# An HKR theorem for factorization homology

Hari Rau-Murthy

###### Abstract.

We prove a Hochschild-Kostant-Rosenberg theorem (“the HKR theorem”) which
computes the factorization homology of certain smooth commutative ring
spectra. In doing so, we fix and generalize a THH computation which was first
conceived as the brainchild of McCarthy-Minasian. A direct application of our
revised HKR theorem is the higher THH of rational $KU$.

##  1\. Introduction

This paper concerns the following classical theorem due to Hochschild, Kostant
and Rosenberg (HKR) [HKR62]. When $R\to A$ is a smooth map of commutative
rings, we have an isomorphism

(1.1) $\mathrm{HH}_{*}(A/R)\to\Omega^{*}(A/R)$

from the Hochschild homology relative to $R$, to the Kähler differential forms
of $A$ relative to $R$.

The idea for generalizing (1.1) to commutative ring spectra is due to
McCarthy-Minasian in their main theorem of [MM03]. In particular, they develop
certain generalizations of a smooth map of classical rings to commutative
rings. They state that

(1.2) $\mathrm{THH}(A/R)\simeq P_{A}(\Sigma\mathrm{TAQ}(A/R))$

is an equivalence for suitable $A$. Here, $P_{A}(-)$ is notation for the free
commutative $A$-algebra on the $A$-module and $\mathrm{TAQ}(A/R)$ denotes the
unaugmented Topological Andre Quillen homology of $A$ relative to $R$,
developed by [Bas99], and discussed in Section 2.3.

We give assumptions under which (1.2) is true. While the precise assumptions
on $R$ and $A$, including the smoothness assumption, and the proof of (1.2)
have flaws in [MM03], McCarthy-Minasian provides the foundations we build on
in this present paper.

The equivalence (1.2) is analogous to the classical HKR theorem (1.1):
informally, the left hand side of (1.2) is Hochschild homology and the right
hand side is the graded commutative algebra over Kähler 1-forms concentrated
in degree 1, which is the Kähler differential forms.

######  Remark 1.1.

We remark that the equivalence (1.2) only implies the classical HKR theorem
(1.1) for $\mathbb{Q}$ algebras. A free commutative algebra in the classical
setting is only rationally a free commutative algebra in spectra, owing to the
homology of the symmetric group. Additionally, the topological Andre Quillen
homology, $\mathrm{TAQ}$, only agrees rationally with algebraic Andre Quillen
homology. This problem is ubiquitous when switching from the setting of
spectral algebraic geometry and derived/simplicial algebraic geometry as
developed by [Lur18, Chapter 25].

Among the errors in [MM03], the first two of which having been noted by
Antieau, Toen and Vesozzi in [AV20][TV08], are the following:

  1. (1)

MccCarthy-Minasian argue that (1.2) is true etale locally and hence it is true
globally. They fail, however, to produce a global map which induces the etale
local equivalences.

  2. (2)

[MM03] MccCarthy-Minasian assert that Theorem 1.2 implies the classical HKR
theorem in all characteristics.

  3. (3)

[MM03] MccCarthy-Minasian assume that two notions of the completion of a
module with respect to an ideal, with one involving smash products over
$A\otimes_{R}S^{1}$, and the other involving smash products over $A$, are the
same.

Actually, the correct hypotheses of the main theorem of [MM03] were stated in
McCarthy-Minasian’s subsequent paper [MM04]. Although [MM04] refers to the
flawed [MM03] for details of a proof, the statements in [MM04] are correct;
moreover [MM04] gives valuable insight, which we use, on how to correct the
first error mentioned above of [MM03].

###  1.1. Results

We aim to fix and generalize (1.2). The smoothness hypothesis we will need is
as follows. Classically a smooth map of rings is one which is locally etale
over an affine space. One of the many ways etale is interpreted in the context
of commutative ring spectra is as follows. For a map $A\to B$ of commutative
rings, $\mathrm{TAQ}(B/A)$ is Topological Andre Quillen homology - a derived
version of differential 1-forms. We say that $A\to B$ is formally TAQ-etale
when $\mathrm{TAQ}(B/A)\simeq 0$. The corresponding notion of smoothness is
formal TAQ smoothness. For details, see Section 5.

######  Theorem 1.2.

Let $R\to A$ be a formally $\mathrm{TAQ}$-smooth map in the category of
connective commutative rings and let $M$ be the realization of a pointed,
connected simplicial set.

Then the factorization homology is given by

(1.3) $\int_{M}(A/R):=A\otimes_{R}M\simeq P_{A}(M\wedge\mathrm{TAQ}(A/R)).$

We will call this theorem ‘the generalized HKR theorem.’ Here $-\otimes_{R}M$
is the Loday functor of [MSV97] discussed in Section 2.1. In the case
$M=S^{1}$ we recover Theorem 1.2.

The Loday functor, $-\otimes_{R}S^{n}$, gives a different notion of smoothness
which we call $S^{n}$-THH smooth (see Section 5).

In this setting, we also prove the following:

######  Theorem 1.3.

Let $R\to A$ be a formally $S^{n}$-$\mathrm{THH}$-smooth map, and let $M$ be
n-connected. Then the conclusion 1.3 of the generalized HKR theorem is
satisfied.

Theorem 1.3 depends on a certain etale descent lemma (Theorem 3.7) which was
independently proven in [LR22, Proposition 2.11] and proven in a special case
in [RSV22, Theorem 7.5].

Theorem 1.2 also includes as special cases $M=\mathbb{T}^{n}$, known as
iterated THH (see [CDD11][Sto20][BHL+19][HKL+22]) and which accesses iterated
K-theory important for the redshift conjecture as noted in [HKL+22][CDD11].
Also included is the case $M={S^{n}}$ which is a topological version of higher
Hochschild homology (see [Pir00][BHL+19]). These special cases were what
initially motivated us to prove Theorem 1.2.

In [Sto20] and [RSV22], Stonek et al. prove that Theorem 1.2 is true for
$R=\mathbb{S}$, $A=KU$, and any connected $M$ by direct computation. Since
[MM03] was released, several authors have proven etale descent results for THH
[Mat17][CM21][Rog08] and have proven etale descent results for higher THH
[Sto20][BHL+19][RSV22].

###  1.2. Organization of the paper

We begin with a background section where we review the definition of
$A\otimes_{R}M$ (Section 2.1) and facts about nonunital commutative algebras
(Section 2.7). In Section 2.4, we define $\mathrm{TAQ}$, the corresponding
notion of etale, its generalization $M$-THH-etale, and other notions of etale.
In Section 3, we state and prove the etale descent lemmas which we need to
prove our generalized HKR theorems. In Section 4, we show that the generalized
HKR theorem is satisfied for free commutative algebras. In Section 5 we
develop a generalized notion of etale covering for commutative ring spectra.
In Section 6, we use this generalized notion of a covering to prove the
generalized HKR Theorems 1.2 and 1.3.

###  1.3. Acknowledgements

We gratefully acknowledge helpful conversations with Mark Behrens, Inbar
Klang, Nikolai Konovolov, Andrew Blumberg, Ayelet Lindenstrauss, Connor Malin,
Lorenzo Riva, Sihao Ma, Claudiu Raicu, John Rognes, and Nat Stapleton. We
acknowledge support from the eCHT graduate research fellowship (NSF Grant
DMS-1547292) for the academic year of 2022-2023 and support from the NSF Grant
DMS-1547292 during the academic year of 2021-2022.

##  2\. Background

###  2.1. The Loday construction

Let $A$ be a commutative $R$-algebra, and let $M$ be the realization of a
pointed simplicial set $M_{\bullet}$. We will define the commutative
$R$-algebra $A\otimes_{R}M$, known as the Loday construction. The foundations
for this construction were first established in [MSV97] and the terminology is
due to [HHL+18]. This construction is also quite standard and general and it
can be found in any discussion on homotopy colimits.

