# Hochschild homology of mod-$p$ motivic cohomology over algebraically closed
fields

Bjørn Ian Dundas, Michael A. Hill, Kyle Ormsby, Paul Arne Østvær

###### Abstract

We perform Hochschild homology calculations in the algebro-geometric setting
of motives. The motivic Hochschild homology coefficient ring contains torsion
classes which arise from the mod-$p$ motivic Steenrod algebra and from
generating functions on the natural numbers with finite non-empty support.
Under the Betti realization, we recover Bökstedt’s calculation of the
topological Hochschild homology of finite prime fields.

##  1 Introduction

Let $\mathscr{R}$ be a motivic ring spectrum such as algebraic cobordism,
homotopy algebraic $K$-theory, or motivic cohomology [31]. Working in the
stable motivic homotopy category $\mathscr{SH}(F)$ of a field $F$, we define
the motivic Hochschild homology $\mathbf{MHH}(\mathscr{R})$ of $\mathscr{R}$
as the derived tensor product

$\mathscr{R}\wedge_{\mathscr{R}\wedge\mathscr{R}^{\text{op}}}\mathscr{R}.$ (1)

The concepts of Hochschild homology for associative algebras and topological
Hochschild homology for structured ring spectra inspire our constructions. In
the event $\mathscr{R}$ is commutative one may equivalently to (1) form the
tensor product in the category of commutative motivic ring spectra with the
simplicial circle

$S^{1}\otimes\mathscr{R}.$ (2)

The primary purpose of this paper is to calculate the homotopy groups of
motivic Hochschild homology of $\mathbf{M}\mathbb{F}_{p}$ over algebraically
closed fields — the Suslin-Voevodsky mod-$p$ motivic cohomology ring spectrum
for $p$ any prime number. When the base field $F$ admits an embedding into the
complex numbers $\mathbb{C}$, the Betti realization functor allows us to
compare our $\mathbf{MHH}$ calculations with Bökstedt’s pioneering work on
topological Hochschild homology of the corresponding topological Eilenberg-Mac
Lane spectrum $\mathbf{H}\mathbb{F}_{p}$. In fact, our calculation for
$\mathbf{MHH}$ specializes to the one for $\mathbf{THH}$ in [8]. Additively,
$\mathbf{THH}(\mathbb{F}_{p})$ splits as a restricted product of Eilenberg-
MacLane spectra in the stable homotopy category. This is not the case,
however, for $\mathbf{MHH}(\mathbb{F}_{p})$, $\mathbf{M}\mathbb{F}_{p}$, and
$\mathscr{SH}(F)$. The source of this extra layer of complexity is the
abundance of $\tau$-torsion elements in the coefficients. Here $\tau$ is a
canonical class in the mod-$p$ motivic cohomology of $F$, which maps to the
unit element in singular cohomology under Betti realization.

We express the coefficient ring $\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ in
terms of algebra generators $\tau$, $\mu_{i}$, $\mathsf{x}_{S,f}$ arising from
the mod-$p$ motivic Steenrod algebra [17], [34], and generating endofunctions
$f\colon\mathbb{N}\circlearrowleft$ with finite non-empty support containing
some subset $S\subset\mathbb{N}$. The infinity of $\tau$-torsion classes
$\mathsf{x}_{S,f}$ is not witnessed in $\mathbf{THH}_{\star}(\mathbb{F}_{p})$.
For example, Kronecker delta functions give rise to such classes (in this
case, $S$ is either empty or a singleton set).

