simplicial ring, but for $q\geq 1$ the algebraic $K$-theory and topological cyclic homology of this simplicial ring appear to be difficult to analyze.

4.3. Descent along $\mathbb{S}\to MU$

Experience from algebraic topology shows that the complex bordism spectrum $MU$ is a convenient stopping point on the way from the sphere spectrum to the integers:

[TABLE]

Here $MU$ is a commutative $\mathbb{S}$-algebra with $1$-connected unit map $\mathbb{S}\to MU$. The coefficient ring $MU_{*}=\pi_{*}(MU)=\mathbb{Z}[x_{k}\mid k\geq 1]$ and the homology algebra $H_{*}(MU)\cong H_{*}(BU)=\mathbb{Z}[b_{k}\mid k\geq 1]$ are explicitly known [Mil60], [Nov62], with $|x_{k}|=|b_{k}|=2k$ for each $k$. The associated cosimplicial resolution

[TABLE]

induces the coaugmented cosimplicial spectrum

[TABLE]

By Theorem 3.7, applied with $R=\mathbb{S}$, $A=\mathbb{S}[\Gamma]$ and $B=MU$, the natural map from $K(\mathbb{S}[\Gamma])$ to the homotopy limit of this cosimplicial spectrum is an equivalence, and there are corresponding equivalences for ${THH}$ and ${TC}$.

From its definition as an $E_{\infty}$ Thom spectrum, $MU$ comes equipped with a Thom equivalence $MU\wedge MU\simeq MU\wedge BU_{+}$. Hence the smash product

[TABLE]

that occurs in codegree $q$ is not significantly more complicated for $q\geq 1$ than for $q=0$. This means that it may be more realistic to study $K(\mathbb{S})$ by descent along $\mathbb{S}\to MU$ than by descent along $\mathbb{S}\to H\mathbb{Z}$. We propose that in order to understand $A(*)=K(\mathbb{S})$ from the chromatic point of view [MRW77], [Rav84], one should study this cosimplicial resolution, starting with $K(MU)$ and continuing with $K(MU\wedge BU^{q}_{+})$ for each $q\geq 0$, together with the associated descent spectral sequence converging to $\pi_{*}K(\mathbb{S})$. Similar remarks apply for $A(X)=K(\mathbb{S}[\Gamma])$ and for the functors ${THH}$ and ${TC}$, see Theorem 3.9.

4.4. (Hopf-)Galois descent

In the language of [Rog08]§12, the map $\mathbb{S}\to MU$ is a Hopf–Galois extension of commutative $\mathbb{S}$-algebras. Our result can be viewed as proving $1$-connected Hopf–Galois descent for $K$, ${THH}$ and ${TC}$, but the actual Hopf $\mathbb{S}$-algebra coaction plays no role in the proof. Analogously, consider a $G$-Galois extension $A\to B$ of commutative $\mathbb{S}$-algebras, in the sense of [Rog08]§4, with $G$ a finite group. The equivalence $B\wedge_{A}B\simeq F(G_{+},B)\cong F(G^{2}_{+},B)^{G}$ induces a level equivalence of cosimplicial resolutions from

[TABLE]

to

[TABLE]

i.e., from the Amitsur resolution $Y^{\bullet}=Y^{\bullet}_{A}(A,B)$ to the cosimplicial commutative $\mathbb{S}$-algebra $F(E_{\bullet}G_{+},B)^{G}$ obtained by mapping out of the free contractible simplicial $G$-set $E_{\bullet}G\colon[q]\mapsto E_{q}G=G^{q+1}$.

Algebraic $K$-theory preserves equivalences and commutes with finite products, so the cosimplicial spectra $K(Y^{\bullet})$, $K(F(E_{\bullet}G_{+},B)^{G})$ and $F(E_{\bullet}G_{+},K(B))^{G}$ are level equivalent. Hence the homotopy limit of $K(Y^{\bullet})$ is equivalent to the homotopy fixed point spectrum $F(EG_{+},K(B))^{G}=K(B)^{hG}$, where $EG=|E_{\bullet}G|$. (The homotopy type of $K(B)^{hG}$ only depends on the homotopy type of $K(B)$ as a spectrum with $G$-action, and the $G$-action is determined by functoriality.) The canonical map $\eta\colon K(A)\to K(B)^{hG}$ is not in general an equivalence, so algebraic $K$-theory does not in general satisfy descent along Galois extensions $A\to B$. However, a recent result of Clausen, Mathew, Naumann and Noel [CMNN]*Thm. 1.7 shows that, after any “periodic localization”, algebraic $K$-theory satisfies descent along all maps $A\to B$ of commutative $\mathbb{S}$-algebras for which $B$ is dualizable as an $A$-module and the restriction map $K_{0}(B)\to K_{0}(A)$ is rationally surjective. Again, the Galois condition plays no role for their proof. See also [Tho85] for the corresponding result for Bott localized algebraic $K$-theory of commutative rings (or schemes).