$A\otimes_{R}M$ is formally defined as the $R$-algebra $\mathrm{colim}_{M}A$
where $M$ is viewed as an $\infty$-category and $A$ is taken to be the
constant functor in $R$-algebras; we have not (yet) used the basepoint of $M$.
By the definition of $A\otimes_{R}M$ as a colimit in an $\infty-$category, we
have that up to equivalence, it only depends on the homotopy type of $M$, and
that $A\otimes_{R}-$ commutes with colimits of spaces. This will be used
throughout the paper, and also gives us an explicit model for the Loday
construction.

Explicitly, as shown below, $A\otimes_{R}M$ is the simplicial set which has a
smash copy of $A$ for each simplex of $M_{\bullet}$. The justification is as
follows: we have that

$\displaystyle A\otimes_{R}M\simeq A\otimes|M_{\bullet}|$ $\displaystyle\simeq
A\otimes_{R}\mathrm{colim}_{\bullet\in\Delta}M_{\bullet}\simeq\mathrm{colim}_{\bullet\in\Delta}A\otimes_{R}M_{\bullet}$
$\displaystyle\simeq|A^{\wedge_{R}M_{\bullet}}|.$

The explicit simplicial set structure on $A^{\wedge_{R}M_{\bullet}}$ comes
from the observation that
$A\otimes_{R}(\mathrm{pt}\sqcup\mathrm{pt}\to\mathrm{pt})$ is identified with
the fold or multiplication map $A\wedge_{R}A\to A$;
$A\otimes_{R}(\emptyset\to\mathrm{pt})$ is identified with the map from the
initial object of $R-\mathrm{Alg}$ to $A$, or the unit map.

This classical Loday construction is a special case of a more general Loday
construction $c\otimes v$ where $C$ is an $\infty$-category enriched in a
monoidal $\infty$-category $V$, and $c$ and $v$ are objects of $C$ and $V$
respectively. We are interested only in the cases where $C=R-\mathrm{Alg}$,
$V=\mathrm{Top}$, and $C=R-\mathrm{Mod}$, $V=\mathrm{Top}$. The construction
is as follows - consider the ($\infty$-)functor

$[c,-]:C_{\mathrm{un}}\to V$

from the underlying $\infty$-category of $C$ to $V$. When the unique left
adjoint exists, it is denoted by $c\otimes-$; we will occasionally write
$c\otimes^{V}_{C}v$ when it is necessary to emphasize $C$ and $V$, or
$c\otimes_{C}v$ in the special case where $V=\mathrm{Top}$. When $C$ is itself
a presentable (unenriched) $\infty$-category and $V=\mathrm{Top}$, the left
adjoint exists because it is given by $v\mapsto\mathrm{colim}_{M}c$; this
expression exists because $C$ is cococomplete. In particular, when
$C=R-\mathrm{Alg}$, $V=\mathrm{Top}$, $c\otimes v$ is the classical Loday
construction above, and using our above notation for rings and spaces, we have
the following useful universal property:

(2.1)
$[A\otimes_{R}M,B]_{R-\mathrm{Alg}}\simeq[M,[A,B]_{R-\mathrm{Alg}}]_{\mathrm{Top}}.$

This abstract framework gives rise to the following concrete application which
we will need. When $C=R-\mathrm{Mod}$ and $V=\mathrm{Top}$,
$c\otimes^{\mathrm{Top}}_{C}v$ is given by $c\wedge(v_{+})$. This, for
example, may be seen as a consequence of the fact that for
$w\in\mathrm{ob}(\mathrm{Top}_{+})$ we have that
$c\otimes^{\mathrm{Top}_{+}}_{C}w\simeq c\wedge w$ and we also have the
existence of the free functor from spaces to pointed spaces. Regarding the
free functor $P_{R}:R-\mathrm{Mod}\to R-\mathrm{Alg}$, we also have by the
universal property of the general Loday constructions that

(2.2) $P_{R}(X\otimes^{\mathrm{Top}}_{R-\mathrm{Mod}}M)\simeq
P_{R}(X)\otimes_{R}^{\mathrm{Top}}M$

for any $R$-module $X$.

Lastly, the Loday construction is important to us as we have the equivalence
$A\otimes_{R}S^{1}\simeq\mathrm{THH}(A/R)$ where the right hand side is
defined via a cyclic bar construction [MSV97]. This follows from the explicit
model for $A\otimes_{R}-$ and the equivalence
$S^{1}_{\bullet}\simeq\mathbb{Z}/(\bullet+1)$. The n-fold iterated THH is
consequently

$(A\otimes_{R}S^{1})\otimes_{R}S^{1}\cdots\simeq
A\otimes_{R}(S^{1}\times\cdots)$

where the last equality follows from $(A^{\wedge_{R}n})^{\wedge_{R}m}\simeq
A^{\wedge_{R}nm}$.

We have that $\mathrm{THH}(A/R)$ is an $A$-algebra via the map $A\to A\otimes
S^{1}$ induced from $0\to S^{1}$. More generally, we would like to understand
the $A$-algebra structure on $A\otimes M$ for pointed $M$ (and the tensor
product $-\otimes^{\mathrm{Top}}_{R-\mathrm{Alg}}-)$), and we will need the
following general fact about morphisms in under categories.

######  Lemma 2.1 ([Lur09, Lemma 5.5.5.12]).

Let $C$ be an $\infty$ category, $c\in\mathrm{ob}(C)$, and ${}_{c\setminus}C$
the undercategory for $c$. Let $\underset{\sim}{d}:c\to d$ and
$\underset{\sim}{e}:c\to e$ be objects in ${}_{c\setminus}C$. We then have a
pullback square

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An immediate application of Lemma 2.1 and the universal property of the Loday
construction (2.1) is the following change of base ring formula:

######  Proposition 2.2.

Let $R\to A\to B$ comprise of algebra maps. Then we have the equivalence

$(B\otimes_{R}M)\wedge_{A\otimes_{R}M}A\simeq B\otimes_{A}M$

###### Proof.

For any $A$-algebra $C$, we have the following equivalences:

$[(B\otimes_{R}M)\wedge_{A\otimes_{R}M}A,C]_{A-\mathrm{Alg}}\simeq[B\otimes_{R}M,C]_{A\otimes_{R}M-\mathrm{Alg}}$
$\simeq\mathrm{fib}([B\otimes_{R}M,C]_{R-\mathrm{Alg}}\to[A\otimes_{R}M,C]_{R-\mathrm{Alg}})$
$\simeq\mathrm{fib}([M,[B,C]_{R-\mathrm{Alg}]_{\mathrm{Top}}}\to[M,[A,C]_{R-\mathrm{Alg}}])$
$\simeq[M,\mathrm{fib}([B,C]_{R-\mathrm{Alg}}\to[A,C]_{R-\mathrm{Alg}})]_{\mathrm{Top}})$
$\simeq[M,[B,C]_{A-\mathrm{Alg}}]_{\mathrm{Top}}\simeq[B\otimes_{A}M,C]_{A-\mathrm{Alg}}$

By the Yoneda lemma we are done ∎

where the $A\otimes_{R}M$ module structure on $A$ is induced from the map
$M\to\mathrm{pt}$.

The dual statement to Lemma 2.1 is following lemma.

######  Lemma 2.3.

Let $C$ be an $\infty$ category, $c\in\mathrm{ob}(C)$, and $C_{/c}$ the
overcategory for $c$. Let $\widetilde{d}:d\to c$ and $\widetilde{e}:e\to c$ be
objects in $C_{/c}$. We then have a pullback square

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######  Lemma 2.4.

There is a canonical equivalence

$[c\otimes_{C}M,\underset{\sim}{d}]_{{}_{c\setminus}C}\simeq[M,[c,d]_{C}]_{{\mathrm{Top}_{+}}}.$

Here, we view $c\otimes_{C}M$ as an object of ${}_{c\setminus}C$ via the map
$c\otimes_{C}\mathrm{pt}\to c\otimes_{C}M$.

###### Proof.

By Lemma 2.1 and by (2.1), we have the following equivalences.