######  Theorem 1.1.

Over an algebraically closed field of exponential characteristic
$\mathsf{e}(F)\neq p$, there is an algebra isomorphism

$\mathbf{MHH}_{\star}(\mathbb{F}_{p})\cong\mathbb{F}_{p}[\tau,\mu_{i},\mathsf{x}_{S,f}]_{i\in\mathbb{N},\,(S\subset\mathrm{supp}\,f,f\colon\mathbb{N}\circlearrowleft)}/\mathcal{I}$
(3)

with the ideal of relations

$\mathcal{I}=\left(\begin{matrix}\mu_{i}^{p}-\tau^{p-1}\mu_{i+1},\\\
\tau^{p-1}\mathsf{x}_{S,f},\\\
\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\underset{u\in\mathrm{supp}(f+g)-S\cup
T}{\sum}\epsilon_{u}\cdot\mathsf{x}_{S\cup
T\cup\\{u\\},f+g}\end{matrix}\right).$

Here the support of $f$ is a finite non-empty subset of the natural numbers
and $S\subset\mathrm{supp}f\subset\mathbb{N}$ does not contain the minimal
element of $\mathrm{supp}f$. The coefficient $\epsilon_{u}\in\mathbb{F}_{p}$
is given explicitly in 2.12. The algebra generators have bidegrees given by
$|\tau|=(0,-1)$, $|\mu_{i}|=(2p^{i},p^{i}-1)$, and

$|\mathsf{x}_{S,f}|=(|S|+1)(-1,p-1)+p\sum_{j\in\mathrm{supp}f}f(j)(2p^{j},p^{j}-1).$

Since the homotopy of $\mathbf{MHH}(\mathbb{F}_{p})$ is not a free module over
the homotopy of $\mathbf{M}\mathbb{F}_{p}$, we deduce a non-splitting of the
motivic Hochschild homology in $\mathbf{M}\mathbb{F}_{p}$-modules.

######  Corollary 1.2.

The motivic Hochschild homology of $\mathbb{F}_{p}$ does not split as a wedge
of suspensions of $\mathbf{M}\mathbb{F}_{p}$.

This gives a surprising obstruction to classical results about topological
Hochschild homology and Thom spectra. Mahowald showed that the
Eilenberg–MacLane spectrum $\mathbf{H}\mathbb{F}_{2}$ is a Thom spectrum of a
double loop map with source $\Omega^{2}S^{3}$ [23]. Behrens–Wilson showed that
an analogous result is true $C_{2}$-equivariantly, with the base now
$\Omega^{2,1}S^{3,1}$ [6]. Blumberg–Cohen–Schlichtkrull showed that the
topological Hochschild homology of Thom spectra are Thom spectra, and when the
topological $\eta$ vanishes, these split as smash products of the original
Thom spectrum and a space related to the classifying space of the base [7].
Equivariantly, classically and $C_{2}$-equivariantly, this splits as a wedge
of smash powers of spheres. Putting this all together, we cannot have all of
these results hold in the motivic setting.

As a guide to this paper, we outline the proof of Theorem 1.1 and explain how
the algebra generators arise in our context. The key idea in proving our
results is to study the $\tau$-inversion and mod-$\tau^{p-1}$ reduction of
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})$. We review some background and set our
notation in Section 2. In Section 3 we divide the proof of Theorem 1.1 into
the following steps.

  1. Step 1

Theorem 3.3 calculates the $\tau$-inverted or étale motivic Hochschild
homology

$\mathbf{MHH}_{\star}(\mathbb{F}_{p})[\tau^{-1}]\cong\mathbb{F}_{p}[\tau^{\pm
1},\mu_{i}]_{i\geq
0}/(\mu_{i}^{p}-\tau^{p-1}\mu_{i+1})\cong\mathbb{F}_{p}[\mu,\tau^{\pm
1}]\cong\mathbf{THH}_{\ast}(\mathbb{F}_{p})[\tau^{\pm 1}].$ (4)

The generator $\mu$ in (4) has bidegree $(2,0)$. Hence all the classes
$\mu_{i}$, $i\in\mathbb{N}$, generate the non-$\tau$-torsion part of
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ subject to the relation
$\mu_{i}^{p}=\tau^{p-1}\mu_{i+1}$.