4.5. Descent along $\mathbb{S}\to X(n)$

There is a sequence of $E_{2}$ ring spectra $X(n)$ interpolating between $\mathbb{S}$ and $MU$, see [DHS88]. Recall that $MU=BU^{\gamma}$ is the Thom spectrum of a virtual vector bundle $\gamma$ over $BU$. For each $n\geq 2$, the Thom spectrum $X(n)=\Omega SU(n)^{\gamma}$, of the pullback of $\gamma$ over the double loop map $\Omega SU(n)\to\Omega SU\simeq BU$, is an $E_{2}$ ring spectrum with $1$-connected unit map $\mathbb{S}\to X(n)$. There are natural maps of $E_{2}$ ring spectra

[TABLE]

connecting these examples to those previously discussed. We identify $H_{*}(X(n))\cong H_{*}(\Omega SU(n))$ with the subalgebra $\mathbb{Z}[b_{1},\dots,b_{n-1}]$ of $H_{*}(MU)\cong H_{*}(BU)=\mathbb{Z}[b_{k}\mid k\geq 1]$.

By Theorem 3.14, the functors $K$, ${THH}$ and ${TC}$ satisfy descent along $\mathbb{S}\to X(n)$ for each $n\geq 2$. There are Thom equivalences $X(n)\wedge X(n)\simeq X(n)\wedge\Omega SU(n)_{+}$, so the study of $K(\mathbb{S})$ by descent along $\mathbb{S}\to X(n)$ leads to the study of the algebraic $K$-theory of $X(n)\wedge(\Omega SU(n))^{q}_{+}$ for $q\geq 0$, and similarly for $K(\mathbb{S}[\Gamma])$, ${THH}$ and ${TC}$. The $E_{2}$ ring spectra $X(n)$ are closer to $\mathbb{S}$ than $MU$, hence $K(X(n))$ can yield finer information about $K(\mathbb{S})$ than $K(MU)$ does. However, like in the case of $\mathbb{S}$, the homotopy groups of $X(n)$ are not explicitly known, so a direct analysis of $\pi_{*}{THH}(X(n))$ and $\pi_{*}{TC}(X(n))$ may be less feasible than in the case of $MU$.

4.6. Trace methods

The cyclotomic trace map $\operatorname{trc}\colon K(B)\to{TC}(B;p)$ introduced in [BHM93], in conjunction with the relative equivalence theorem from [Dun97], is the main method available for calculating the algebraic $K$-groups of connective $\mathbb{S}$-algebras other than (simplicial) rings. In the case of the sphere spectrum, ${TC}(\mathbb{S};p)$ is $p$-adically equivalent to $\mathbb{S}\vee\Sigma\mathbb{C}P^{\infty}_{-1}$, so calculations of $\pi_{*}K(\mathbb{S})$ are possible in a moderate range of degrees [Rog02], [Rog03] (see also recent work of Blumberg–Mandell [BM] at irregular primes). Nonetheless, complete calculations are at least as hard as those for $\pi_{*}(\mathbb{S})$, hence appear to be out of reach.

The difficulty of understanding the stable homotopy groups of spheres can be formulated as the difficulty of understanding the Adams–Novikov spectral sequence [Nov67]

[TABLE]

i.e., to understand the descent spectral sequence

[TABLE]

associated with the cosimplicial commutative $\mathbb{S}$-algebra $Y^{\bullet}=Y^{\bullet}_{\mathbb{S}}(\mathbb{S},MU)$, with $Y^{q}=MU\wedge\dots\wedge MU\simeq MU\wedge BU^{q}_{+}$. An advantage of this approach is that chromatic phenomena in $\pi_{*}(\mathbb{S})$ are more readily visible at the $E_{2}$-term of the Adams–Novikov spectral sequence.