$\displaystyle[c\otimes_{C}M,\underset{\sim}{d}]_{{}_{c\setminus}C}\simeq\mathrm{fib}([M,[c,d]_{C}]_{\mathrm{Top}}\to[\mathrm{pt},[c,d]_{C}]_{\mathrm{Top}})$
$\displaystyle\simeq[M,[c,d]_{C}]_{\mathrm{Top}_{+}}.$

∎

######  Lemma 2.5.

Dually, we have the canonical equivalence

$\widetilde{c}\otimes_{C_{/c}}M\simeq c\otimes_{C}M.$

Here, the right hand side is interpreted as an object of $C_{/c}$ via the map
$M\to\mathrm{pt}$, and $\widetilde{c}$ is notation for the object in $C_{/c}$
given by $c\xrightarrow{\mathrm{Id}}c$.

###### Proof.

By Lemma 2.3 and by (2.1), we have the following equivalences.

$\displaystyle[c\otimes_{C}M,\widetilde{d}]_{C_{/c}}\simeq\mathrm{fib}([M,[c,d]_{C}]_{\mathrm{Top}}\to[M,[c,c]_{C}]_{\mathrm{Top}})$
$\displaystyle\simeq[M,[\widetilde{c},\widetilde{d}]_{C_{/c}}]_{\mathrm{Top}}\simeq[c\otimes_{C_{/c}}M,\widetilde{d}]_{\mathrm{Top}}.$

∎

As an immediate corollary of Lemma 2.4, we obtain a description for the
$A$-algebra structure on $A\otimes_{R}M$.

######  Proposition 2.6.

The $A$-algebra structure on $A\otimes_{R}M$ may be understood via the
following adjunction

$[A\otimes_{R}M,B]_{A-\mathrm{Alg}}\simeq[M,[A,B]_{R-\mathrm{Alg}}]_{\mathrm{Top}_{+}}$

where $[A,B]_{R-\mathrm{Alg}}$ has the base point given by the unique
$A$-algebra map to $B$.

Lastly, for an $n$ manifold $M$, we have an equivalence

(2.3) $\int_{M}A\to A\otimes_{\mathbb{S}}M$

from the $E_{n}$ factorization homology to the Loday construction relative to
the sphere spectrum [AF15, Proposition 5.1].

The proof is simple: since $A\otimes_{\mathbb{S}}-$ commutes with colimits of
spaces, it is an excisive functor. $E_{n}$ factorization homology is excisive
for embedded framed $n$-disks by definition. Constructing the map (2.3) is the
same as constructing the map for $M=\mathbb{R}^{n}$ via

$\int_{\mathbb{R}^{n}}A\simeq A\simeq A\otimes_{\mathbb{S}}\mathrm{pt}\simeq
A\otimes_{\mathbb{S}}\mathbb{R}^{n}.$

Since the above map is an equivalence, (2.3) is an equivalence.

In particular we have an equivalence

$\int_{S^{n}}A\xrightarrow{\simeq}A\otimes_{\mathbb{S}}S^{n}$

from $E_{n}$ factorization homology over $S^{n}$ to $n$-higher THH.

###  2.2. Nonunital Commutative Algebras

We give the basic properties of the $\infty$-category of nonunital commutative
algebras, after [Lur17, Chapter 3] and [Bas99, Section 1]. This will be used
in the next section to define topological Andre Quillen homology, and to
define the completions used throughout this paper.

Let $A$ be a commutative ring.

######  Definition 2.7.

The category $A$-NUCA of $A$-nonunital commutative algebras is defined as the
$\infty$-category of symmetric monoidal functors of $\infty$-categories

$(\mathrm{Fin}_{\mathrm{surj}},\sqcup)\to\mathrm{Spectra}$

from finite sets with surjections to spectra.

To illuminate this definition, we compare to the definition of a commutative
algebra. The unit map of a commutative algebra is induced from
$[0]=\emptyset\to[1]\in\mathrm{Fin}$; this is exactly the structure we remove
to obtain a nonunital commutative algebra as in Definition 2.7.

A related construction is the category of augmented $A$ algebras, defined by
the slice category $A-\mathrm{Alg}/A$. Explicitly, an object of this category
is an $A$ algebra $U$ with a module equivalence $U\simeq A\vee I$ for some $A$
module $I$. The commutative algebra structure on $A$ gives rise to an
$A-\mathrm{NUCA}$ structure on $I$. The functor which assigns to $A\vee I$,
the $A-\mathrm{NUCA}$ $I$ is referred to as the augmentation ideal. This
functor gives rise to the following equivalence [Lur17, Proposition 5.4.4.10]
of $\infty$-categories

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0.0pt\hbox{{}{}{}{{}{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
90.78278pt\raise-5.49818pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}{}{{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
48.38889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 78.38889pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left.A-\mathrm{Alg}\middle/A\right.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}{\hbox{\kern
37.5139pt\raise 12.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\text{aug.
ideal}}$}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{}{{}{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
24.38412pt\raise 8.58209pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}{}{{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

Thus both adjoints are simultaneously left and right adjoints.

An important augmented $A$-algebra is $A\otimes_{R}M$, with the augmentation
induced from the map $M\to\mathrm{pt}$. Denote the augmentation ideal by
$I_{M}^{A}$. For instance, $I_{S^{0}}^{A}=\mathrm{fib}(A\wedge_{R}A\to A)$ is
the fiber of the multiplication map. We remark that this notation does not
make explicit the base ring $R$; throughout this paper, the base ring will
change and be understood from the context.

The augmented algebra structure on $A\otimes_{R}M$ is best understood via the
following proposition.

######  Proposition 2.8.

For any augmented $A$-algebra $U$, we have the following equivalence

$[A\otimes_{R}M,U]_{A-\mathrm{Alg}/A}\simeq[M,[A,U]_{R-\mathrm{Alg}/A}]_{\mathrm{Top}_{+}}.$

###### Proof.

By applying Lemma 2.5 and then Lemma 2.5, we obtain

$\displaystyle[A\otimes_{R}M,U]_{A-\mathrm{Alg}/A}\simeq[A\otimes_{R-alg/A}M,U]_{{}_{A\setminus}R-\mathrm{Alg}_{/A}}$
$\displaystyle\simeq[M,[A,U]_{R-\mathrm{Alg}_{/A}}]_{\mathrm{Top}_{+}}$

∎

######  Definition 2.9.

Let $N$ be an $A$-module, and let $I$ be a $A$-NUCA, with an associative map
$I\wedge_{A}N\to N$. Define $N/I^{n}$ via the cofiber sequence:

$I^{\wedge_{A}n}\wedge_{A}N\to N\to N/I^{n}$

Examples of definition 2.9 are following:

######  Example 2.10.

  1. (1)

the $A$-module

$A\otimes_{R}M/(I_{M}^{A})^{n}$

where $N=A\otimes_{R}M$, $I=I^{A}_{M}$

  2. (2)

the $A$-module

$I_{M}^{A}/(I_{M}^{A})^{n}$

where $N=I_{M}^{A}$, $I=I_{M}^{A}$

  3. (3)

$(I_{M}^{A})^{n}/(I_{M}^{A})^{n+1}$

where $N=(I_{M}^{A})^{\wedge_{A}n}$ is given an $A$-module structure via any
of the factors, and $I=I_{M}^{A}$. It is also true that we have the
equivalence

$(I_{M}^{A})^{n}/(I_{M}^{A})^{n+1}\simeq\mathrm{fib}(I_{M}^{A}/(I_{M}^{A})^{n+1}\to
I_{M}^{A}/(I_{M}^{A})^{n}).$

Definition 2.9 also tells us how to make sense of completions:

######  Definition 2.11.

In the setting of Definition 2.9, let

$N^{\wedge}_{I}:=\mathrm{lim}_{n}{N/I^{n}}.$

######  Proposition 2.12.

If $A$ is connective, $M$ is connected, and $N$ is bounded below, we then have
the equivalence

$N\xrightarrow{\simeq}N^{\wedge}_{I_{M}^{A}}.$

###### Proof.

The map $A\otimes_{R}M\to A$ is an equivalence on $\pi_{0}$, since $M$ can be
taken to be a simplicial set with a unique zero simplex. Hence $I_{M}^{A}$ is
connected, since $A$, $A\otimes_{R}M$ are connective. Whence we obtain that
$(I_{M}^{A})^{\wedge_{A}n}$ is $n$-connected.