  2. Step 2

Theorem 3.6 calculates the coefficients of mod-$\tau^{p-1}$ motivic Hochschild
homology

$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}\cong(\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\bar{\mu}_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1}))\otimes\mathbb{F}_{p,\tau}.$
(5)

The bidegrees of the generators are $|\bar{\lambda}_{i}|=(2p^{i}-1,p^{i}-1)$,
$|\bar{\mu}_{i}|=(2p^{i},p^{i}-1)$, and $\mathbb{F}_{p,\tau}$ is shorthand for
$\mathbb{F}_{p}[\tau]/\tau^{p-1}$. The divided powers algebra generator
$\bar{\mu}_{i}$ is the image of
$\mu_{i}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})$. We note that (5) coincides
with the $E^{2}$ page of the Tor spectral sequence for
$\mathbf{MHH}(\mathbb{F}_{p})/\tau^{p-1}$. In fact, the said Tor spectral
sequence collapses at $E^{2}$ with no multiplicative extensions.

  3. Step 3

Lemma 3.9 shows the $\tau^{p-1}$-Bockstein of $\gamma_{j}\bar{\mu}_{i}$ equals
$\bar{\lambda}_{i+1}\gamma_{j-p}\bar{\mu}_{i}$. First we establish the case
$j=p$, and the rest follows by shuffle products in the bar construction of
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$. Here, the
$\tau^{p-1}$-Bockstein on $\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ is the
composite of the canonical boundary and quotient maps in

$\bar{\partial}\colon\mathbf{MHH}_{\ast+1,\ast}(\mathbb{F}_{p})/\tau^{p-1}\xrightarrow{\partial}\mathbf{MHH}_{\ast,\ast+p-1}(\mathbb{F}_{p})\xrightarrow{q}\mathbf{MHH}_{\ast,\ast+p-1}(\mathbb{F}_{p})/\tau^{p-1}.$
(6)

In Corollary 3.10, we conclude the Bockstein homology of
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ is isomorphic to the graded
commutative $\mathbb{F}_{p,\tau}$-algebra $\bigoplus_{i\geq
0}\Lambda_{\mathbb{F}_{p,\tau}}(\bar{\mu}_{i})$.

  4. Step 4

Lemma 3.13 shows the $\tau$-torsion classes in
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ inject into
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ with image that of the
$\tau^{p-1}$-Bockstein $\bar{\partial}$ (degrees are made explicit through
generating functions). Moreover, the reduction map $q$ sends the image of the
boundary $\partial$ isomorphically to the image of the Bockstein
$\bar{\partial}$.

  5. Step 5

If $f\colon\mathbb{N}\circlearrowleft$ has finite support and
$S\subseteq\mathrm{supp}f$, we set

$\chi_{S,f}=\left(\prod_{m\in
S}\bar{\lambda}_{m+1}\gamma_{pf(m)-p}\bar{\mu}_{m}\right)\left(\prod_{n\not\in
S}\gamma_{pf(n)}\bar{\mu}_{n}\right)\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}.$

We define the $\tau$-torsion algebra generators in Theorem 1.1 by

$\mathsf{x}_{S,f}=\partial\chi_{S,f}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p}).$

In particular, $\chi_{\emptyset,0}=1$,
$\chi_{\emptyset,p^{j}\delta_{n}}=\gamma_{p^{j+1}}\bar{\mu}_{n}$ and
$\chi_{\\{m\\},\delta_{m}}=\lambda_{m+1}$. Here
$\delta_{n}:\mathbb{N}\circlearrowleft$ is zero except for $\delta_{n}(n)=1$.
The Bockstein yields
$\mathsf{x}_{S,f}=\sum_{n\in\mathrm{supp}(f)-S}\chi_{S\cup\\{n\\},f}$ since
$\bar{\partial}\gamma_{n}\bar{\mu}_{i}=\bar{\lambda}_{i+1}\gamma_{n-p}\bar{\mu}_{i}$,
$\bar{\partial}\bar{\lambda}_{i}=0$, and $\bar{\partial}$ is a derivation.
Since the classes $\bar{\mu}_{i}$, $\chi_{S,f}$, and the $\bar{\partial}$
cycles $\lambda_{i+1}=\bar{\partial}\gamma_{p}\bar{\mu}_{i}$ generate
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$, the classes $\bar{\mu}_{i}$
and $\mathsf{x}_{S,f}$ generate the boundary.