By analogy, the difficulty of understanding $\pi_{*}K(\mathbb{S})$ and $\pi_{*}{TC}(\mathbb{S};p)$ can be separated into two parts: first that of understanding the cosimplicial objects $[q]\mapsto\pi_{*}K(Y^{q})$ and $[q]\mapsto\pi_{*}{TC}(Y^{q};p)$, and secondly that of understanding the behavior of the descent spectral sequences

[TABLE]

and

[TABLE]

The first aim of understanding $\pi_{*}K(MU)$ and $\pi_{*}{TC}(MU;p)$, corresponding to $q=0$, then plays an analogous role to that of understanding $MU_{*}=\pi_{*}(MU)$. An optimist may seek to discern chromatic phenomena in $\pi_{*}K(\mathbb{S})$ and $\pi_{*}{TC}(\mathbb{S};p)$ at the level of these $E_{2}$-terms.

4.7. Descent for ${THH}$

As an illustration, Theorem 3.9 for ${THH}$ at $\mathbb{S}$ along $\mathbb{S}\to MU$ gives a strongly convergent descent spectral sequence

[TABLE]

There is an equivalence ${THH}(MU)\simeq MU\wedge SU_{+}$, by [BCS10]*Cor. 1.1, and the Atiyah–Hirzebruch spectral sequence $E^{2}_{*,*}=H_{*}(SU;MU_{*})\Longrightarrow\pi_{*}(MU\wedge SU_{+})$ collapses at $E^{2}$ to give the algebra isomorphism

[TABLE]

of [MS93]*Rmk. 4.3. Here $L=MU_{*}=\mathbb{Z}[x_{k}\mid k\geq 1]$ is the Lazard ring, $E=E(e_{k}\mid k\geq 1)$ is the exterior algebra over $\mathbb{Z}$ on a sequence of generators $e_{k}$, with $|e_{k}|=2k+1$, and $LE=L\otimes E$ is shorthand notation for their tensor product. Similarly,

[TABLE]

is flat as a left $MU_{*}$-module, where $LB=MU_{*}MU$ and $B=H_{*}(MU)\cong\mathbb{Z}[b_{k}\mid k\geq 1]$, and

[TABLE]

In general, $\pi_{*}{THH}(Y^{q})\cong LE\otimes(BE)^{\otimes q}$, and $[q]\mapsto\pi_{*}{THH}(Y^{q})$ is the cobar construction associated to the split Hopf algebroid $(LE,LE\otimes BE)$, see [Rav86]*App. A1. Hence the descent spectral sequence for ${THH}$ takes the form

[TABLE]

The unit inclusion $\mathbb{Z}\to E$ induces an equivalence

[TABLE]

of Hopf algebroids, which induces an isomorphism from the Adams–Novikov spectral sequence (7) to the descent spectral sequence (8). This is of course compatible with the identity $\mathbb{S}={THH}(\mathbb{S})$, and shows, somewhat tautologically, that the descent spectral sequence for ${THH}$ at $\mathbb{S}$ along $\mathbb{S}\to MU$ has an $E_{2}$-term that is susceptible to chromatic analysis along the lines of [MRW77] and [Rav86]*Ch. 5.

By contrast, descent for ${THH}$ at $\mathbb{S}$ along $\mathbb{S}\to H\mathbb{Z}$ leads to the study of $\pi_{*}{THH}(H\mathbb{Z}\wedge\dots\wedge H\mathbb{Z})\cong\pi_{*}({THH}(\mathbb{Z})\wedge\dots\wedge{THH}(\mathbb{Z}))$, with $(q+1)$ copies of $H\mathbb{Z}$ or ${THH}(\mathbb{Z})$, and while $\pi_{*}{THH}(\mathbb{Z})$ is explicitly known [BM94]*§5, the term $\pi_{*}({THH}(\mathbb{Z})\wedge{THH}(\mathbb{Z}))$ is not flat over $\pi_{*}{THH}(\mathbb{Z})$, and there is no description of the resulting $E_{1}$-term as the cobar complex of a Hopf algebroid. If we instead were to study the functor $F(A)={THH}(A)\wedge S/p$, where $S/p$ is the mod $p$ Moore spectrum, then $\pi_{*}F(H\mathbb{Z}\wedge H\mathbb{Z})=\pi_{*}({THH}(\mathbb{Z})\wedge{THH}(\mathbb{Z});\mathbb{Z}/p)$ is a flat Hopf algebroid over $\pi_{*}F(H\mathbb{Z})=\pi_{*}({THH}(\mathbb{Z});\mathbb{Z}/p)$. The associated cobar complex is isomorphic to the $E_{1}$-term of the descent spectral sequence for $F$ at $\mathbb{S}$ along $\mathbb{S}\to\mathbb{Z}$, which in turn is isomorphic to the ${THH}(\mathbb{Z})$-based Adams spectral sequence for $S/p$, see [MNN17]*Prop. 2.14. The canonical ${THH}(\mathbb{Z})$-based tower for $S/p$ is also a mod $p$ Adams tower for $S/p$, so these spectral sequences are therefore isomorphic to the mod $p$ Adams spectral sequence for $S/p$, from the $E_{2}$-term and onwards.