Thus the map

$N\to N^{\wedge}_{I_{M}^{A}}$

is $n$-connected for all $n$ since $N$ is bounded below, and is therefore a
weak equivalence. ∎

###  2.3. Topological Andre Quillen homology

We define Topological Andre Quillen homology and discuss its basic properties.
This will be used to define brave new generalizations of the etale condition.
For a good complementary exposition, see Richter’s survey paper [Ric22] and
for comparisons of different approaches, see [RS20].

The topological Andre Quillen homology of $B$ relative to $A$ for a map $A\to
B$ of commutative ring spectra, denoted $\mathrm{TAQ}(B/A)$, is designed to be
a derived version of the classical Kahler 1-forms $\Omega^{1}(B/A)$ where
$A\to B$ is a map of classical rings. In the classical setting,
$\Omega^{1}(B/A)\simeq I/I^{2}$ where $I$ is the augmentation ideal of the
multipilcation map $B\otimes_{A}B\to B$. This leads us the following
definition for commutative ring spectra:

######  Definition 2.13.

$\mathrm{TAQ}(B/A):=I^{B}_{S^{0}}/(I^{B}_{S^{0}})^{2}.$

Regarding the motivation of Definition 2.13, we recall from the previous
section that $I^{B}_{S^{0}}$ is the augmentation ideal of the map
$B\wedge_{A}B\to B$.

Another important property of $\mathrm{TAQ}(B/A)$ we will need is that it
represents derivations:

######  Proposition 2.14.

For any $V\in B-\mathrm{Mod}$, we have the equivalence

(2.5) $[\mathrm{TAQ}(B/A),V]_{B-\mathrm{Mod}}\simeq[B,B\vee
V]_{A-\mathrm{Alg}/B}.$

The right hand side of (2.5) may be interpreted as derivations of $B$ over $A$
with values in $V$.

###### Proof.

We have the following diagram of adjunctions

|  |  |  |  |
---|---|---|---|---|---
$\textstyle{B-\mathrm{Alg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$forget$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A-\mathrm{Alg}}$$\scriptstyle{-\otimes_{A}B}$$\textstyle{B-\mathrm{Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{triv}}$$\textstyle{B-\mathrm{NUCA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I/I^{2}}$$\scriptstyle{B\vee-}$$\textstyle{\left.B-\mathrm{Alg}\middle/B\right.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$aug.
idealforget$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\left.A-\mathrm{Alg}\middle/B\right.}$$\scriptstyle{-\otimes_{A}B}$

Here the middle adjunction is (2.4) and as noted there, both arrows are
simultaneously left and right adjoints. Observe that $\mathrm{TAQ}(B/A)$ is
the composite of the arrows from $A-\mathrm{Alg}/B\to B-\mathrm{Mod}$ applied
to the $A$-algebra $B$. We can then unravel the adjunctions to obtain that

$[\mathrm{TAQ}(B/A),V]_{B-\mathrm{Mod}}\simeq[(I/I^{2})\circ(\text{aug.
ideal})\circ(-\otimes_{A}B)(B),V]_{B-\mathrm{Mod}}$
$\simeq[B,\text{forget}\circ(B\vee-)\circ\mathrm{triv}(V)]_{A-\mathrm{Alg}/B}\simeq[B,B\vee
V].$

∎

###  2.4. Notions of etale:

In what follows, we define several brave new versions of etale. We will show
that these fit in the following dependency diagram:

(2.6) classically etale$\scriptstyle{(1)}$Lurie-etale$\scriptstyle{(2)}$THH-
etale descent$\scriptstyle{(3)}$THH-etale$\scriptstyle{(4)}$TAQ-
etale$\scriptstyle{(5)}$

Here, the implications (1),(2),(3),(4) always hold, and (5) holds under a
connectivity hypothesis. A classically etale map of classical rings is etale
in every other sense by (2.6), justifying the terminology. A good summary of
all of these notions of etale are given in Richter [Ric22, Sections 8.2,8.3].

######  Definition 2.15.

A ring map $A\to B$ is formally TAQ-etale iff $\mathrm{TAQ}(B/A)\simeq 0$. For
brevity we will simply say TAQ-etale.

When we have a ring map $R\to A$, the condition $\mathrm{TAQ}(B/A)\simeq 0$
implies the equivalence

(2.7) $\mathrm{TAQ}(A/R)\wedge_{A}B\cong\mathrm{TAQ}(B/R);$

this is true by the exact sequence/cofibration sequence of relative
differential forms
$\mathrm{TAQ}(A/R)\wedge_{A}B\to\mathrm{TAQ}(B/R)\to\mathrm{TAQ}(B/A)$,
discussed in [Bas99, Proposition 4.2].

We remark that when the simplicial cotangent complex vanishes for a map of
classical rings $\textsc{a}\to\textsc{b}$, the map is typically called
formally etale.

######  Definition 2.16.

$A\to B$ is formally $S^{1}$-THH-etale if $B\simeq B\otimes_{A}\mathrm{pt}\to
B\otimes_{A}S^{1}:=\mathrm{THH}(B/A)$ is an equivalence. For brevity, we will
say THH-etale.

######  Definition 2.17.

Given a pointed space $M$, we will say $A\to B$ is $M$-THH-etale if

(2.8) $B\simeq B\otimes_{A}\mathrm{pt}\to B\otimes_{A}M\simeq B$

is an equivalence.

We remark that Bobkova et al in [BHL+19] call this notion $M$-etale.

######  Remark 2.18.

It is equivalent to ask that $B\otimes_{A}M\to B\otimes_{A}\mathrm{pt}$ is an
equivalence, as done in [MM03]. Thus definition 2.17 does not depend on the
basepoint of $M$.

###### Proof of Remark 2.18.

Consider the following diagram:

$\textstyle{B\simeq
B\otimes_{A}\mathrm{pt}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(3)}$$\scriptstyle{(1)}$$\textstyle{B\otimes_{A}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2)}$$\textstyle{B\otimes_{A}\mathrm{pt}}$

The arrow (3) is always the identity map. Thus by the two out of three
property of equivalences, either of the other requirements implies the other.
∎

######  Lemma 2.19.

For any $n$, $S^{n}$-$\mathrm{THH}$-etale implies formally
$\mathrm{TAQ}$-etale.

###### Proof.

By hypothesis, we have that the augmentation ideal is zero:
$I_{S^{n}}^{B}\simeq 0$. Thus $0\simeq I_{S^{n}}^{B}/(I_{S^{n}}^{B})^{2}\simeq
S^{n}\wedge\mathrm{TAQ}(B/A)$ ; here the second equality is true by Lemma
3.11, proven in Section 3.

∎

In particular, we have implication (4) from our dependency diagram (2.6).

The $S^{0}$-THH-etale condition is very strong. It is not satisfied for Galois
extensions of (classical) fields. It is satisfied for the localization of a
commutative (or even an $E_{2}$) ring at a homotopy group.

######  Proposition 2.20.

Let $A$ be a commutative ring and let $x\in\pi_{k}A$. Let
$A[x^{-1}]:=\mathrm{colim}\,A\xrightarrow{x}\Sigma^{-k}A\xrightarrow{x}...$
where the arrows are given by left multiplication by $x$. Then the map $A\to
A[x^{-1}]$ is $S^{0}-\mathrm{THH}$-etale.

###### Proof.

Note that $A[x^{-1}]$ has an action of $A$ on the right. Using the commutative
structure, we see that it also has an action on the left, through the right
action. Hence we can write $A[x^{-1}]\wedge_{A}A[x^{-1}]=(\text{colim
}A\xrightarrow{x}\Sigma^{-k}A\xrightarrow{x}...)\wedge
A[x^{-1}]\simeq\text{colim
}A[x^{-1}]\xrightarrow{x}\Sigma^{-k}A[x^{-1}]\xrightarrow{x}...\simeq
A[x^{-1}].$

∎

The last version of etale we consider is that of Lurie [Lur17, Chapter 7,
Section 5].