  6. Step 6

By combining the $\tau$-inverted and mod-$\tau^{p-1}$ calculations we finally
deduce (3). The relation $\mu_{i}^{p}=\tau^{p-1}\mu_{i+1}$ is rooted in the
mod-$p$ motivic Steenrod algebra. The Bockstein calculation shows the
vanishing $\tau^{p-1}\mathsf{x}_{S,f}=0$. Corollary 2.15 shows the
multiplicative relation between the $\mathsf{x}_{S,f}$ classes follows from a
similar formula for the $\chi_{S,f}$ classes. We refer to 2.12 for the entity
$\epsilon_{u}$.

For example, at the prime $p=2$, we obtain the relations

$\mathsf{x}_{\delta_{0}+\delta_{1}}\mathsf{x}_{\delta_{2}}+\mathsf{x}_{\delta_{1}+\delta_{2}}\mathsf{x}_{\delta_{0}}+\mathsf{x}_{\delta_{2}+\delta_{0}}\mathsf{x}_{\delta_{1}}=0,$
$\mathsf{x}_{\delta_{0}+\delta_{1}}\mathsf{x}_{\delta_{1}+\delta_{2}}=\mathsf{x}_{2\delta_{1}}\mathsf{x}_{\delta_{0}+\delta_{2}}.$

Theorem 1.1 admits a succinct reformulation in terms of naturally induced
pullback squares of commutative $\mathbb{F}_{p}[\tau]$-algebras given in
Section 3.3.1 and Section 3.4. For example, when $p=2$, we note the pullback
square of commutative $\mathbb{F}_{p}[\tau]$-algebras

$\textstyle{\mathbf{MHH}_{\star}(\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}[\tau,\mu_{i}]/(\mu_{i}^{2}-\tau\mu_{i+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{2}[\tau,\bar{\mu}_{i},\mathsf{x}_{S,f}]/\mathcal{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{2}[\tau,\bar{\mu}_{i}]/(\bar{\mu}_{i}^{2},\tau)}$

where the ideal of relations is given by

$\mathcal{I}=\left(\tau,\bar{\mu}_{i}^{2},\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\sum_{t_{f+g}\not=u\in\mathrm{supp}(f+g)-S\cup
T}\epsilon_{u}\cdot\mathsf{x}_{S\cup T\cup\\{u\\},f+g}\right).$

Our calculation shows the left vertical map in the pullback is an isomorphism
on $\tau$-torsion classes. Furthermore, the upper horizontal map is an
injection on non-$\tau$-torsion classes. An analogous result holds for all odd
primes.

###  1.1 Notation

This paper uses the following notation.

$p$, $F$ | prime number, base field of exponential characteristic $\mathsf{e}(F)\neq p$
---|---
$\operatorname{Sm}_{F}$ | smooth separated schemes of finite type over $\operatorname{Spec}(F)$
$\mathscr{SH}(F)$ | stable motivic homotopy category of $F$
$\mathscr{R}$ | motivic ring spectrum
$H^{\ast,\ast}$, $h^{\ast,\ast}$ | integral, mod-$p$ motivic cohomology groups of $F$
$K^{M}_{\ast}$, $k^{M}_{\ast}$ | integral, mod-$p$ Milnor $K$-groups of $F$
$\mathbb{M}_{\star}$ | mod-$p$ motivic homology ring of $F$
$\mathcal{A}_{\star}$ | dual motivic Steenrod algebra of $F$ at $p$
$\mathbb{F}_{p,\tau}$ | shorthand for $\mathbb{F}_{p}[\tau]/\tau^{p-1}$
$\Gamma$, $\Lambda$, S | divided power, exterior, symmetric algebras

##  2 Motivic Hochschild homology

######  Definition 2.1.

Let $\mathscr{R}$ be a motivic ring spectrum. The _motivic Hochschild
homology_ of an $\mathscr{R}$-bimodule $\mathscr{M}$ is the derived smash
product

$\mathbf{MHH}(\mathscr{R};\mathscr{M}):=\mathscr{M}\wedge_{\mathscr{R}\wedge\mathscr{R}^{\text{op}}}\mathscr{R}$
(7)

in $\mathscr{SH}(F)$.