4.8. Fixed points of ${THH}$

The next steps toward analyzing descent for ${TC}$ along $\mathbb{S}\to MU$ would be to study descent for the fixed point functor ${THH}(-)^{C}$ along $\mathbb{S}\to MU$, for each $C=C_{p^{m}}$. Let ${THH}(A)^{tC}=[\widetilde{EC}\wedge F(EC_{+},{THH}(A))]^{C}$ denote the $C$-Tate construction on ${THH}(A)$, i.e., the $C$-fixed points of the Tate $C$-spectrum of [GM95]*pp. 3-4. The comparison maps

[TABLE]

are known to be equivalences after $p$-completion, by [LNR11]*Thm. 1.1 in the case $m=1$, hence also for the cases $m\geq 2$ by [Tsa98]*Thm. 2.4 or [BBLNR14]*Thm. 2.8. It is very likely that the same methods will apply when $Y^{0}=MU$ is replaced by $Y^{q}=MU\wedge\dots\wedge MU$ for $q\geq 1$. If so, the $p$-completed spectral sequence

[TABLE]

can equally well be analyzed using $C_{p^{m}}$-homotopy fixed points or $C_{p^{m+1}}$-Tate constructions in place of $C_{p^{m}}$-fixed points. A first step in this direction would be to determine the differential structure of the $C_{p}$-Tate spectral sequence

[TABLE]

By truncation to the second quadrant, this would determine the $C_{p}$-homotopy fixed point spectral sequence converging to $\pi_{*}{THH}(MU)^{hC_{p}}$, hence also $\pi_{*}{THH}(MU)^{C_{p}}$ after $p$-completion.

The comparison maps $\Gamma_{m}\colon{THH}(\mathbb{S})^{C_{p^{m}}}\to{THH}(\mathbb{S})^{hC_{p^{m}}}$ and $\hat{\Gamma}_{m}\colon{THH}(\mathbb{S})^{C_{p^{m-1}}}\to{THH}(\mathbb{S})^{tC_{p^{m}}}$ are also $p$-adic equivalences, by the proven Segal conjecture [Car84], but even in the case $m=1$ the differential structure of the $C_{p}$-Tate spectral sequence converging to $\pi_{*}{THH}(\mathbb{S})^{tC_{p}}\cong\pi_{*}(\mathbb{S})$ is not known [Ada74]. In the case of ${THH}(\mathbb{Z})$ the maps $\Gamma_{m}$ and $\hat{\Gamma}_{m}$ are not quite equivalences, but they do induce isomorphisms in homotopy with mod $p$ coefficients in non-negative degrees. The structure of the $C_{p^{m}}$-Tate spectral sequence converging to $\pi_{*}({THH}(\mathbb{Z})^{tC_{p^{m}}};\mathbb{Z}/p)$ is known [BM95], [Rog99], but these calculations appear to be difficult to extend to the case of ${THH}(H\mathbb{Z}\wedge\dots\wedge H\mathbb{Z})$.

4.9. Topological periodic homology

The limiting maps

[TABLE]

are also equivalences after $p$-completion, cf. [BM95]*(2.11) and [AR02]*Thm. 5.7. Thus the $E^{2}$-terms of the $\mathbb{T}$-homotopy fixed point and $\mathbb{T}$-Tate spectral sequences

[TABLE]

are known, and both converge to $\pi_{*}{TF}(MU;p)$ after $p$-completion. Note that ${TP}(MU)={THH}(MU)^{t\mathbb{T}}$ is the arithmetically interesting topological periodic homology studied by Hesselholt [Hes], also known as periodic topological cyclic homology or topological de Rham homology. As in the case of ${TF}(\mathbb{S};p)$, the inverse Frobenius operator $\varphi^{-1}=R\colon{TF}(MU;p)\to{TF}(MU;p)$ extends to ${TP}(MU)$ after $p$-completion, without any further localization.