######  Definition 2.21.

We will say that a map of commutative rings $A\to B$ is Lurie-etale if

  1. (1)

The map on $\pi_{0}$ is etale.

  2. (2)

The map

$\pi_{*}(A)\otimes_{\pi_{0}(A)}\pi_{0}(B)\xrightarrow{\simeq}\pi_{*}(B)$

is an isomorphism.

We note by definition that a classically etale map of rings
$\textsc{a}\to\textsc{b}$ is Lurie-etale, giving us implication (1) in our
dependency diagram 2.6.

As we will see, $M$-THH-etale is a particular case of etale descent; thus to
further study the basic properties of the higher $S^{n}$-THH-etale, $M$-THH-
etale, Lurie-etale conditions and the implications (1),(2),(3),(5) in our
dependency diagram, it is most expedient to study etale descent and its
properties.

##  3\. Etale descent

To prove Theorem 1.2 we will discuss a computation establishing it for the
case $A=P_{R}X$, with $X$ an $R$-module. We will then extend to the general
case by proving that THH and its analogues satisfy etale descent. As in the
last section, $R\to A\to B$ will be maps of commutative rings and $M$ will be
a pointed space.

######  Definition 3.1.

We say that $\otimes_{R}M$ satisfies etale descent along $A\to B$ whenever the
map

(3.1) $(A\otimes_{R}M)\wedge_{A}B\xrightarrow{\simeq}B\otimes_{R}M.$

is an equivalence.

To see that we need an etale-type hypothesis, we observe the dependence of
etale descent on the base ring $R$.

######  Lemma 3.2.

Let $R\to S\to A\to B$ be maps of commutative rings, giving $A$ and $B$ the
simultaneous structure of $R$-algebras and $S$-algebras. Then if (3.1) holds
when the base ring is $R$, it holds when the base ring is $S$.

###### Proof.

By Proposition 2.2, we have lo

$(A\otimes_{R}M)\wedge_{S\otimes_{R}M}S\simeq A\otimes_{S}M.$

Applying $\wedge_{S\otimes_{R}M}S$ to both sides of (3.1), and using
Proposition 2.2, we obtain that

$(A\otimes_{S}M)\wedge_{A}B\to B\otimes_{S}M$

is an equivalence. ∎

######  Corollary 3.3.

If $-\otimes_{R}M$ satisfies etale descent along $A\to B$ (3.1), then $A\to B$
is $M$-$\mathrm{THH}$ etale

###### Proof.

By Lemma 3.2, we may let the base ring $R=A$ to deduce that

$(A\otimes_{A}M)\wedge_{A}B\to B\otimes_{A}M$

is an equivalence. We have that $A\otimes_{A}M\simeq A$ since
$A^{\wedge_{A}n}\simeq A$. Thus

$B\to B\otimes_{A}M$

is an equivalence. ∎

This proves implication (3) of our dependency diagram (2.6). Taken together
with implication (4) proven in Lemma 2.19, we have the following implication
which explains the namesake of the term etale descent: when etale descent
holds along $A\to B$ for $M=S^{n}$, we obtain that $\mathrm{TAQ}(B/A)\simeq
0$.

Next, we study the effect of the space $M$ in (3.1).

######  Lemma 3.4.

Suppose (3.1) holds for spaces $X,Y,\text{ and }Z$. Then (3.1) is satisfied
for $M\simeq X\cup_{Y}Z$.

###### Proof.

We have the following equivalence:

$B\wedge_{A}(A\otimes_{R}(X\cup_{Y}Z))\simeq
B\wedge_{A}(A\otimes_{R}X\wedge_{A\otimes_{R}Y}A\otimes_{R}Z)$ $\simeq
B\wedge_{A}(A\otimes_{R}X)\wedge_{B\wedge_{A}(A\otimes
Y)}B\wedge_{A}(A\otimes_{R}Z)\simeq
B\otimes_{R}X\wedge_{B\otimes_{R}Y}B\otimes Z$ $\simeq
B\otimes_{R}(X\cup_{Y}Z)$

∎

######  Proposition 3.5.

Suppose (3.1) is true for $M=S^{n}$, $n\geq 0$. Then etale descent (3.1) holds
for any (n-1)-connected $M$.

###### Proof.

This proof is via Lemma 3.4 and induction on cellular dimension. Etale descent
always holds for $M\simeq\mathrm{pt}$. Etale descent holds for cell complexes
concentrated in degree $k$, since $\vee_{\alpha}S^{k}$ is built using $X\simeq
Z\simeq S^{k},Y\simeq\mathrm{pt}$. Any $(k-1)$-connected $(n+1)$-skeletal
space can be built as $X\cup_{Y}Z$ where $X\simeq\mathrm{pt}$ and $Y,Z$ are
$(k-1)$-connected $n$ skeletal spaces. Assuming Proposition 3.5 is true for
$n$-skeletal spaces which are $(k-1)$-connected, we obtain that its true for
$(n+1)$-skeletal spaces which are $(k-1)$-connected. By induction, etale
descent holds for all $(k-1)$-connected spaces. ∎

######  Remark 3.6.

Proposition 3.5 in particular shows that $M-\mathrm{THH}$-etale descent holds
for connected $M$ whenever etale descent holds for $\mathrm{THH}$.

We now prove the etale descent lemma needed for Theorem 1.3. The following
theorem was proven independently for $n=0$ in [RSV22, Theorem 7.5] and for
arbitrary $n$ in [LR22, Proposition 2.11].

######  Theorem 3.7.

Let $A\to B$ be $S^{n}$-$\mathrm{THH}$-etale. Then etale descent (3.1) holds
for $M=S^{n+1}$, and hence whenever $M$ is $n$-connected by Proposition 3.5.

###### Proof.

$B\wedge_{A}(A\otimes_{R}S^{n+1})\simeq
B\wedge_{A}A\wedge_{A\otimes_{R}S^{n}}A$ $\simeq
B\wedge_{A\otimes_{R}S^{n}}A\simeq
B\wedge_{B\otimes_{R}S^{n}}B\otimes_{R}S^{n}\wedge_{A\otimes_{R}S^{n}}A$
$\simeq B\wedge_{B\otimes_{R}S^{n}}B\otimes_{A}S^{n}\simeq
B\wedge_{B\otimes_{R}S^{n}}B$ $\simeq B\otimes_{R}S^{n+1}$

∎

In particular if $A\to B$ is $S^{0}$-THH-etale, then THH-etale descent holds
along $A\to B$. This implication was used in [RSV22]. In their paper,
$S^{0}$-THH-etale extensions are called ‘solid’.

A byproduct of this theory on etale descent is nontrivial results for $M$-THH-
etaleness. In particular we have the following corollary.

######  Corollary 3.8.

  1. (1)

If $A\to B$ is $X$,$Y$, and $Z$-$\mathrm{THH}$-etale, then $A\to B$ is
$M$-$\mathrm{THH}$-etale where $M\simeq X\cup_{Y}Z$

  2. (2)

If $A\to B$ is $S^{n}$-THH-etale, then $A\to B$ is $M$-$\mathrm{THH}$-etale
whenever $M$ is $(n-1)$ connected.

We note that Corollary 3.8 (2) is stronger than Theorem 3.7 in the setting of
$M$-THH-etale.

Our next focus is on the remaining implications in our dependency diagram
(2.6). If $A\to B$ is Lurie-etale, etale descent (3.1) holds along $A\to B$
for $M=S^{1}$ and any choice of base ring by [Mat17]; from this we obtain
implication (2). In addition, $M-\mathrm{THH}$-etale descent for any connected
$M$ holds along $A\to B$ by Proposition 3.5. The last implication we have not
proven yet is implication (5) which is exactly the content of the next
theorem. This etale descent theorem is also the main ingredient for Theorem
1.2.

######  Theorem 3.9.

Let $R\to A\to B$ be maps of commutative rings with $A,B$ connective. Suppose
further that $\mathrm{TAQ}(B/A)\simeq 0$. Then etale descent (3.1) holds for
$\mathrm{THH}(-/R)$ and equivalently for $-\otimes_{R}M$ with $M$ connected.

######  Remark 3.10.

The version of Theorem 3.9 proven in the setting of a Lurie-etale extension in
[Mat17] and [CM21] is neither a special case nor a generalization of the
connective, formal TAQ-etale setting here. Our proof follows the ideas of
[MM03].

Proof of Theorem 3.9:

We have a filtration on $A\otimes_{R}M$ by the powers of the augmentation
ideal $I_{M}^{A}$ (Section 2.7 for a definition), giving us the following
tower:

$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(I_{M}^{A})^{2}/(I_{M}^{A})^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{R}M/(I_{M}^{A})^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{M}^{A}/(I_{M}^{A})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{R}M/(I_{M}^{A})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{R}M/(I_{M}^{A})}$

We now show that the comparison map

$(A\otimes_{R}M)\wedge_{A}B\to B\otimes_{R}M$

(3.1) induces equivalent ‘associated graded’s’ or more precisely equivalences
on the $n^{th}$ stages of the towers for $(A\otimes_{R}M)\wedge_{A}B$ and
$B\otimes_{R}M$.

For $n=1$, we note the following lemma:

######  Lemma 3.11.

$I_{M}^{A}/(I_{M}^{A})^{2}\simeq M\wedge\mathrm{TAQ}(A/R).$

In other words $I_{M}^{A}/(I_{M}^{A})^{2}$ is linear in $M$.

###### Proof.

We have the equivalences

$[I_{M}^{A}/(I_{M}^{A})^{2},V]_{A-\mathrm{Mod}}\simeq[I_{M}^{A},V]_{A-\mathrm{NUCA}}\simeq[A\otimes_{R}M,A\vee
V]_{A-\mathrm{Alg}/A}.$

By Proposition 2.8, this is equal to $[M,[A,A\vee
V]_{R-\mathrm{Alg}/A}]_{\mathrm{Top}_{+}}$. On the other hand, we have the
equivalences

$[M\wedge\mathrm{TAQ}(A/R),V]_{A-\mathrm{Mod}}\simeq[M,[\mathrm{TAQ}(A/R),V]_{A-\mathrm{Mod}}]_{\mathrm{Top}_{+}}$
$\simeq[M,[A,A\vee V]_{R-\mathrm{Alg}/A}]_{\mathrm{Top}_{+}}.$

For the last equality, see Proposition 2.14. Since this is true for all $V$,
we are done by Yoneda’s lemma. ∎

The comparison map (3.1) induces an equivalence on the first stages of our
towers:

$I_{M}^{A}/(I_{M}^{A})^{2}\wedge_{A}B\simeq
M\wedge\mathrm{TAQ}(A/R)\wedge_{A}B$ $\simeq M\wedge\mathrm{TAQ}(B/R)\simeq
I_{M}^{B}/(I_{M}^{B})^{2}.$

Here we use Lemma 3.11, and that $\mathrm{TAQ}$ satisfies etale descent, by
(2.7). For arbitrary $n$, we need the following lemma.

######  Lemma 3.12.

Let $I$ be an $A$-$\mathrm{NUCA}$. Then

$I^{n}/I^{n+1}\simeq(I/I^{2})^{\wedge_{A}n}_{h\Sigma_{n}}$

###### Proof.

This is the main result of [Min03, Proposition 2.4]. The intuition is that the
left hand side consists of products of $n$ elements which can not be written
as a product of more elements. The right hand side consists of products of $n$
indecomposables modulo the order in which they are written.

∎

######  Lemma 3.13.

Etale descent holds up to completion; more precisely, we have the following
equivalence

$\left((A\otimes_{R}M)\wedge_{A}B\right)^{\wedge}_{I_{M}^{A}}\simeq(B\otimes_{R}M)^{\wedge}_{I_{M}^{B}}.$

###### Proof.

By Lemma 3.12, we have for arbitrary $n$ that

$(I_{M}^{A})^{n}/(I_{M}^{A})^{n+1}\wedge_{A}B\simeq\left((I_{M}^{A})/(I_{M}^{A})^{2}\right)^{\wedge_{A}n}_{h\Sigma_{n}}\wedge_{A}B$
$\simeq\left(M\wedge\mathrm{TAQ}(A/R)\wedge_{A}B\right)^{\wedge_{B}n}_{h\Sigma_{n}}\simeq\left(M\wedge\mathrm{TAQ}(B/R)\right)^{\wedge_{B}n}_{h\Sigma_{n}}$
$\simeq(I_{M}^{B})^{n}/(I_{M}^{B})^{n+1}.$

Hence we have proven that the comparison map (3.1) induces an equivalence on
associated graded’s and hence an equivalence on completion.

∎

Apply Proposition 2.12 to both sides of Lemma 3.13 to obtain (3.1). This
completes the proof of Theorem 3.9. ∎

##  4\. Proof of Theorem 0.1 for polynomial algebras

We will prove Theorem 0.1 when $A=P_{R}X$ where $X$, as before, is an
(arbitrary) $R$-module, and $M$ is a pointed space.

######  Lemma 4.1.

The functor $P_{R}(-)$ takes the tensor in the category $R-\mathrm{Mod}$ with
an unpointed space, to the tensor in the category $R-\mathrm{Alg}$, with the
same unpointed space. That is

$P_{R}(X\wedge M_{+})\simeq P_{R}(X)\otimes_{R}M.$

###### Proof.

This is a direct consequence of (2.2). ∎

Thus we have that

$\displaystyle P_{R}(X)\otimes_{R}M\simeq P_{R}(X\wedge M_{+})$
$\displaystyle\simeq P_{R}(X\wedge M\vee X)\simeq
P_{R}(X)\wedge_{R}P_{R}(X\wedge M)$ $\displaystyle\simeq
P_{P_{R}X}(P_{R}X\wedge_{R}X\wedge_{R}M).$

Next we write down the right hand side using $\mathrm{TAQ}$ (see also [BGR08,
Proposition 1.6]).

######  Proposition 4.2.

We have the equivalence

$\mathrm{TAQ}(P_{R}(X)/R)\simeq P_{R}(X)\wedge_{R}X.$

Intuitively, this is true because the left hand side is reminiscent of
differential forms. On the RHS, $P_{R}X$ reminds us of the coefficients of the
differential forms, and $X$ reminds us of the span of the $dx_{1}...dx_{n}$.

To prove this proposition, we will need the following lemma.

######  Lemma 4.3.

Let $A$ be an augmented $R$ algebra. We then have an adjunction

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.33505pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R-\mathrm{Alg}/A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}{\hbox{\kern
48.0573pt\raise-12.0pt\hbox{\hbox{\kern
0.0pt\raise-1.70833pt\hbox{$\scriptstyle{I_{R}}$}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
95.93169pt\raise-2.99615pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}{}{{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
48.33505pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 78.33505pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R-\mathrm{NUCA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}{\hbox{\kern
45.27951pt\raise 12.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{-\vee
A}$}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{}{{}{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
24.33435pt\raise 8.67218pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}{}{{{}}{{}}{}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

Here $I_{R}$ is the functor $R-\mathrm{Alg}/A\to R-\mathrm{Alg}/R\to
R-\mathrm{NUCA}$.