4.10. Rational analysis

The rational algebraic $K$-groups of $MU$, i.e., $\pi_{*}K(MU)\otimes\mathbb{Q}$, were determined in [AR12]*Thm. 4.2 by using Goodwillie’s theorem [Goo86]. The Poincaré series is

[TABLE]

where

[TABLE]

A similar result holds for each $\pi_{*}K(Y^{q})\otimes\mathbb{Q}$, where $Y^{q}=MU\wedge\dots\wedge MU$, with $(q+1)$ copies of $MU$. Since $\pi_{0}Y^{q}=\mathbb{Z}$ and $Y^{q}$ is connective and of finite type, we know by an easy generalization of [Dwy80]*Prop. 1.2 that $E_{1}^{s,t}=\pi_{t}K(Y^{s})$ is a finitely generated abelian group in each bidegree $(s,t)$. By the following result $E_{2}^{s,t}=\pi^{s}\pi_{t}K(Y^{\bullet})$ is in fact finite in each bidegree, except at the edge $s=0$, where $E_{2}^{0,t}$ is rationally isomorphic to $\pi_{t}K(\mathbb{Z})$.

Proposition 4.1**.**

The rationalized descent spectral sequence

[TABLE]

for algebraic $K$-theory at $\mathbb{S}$ along $\mathbb{S}\to MU$, collapses at the $E_{2}$-term to the edge $s=0$.

Proof.

The vanishing line and strong convergence of the descent spectral sequence $E_{2}^{s,t}\Longrightarrow_{s}\pi_{t-s}K(\mathbb{S})$ from Theorem 3.9 imply the same vanishing line and strong convergence for the rationalized spectral sequence.

The unit map $\mathbb{S}\to Y^{q}$ and the zeroth Postnikov section $Y^{q}\to H\mathbb{Z}$ induce maps $K(\mathbb{S})\to K(Y^{\bullet})\to K(\mathbb{Z})$ that, after rationalization, split off a copy of $K(\mathbb{S})$ from $K(Y^{\bullet})$. It remains to prove that the remainder of the rationalized spectral sequence, associated to the cosimplicial spectrum with the homotopy fiber of $K(Y^{q})\to K(\mathbb{Z})$ in codegree $q$, collapses to zero at the $E_{2}$-term.

The Hurewicz homomorphism $MU_{*}=\pi_{*}(MU)\to H_{*}(MU)\cong B$ is a rational equivalence, so $\pi_{*}{THH}(MU)$ is rationally isomorphic to $HH_{*}(B)=BE$, where $B=\mathbb{Z}[b_{k}\mid k\geq 1]$ and $E=E(e_{k}\mid k\geq 1)$. Connes’ $B$-operator on $HH_{*}(B)$ corresponds to the suspension operator $\sigma$, which is the differential and derivation given by $\sigma(b_{k})=e_{k}$ for each $k\geq 1$. Hence the rationalized de Rham homology $H^{dR}_{*}(B)\otimes\mathbb{Q}=H_{*}(BE,\sigma)\otimes\mathbb{Q}=\mathbb{Q}$ is trivial in positive degrees. By [AR12]*Cor. 2.4, which is a consequence of [Goo86]*II.3.4, it follows that the trace map $K(MU)\to{THH}(MU)$ identifies the kernel of $\pi_{*}K(MU)\to\pi_{*}K(\mathbb{Z})$ with the image of $\sigma\colon BE\to BE$, after rationalization.

Similarly, $\pi_{*}{THH}(Y^{\bullet})$ is rationally isomorphic to the cosimplicial resolution $[q]\mapsto HH_{*}(B^{\otimes q+1})\cong(BE)^{\otimes q+1}$ associated to $\eta\colon\mathbb{Z}\to BE$. By [AR12]*Cor. 2.4 again, the trace map identifies the kernel of $\pi_{*}K(Y^{q})\to\pi_{*}K(\mathbb{Z})$ with the image of $\sigma\colon(BE)^{\otimes q+1}\to(BE)^{\otimes q+1}$, after rationalization. Let $(C^{*},\delta)$ and $(D^{*},\delta)$ be the associated cochain complexes, with $C^{q}=(BE)^{\otimes q+1}$ and $D^{q}=\operatorname{im}(\sigma)\subset C^{q}$. The $E_{2}$-term of the remainder of the descent spectral sequence is given by the cohomology of $(D^{*},\delta)$, after rationalization.

The augmentation $\epsilon\colon BE\to\mathbb{Z}$ induces a cochain contraction $\epsilon\otimes id_{BE}^{\otimes q}$ of $\eta\colon\mathbb{Z}\to C^{*}$, mapping $x_{0}\otimes x_{1}\otimes\dots\otimes x_{q}\in C^{q}$ to $\epsilon(x_{0})x_{1}\otimes\dots\otimes x_{q}\in C^{q-1}$. It restricts to a contraction of $0\to D^{*}$, since $\epsilon\otimes id_{BE}^{\otimes q}$ commutes with $\sigma$. Hence the cohomology of $(D^{*},\delta)$ is zero, as claimed. ∎