###### Proof.

Note that we assert $I_{R}$ is left-adjoint to extension by $A$. First we use
the equivalence of categories

$[I_{R}(B),J]_{R-\mathrm{NUCA}}\simeq[B,R\vee J]_{R-\mathrm{Alg}/R}$

from NUCA’s to augmented algebras in 2.4. If we take this together with the
equivalence

$[B,R\vee J]_{\mathrm{Alg}_{R}/R}\simeq[B,A\vee J]_{\mathrm{Alg}_{R}/A}$

we obtain the lemma. ∎

###### Proof of Proposition 4.2.

We have that TAQ controls square zero extensions, as witnessed in the
following equation

$[\mathrm{TAQ}(A/R),V]_{A-\mathrm{Mod}}\simeq[A,A\vee V]_{R-\mathrm{Alg}/A}$

and discussed in (2.14). Let $A=P_{R}X$. The algebra $A$ is an augmented $R$
algebra and hence by Lemma 4.3 we have the equivalences

$[A,A\vee Z(V)]_{R-\mathrm{Alg}/A}\simeq[I_{R}(A),Z(V)]_{R-\mathrm{NUCA}}$
$\simeq[I_{R}(A)/I_{R}(A)^{2},V]_{R-\mathrm{Mod}}.$

In our case, this can be related back to $A-\mathrm{Mod}$ via the equivalences

$\simeq[X,V]_{R-\mathrm{Mod}}\simeq[X\wedge_{R}P_{R}X,V]_{P_{R}X-\mathrm{Mod}}$

and this establishes Proposition 4.2, by Yoneda’s lemma.

∎

Thus we have that $P_{P_{R}X}(P_{R}X\wedge_{R}X\wedge M)\simeq
P_{P_{R}X}(M\wedge\mathrm{TAQ})$ as desired. This proves Theorem 1.2 in this
special case.

##  5\. Notions of coverings and smoothness

We define the hypotheses used in Theorems 1.2 and 1.3. These involve brave new
notions of smooth; these notion of smoothness will depend on certain
generalized notions of etale coverings that are engineered to rectify the main
error of McCarthy and Minasian’s first paper [MM03] noted in our introduction.
As these notions of coverings may at first feel unnatural, we will also show
that under mild hypotheses, these generalized notions of coverings are all
equivalent to the classical notion of an etale covering, when working in the
underived classical setting.

In this section, $Y$ and $Z$ will denote $A$-modules. As in the rest of the
paper, a will denote a classical ring, and y will denote a classical a-module.
Generalized etale will mean any of the generalized notions of etale from
Section 2.4 including respectively TAQ, THH, M-THH, or Lurie etale. Similarly
for generalized etale cover, and generalized smooth, which will be defined
presently.

######  Definition 5.1.

Say that a collection of generalized etale commutative ring maps $\\{A\to
A_{\alpha}\\}_{\alpha}$ is a generalized etale global covering if whenever an
$A$ module map splits $A_{\alpha}$-locally, there is a splitting globally.

Regarding Definition 5.1, we will use the following standard notation. Given
an $A$-module $Y$, we will let $Y_{\alpha}$ denote $Y\wedge_{A}A_{\alpha}$.
Given an $A$-module map $f:Y\to Z$, we will let $f_{\alpha}$ denote the
induced map $f_{\alpha}:Y_{\alpha}\to Z_{\alpha}$. In this notation,
Definition 5.1 can be formulated as follows: a collection of generalized etale
maps $\\{A\to A_{\alpha}\\}_{\alpha}$ is a generalized etale global covering
if whenever $f:Y\to Z$ has $s_{\alpha}:Z_{\alpha}\to Y_{\alpha}$ such that
$f_{\alpha}\circ s_{\alpha}\simeq\mathrm{Id}_{Z_{\alpha}}$, there is a map
$s:Z\to Y$ such that $f\circ s\simeq\mathrm{Id}_{Z}$. We note that Definition
5.1 is the one given in McCarthy-Minasian’s second paper [MM04, Definition
6.9].

######  Definition 5.2.

  1. (1)

Say that a map $R\to A$ is generalized smooth if there is a generalized etale
global covering $\\{A_{\alpha}\\}_{\alpha}$ of $A$ such that for each
$A_{\alpha}$, there an $R$-module $X$ and a factorization

$R\to P_{R}X\xrightarrow{\phi}A_{\alpha}$

with $\phi$ generalized etale.

In other words $A$ is locally etale over $R$-affines.

  2. (2)

Given such maps, we say that $R\to A$ is generalized smooth in the category of
connective commutative rings if $R,A,A_{\alpha}$ are all connective and for
each $A_{\alpha}$, the corresponding $P_{R}X$ is also connective.

In order to compare this notion of a covering with more classical notions of a
covering, we introduce another generalized notion of cover. This is covering
used in McCarthy-Minasian’s first paper [MM03].

######  Definition 5.3.

Say that a collection of generalized etale commutative ring maps $\\{A\to
A_{\alpha}\\}_{\alpha}$ is a faithful generalized etale covering if whenever a
map $Y\to Z$ induces an equivalence $A_{\alpha}$-locally, it is itself an
equivalence.

Definitions 5.1, 5.2, 5.3 all have analogues for classical ring maps; we
obtain these analogues by defining ‘etale local’ using underived tensor
products. The maps $\textsc{a}\to\textsc{a}_{\alpha}$ will always be required
to be classically etale.

Recall that $\textsc{r}\to\textsc{a}$ is classically smooth if it satisfies
the conditions of Definition 5.2 where in place of a global covering, we use a
classical etale covering:

######  Definition 5.4.

A classical etale covering is a collection of etale maps
${\textsc{a}\to\textsc{a}_{\alpha}}$ inducing a surjection $\cup\text{Spec
}(\textsc{a}_{\alpha})\to\text{Spec }\textsc{a}$.

######  Proposition 5.5.

In this classical setting the following is true.

  1. (1)

A faithful covering is a classical etale covering and vice versa.

  2. (2)

A global covering is always a faithful covering.

  3. (3)

A faithful covering satisfies the requirements of a global covering when z(of
Definition 5.3) is finitely presented.

We remind the reader that all of the maps
$\\{\textsc{a}\to\textsc{a}_{\alpha}\\}$ are required to be etale in all the
notions of covering in Proposition 5.5. In particular, the map
$\textsc{a}\to\underset{\alpha}{\Pi}\textsc{a}_{\alpha}$ will be flat.

Proof of Proposition 5.5 (1):

We have that the flat map $\textsc{a}\to\Pi\textsc{a}_{\alpha}$ is faithfully
flat iff any map between $A$-modules $f:\textsc{y}\to\textsc{z}$ is an
isomorphism whenever $f\otimes_{\textsc{a}}\Pi\textsc{a}_{\alpha}$ is an
isomorphism. This happens iff $\\{\textsc{a}\to\textsc{a}_{\alpha}\\}$ is a
faithful covering. We also have that the flat map
$\textsc{a}\to\Pi\textsc{a}_{\alpha}$ is faithfully flat iff the induced map
on Spec is surjective. This happens iff
$\\{\textsc{a}\to\textsc{a}_{\alpha}\\}$ is an etale covering.∎
Proof of Proposition 5.5 (2):

Suppose we are given a map $f:\textsc{y}\to\textsc{z}$ such that $f_{\alpha}$
is an isomorphism for each $\alpha$. From the condition of a global cover, we
have that the existence of the $f_{\alpha}$’s imply that $f$ is split
surjective and split injective, and hence is an isomorphism. ∎

######  Remark 5.6.

The proof of Proposition 5.5 (2) also implies that global coverings are
faithful coverings in the setting of commutative ring spectra.

Proof of Proposition 5.5 (3):

We prove the following: Let z be finitely presented. Then any map
$\textsc{y}\to\textsc{z}$ is split surjective iff it is locally split
surjective.

Suppose $f:\textsc{y}\to\textsc{z}$ is locally split surjective. This implies
that $f$ is locally surjective, and since surjectivity can be checked on
stalks, this implies $f$ is surjective. Thus, let k fit in the exact sequence

$0\to\textsc{k}\to\textsc{y}\to\textsc{z}\to 0.$

We will show that the corresponding element
$\xi\in\mathrm{Ext}^{1}(\textsc{z},\textsc{k})$ is 0. Since z is finitely
presented, by definition it has a resolution by projectives

$\textsc{p}_{1}\to\textsc{p}_{0}\to\textsc{z}$

with $\textsc{p}_{1},\textsc{p}_{0}$ finitely generated. The map

$\mathrm{Hom}(\textsc{p}_{i},\textsc{k})\otimes_{\textsc{a}}\textsc{a}_{\alpha}\to\mathrm{Hom}_{\textsc{a}_{\alpha}}(\textsc{p}_{i}\otimes_{\textsc{a}}\textsc{a}_{\alpha},\textsc{k}\otimes_{\textsc{a}}\textsc{a}_{\alpha})$

is an isomorphism for $i\simeq 0,1$ since the map is an isomorphism when
$\textsc{p}_{i}$ is replaced by a finitely generated free module. After taking
cohomology, we deduce that the map

$\mathrm{Ext}^{1}(\textsc{z},\textsc{k})\otimes_{\textsc{a}}\textsc{a}_{\alpha}\to\mathrm{Ext}^{1}(\textsc{z}\otimes_{\textsc{a}}\textsc{a}_{\alpha},\textsc{k}\otimes_{\textsc{a}}\textsc{a}_{\alpha})$

is an injection. By hypothesis, the image of $\xi$ is zero in the right hand
side, and hence in the left hand side. Consider the submodule
$\langle\xi\rangle\subset\mathrm{Ext}^{1}(\textsc{z},\textsc{k})$. We have
that the map $\langle\xi\rangle\to 0$ is an isomorphism
$\textsc{a}_{\alpha}$-locally. Hence the module $\langle\xi\rangle$ vanishes.

∎

##  6\. Proof of Theorems 1.2 and 1.3

Consider $R,A,A_{\alpha},P_{R}X$ as in Definitions 5.1 and 5.2. We will show
that if etale descent (3.1) holds along each $A\to A_{\alpha}$ and $P_{R}X\to
A_{\alpha}$, and that $\\{A_{\alpha}\\}_{\alpha}$ form a global covering, that
equivalence (1.3) holds. The hypotheses for Theorem 1.2 and Theorem 1.3
guarantee that these conditions hold by Section 3. By Section 4, we have the
equivalence

$P_{R}X\otimes_{R}M\simeq P_{P_{R}X}(M\wedge\mathrm{TAQ}(P_{R}X/R)).$

Since etale descent holds, we have the equivalence

$\left(A_{\alpha}\wedge_{P_{R}X}P_{R}X\otimes_{R}M\right)\simeq\left(A_{\alpha}\otimes_{R}M\right)$

whence we obtain the equivalences

(6.1) $\begin{split}P_{A_{\alpha}}(M\wedge\mathrm{TAQ}(A_{\alpha}/R))\simeq
A_{\alpha}\wedge_{P_{R}X}P_{P_{R}X}(M\wedge\mathrm{TAQ}(P_{R}X/R))\\\ \simeq
A_{\alpha}\wedge_{P_{R}X}P_{R}X\otimes_{R}M\xrightarrow{\simeq}\left(A_{\alpha}\otimes_{R}M\right).\end{split}$

Hence we obtain a map, denoted by

$s_{\alpha}:M\wedge\mathrm{TAQ}(A_{\alpha}/R)\to\left(A_{\alpha}\otimes_{R}M\right).$

######  Lemma 6.1.

We have that $s_{\alpha}$ is a splitting of the natural map, denoted by

$D_{A_{\alpha}}:(A_{\alpha}\otimes_{R}M)\to
I_{M}^{A_{\alpha}}\to\left(I_{M}^{A_{\alpha}}/(I_{M}^{A_{\alpha}})^{2}\right)\simeq
M\wedge\mathrm{TAQ}(A_{\alpha}/R).$

###### Proof.

The maps making up $D_{A_{\alpha}}$ respect the filtration of the towers in
Lemma 3.13 since this map is the projection onto the first layer. The map
$s_{\alpha}$ also respects the filtration of the these towers; the
equivalences of equation (6.1) were proven using $M-\mathrm{THH}$-etale
descent which was in turn proven by using the filtration given by this tower.
Hence every map that makes up $s_{\alpha}$ and $D_{A_{\alpha}}$ is an
equivalence on the first layer. THus we have that the map $D_{A_{\alpha}}\circ
s_{\alpha}$ is an equivalence. ∎

Consider the map

$D_{A}:A\otimes_{R}M\to M\wedge\mathrm{TAQ}(A/R).$

We have the equivalence $D_{A}\wedge_{A}A_{\alpha}\simeq D_{A_{\alpha}}$ since
again, etale descent respects the layers of the $\mathrm{TAQ}$-tower functor
for the map of $R$-algebras $A\to A_{\alpha}$. Thus $s_{\alpha}$ is a
splitting of $D_{A}\wedge_{A}A_{\alpha}$. Hence, by the definition of a global
covering (Definition 5.1), there is a (global) splitting of $D_{A}$, denoted
by

$s:M\wedge\mathrm{TAQ}(A/R)\to A\otimes_{R}M.$

This $A$-module map induces the $A$-algebra map

(6.2) $P_{A}(M\wedge\mathrm{TAQ}(A/R))\to A\otimes_{R}M.$

The map (6.2) induces the map

$P_{A_{\alpha}}(M\wedge\mathrm{TAQ}(A_{\alpha}/R))\simeq
A_{\alpha}\wedge_{A}P_{A}(M\wedge\mathrm{TAQ})\to
A_{\alpha}\wedge_{A}A\otimes_{R}M\simeq A_{\alpha}\otimes_{R}M$

which is exactly the equivalence of (6.1). Since global coverings are faithful
by Remark 5.6, we have that (6.2) is an equivalence. This finishes the proof
of Theorems 1.2 and 1.3. ∎

######  Remark 6.2.

We named the map $D_{A}$ above as it is the linearization, in the sense of
Goodwillie calculus of the functor $-\otimes_{R}M:R-\mathrm{Alg}\to
R-\mathrm{Alg}$ given by $C\mapsto C\otimes_{R}M$ at the algebra $A$. Thus
what we showed above, is that the Goodwillie tower for $-\otimes_{R}M$ splits
at $A$, when $A/R$ is formally TAQ-smooth.

###### Proof.

For a proof that the tower above is actually the Goodwillie tower of the
functor $-\otimes_{R}M$, see [Min03, Remark/Claim 2.6]. ∎

##  7\. Application to rational higher THH

We calculate $\mathrm{THH}(KU_{\mathbb{Q}}/\mathbb{Q})$ as an application of
our HKR theorem, specifically Theorem 1.3.

######  Theorem 7.1.

We have the following equivalences of $KU_{\mathbb{Q}}$-algebras which are
natural in $M$:

$\mathrm{THH}(KU_{\mathbb{Q}}/\mathbb{Q})\simeq
P_{KU_{\mathbb{Q}}}{\Sigma^{3}H\mathbb{Q}}$
$KU_{\mathbb{Q}}\otimes_{\mathbb{Q}}M\simeq
P_{KU_{\mathbb{Q}}}(M\wedge\Sigma^{2}H\mathbb{Q}).$

###### Proof.

Rationally, we have $KU_{\mathbb{Q}}\simeq\mathbb{Q}[\beta,\beta^{-1}]$
$|\beta|=2$, is an Eilenberg Maclane spectrum. We now show that
$KU_{\mathbb{Q}}$ is formally $S^{0}$-THH-smooth over $\mathbb{Q}$:

We have the equivalence $\mathbb{Q}[\beta]\simeq
P_{\mathbb{Q}}(\Sigma^{2}H\mathbb{Q})$. The extension
$\mathbb{Q}[\beta]\to\mathbb{Q}[\beta,\beta^{-1}]$ is $S^{0}$-THH-etale by
2.20. Since we have only one $A_{\alpha}=A$ in our example, $\mathbb{Q}\to
KU_{\mathbb{Q}}$ is $S^{0}$-THH smooth. Hence, we are done by Theorem 1.3.

∎

######  Corollary 7.2.

The rational $nth$-higher THH and the $nth$ iterated THH of $KU$ is given by

$KU_{\mathbb{Q}}\otimes_{\mathbb{Q}}S^{n}\simeq
P_{KU_{\mathbb{Q}}}(\Sigma^{n+2}H\mathbb{Q})$
$KU_{\mathbb{Q}}\otimes_{\mathbb{Q}}(S^{1})^{n}\simeq
P_{KU_{\mathbb{Q}}}((S^{1})^{n}\wedge\Sigma^{n+2}H\mathbb{Q}).$

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