When $\mathscr{R}=\mathscr{M}$, the derived tensor product (7) specializes to
$\mathbf{MHH}(\mathscr{R})$ in (1) or equivalently (2) in the event
$\mathscr{R}$ is commutative. If $\mathscr{R}\rightarrow\mathscr{Q}$ is a map
of motivic ring spectra and $\mathscr{M}$ is a $\mathscr{Q}$-$\mathscr{R}$
bimodule, then reassociating the smash factors implies the equivalence

$\mathbf{MHH}(\mathscr{R};\mathscr{M})\simeq\mathscr{M}\wedge_{\mathscr{Q}\wedge\mathscr{R}^{\text{op}}}\mathscr{Q}.$
(8)

In the following, we assume that $\mathscr{R}$ is a cofibrant commutative
motivic ring spectrum in any of the model categorical approaches to
$\mathscr{SH}(F)$ as in [10], [15], [18], [22] (this assumption is superfluous
in the $\infty$-category of motivic spectra [26]). Commutative motivic ring
spectra are cotensored over motivic spaces by the usual adjunctions. We will
only need the special case of simplicial sets or topological spaces. The case
of finite simplicial sets is particularly transparent since it derives from
the relation $\\{1,\dots,n\\}\otimes\mathscr{R}=\mathscr{R}^{\wedge n}$. The
assignment $\mathscr{X}\mapsto\mathscr{X}\otimes\mathscr{R}$ from motivic
spaces to motivic ring spectra has several useful properties which generalize
from the topological setting and which we will use freely.

  * •

$\mathscr{X}\mapsto\mathscr{X}\otimes\mathscr{R}$ is homotopy invariant and
preserves coproducts (and so in particular sends pushouts to smashes).

  * •

$*\otimes\mathscr{R}\cong\mathscr{R}$,
$S^{0}\otimes\mathscr{R}\cong\mathscr{R}\wedge\mathscr{R}$ and (since $S^{1}$
is the derived pushout of $*\leftarrow S^{0}\to*$)
$\mathbf{MHH}(\mathscr{R};\mathscr{M})\simeq\mathscr{M}\wedge_{\mathscr{R}}(S^{1}\otimes\mathscr{R})$.

  * •

The product on $\mathscr{X}\otimes\mathscr{R}$ is induced by the fold
$\mathscr{X}\coprod\mathscr{X}\to\mathscr{X}$.

  * •

Choosing a point $*\to\mathscr{X}$ makes $\mathscr{X}\otimes\mathscr{R}$ an
augmented commutative $\mathscr{R}$-algebra.

  * •

The inclusion $\\{-1,1\\}\subseteq\\{-1,0,1\\}\cong\\{0,-1\\}\vee\\{0,1\\}$
induces the comultiplication
$\mathscr{R}\wedge\mathscr{R}\to\mathscr{R}\wedge\mathscr{R}\wedge\mathscr{R}\cong(\mathscr{R}\wedge\mathscr{R})\wedge_{\mathscr{R}}(\mathscr{R}\wedge\mathscr{R})$
and the nontrivial map $\\{-1,1\\}\to\\{-1,1\\}$ gives the anti-involution of
the “dual Steenrod $\mathscr{R}$-Hopf algebroid”
$S^{0}\otimes\mathscr{R}=\mathscr{R}\wedge\mathscr{R}$ (alge _broid_ since the
maps involved are not pointed, and so there is no guarantee that the units
corresponding to the two choices of base points will coincide). The suspension
of these maps give the pinch map

$S^{1}\cong[-1,1]\coprod_{\\{-1,1\\}}*\to[-1,1]\coprod_{\\{-1,0,1\\}}*\cong
S^{1}\vee S^{1}$

and the flip map $S^{1}\to S^{1}$ both of which are pointed maps, inducing the
$\mathscr{R}$-Hopf algebra structure

$\psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.32361pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S^{1}\otimes\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.32361pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.32361pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(S^{1}\vee
S^{1})\otimes\mathscr{R}\cong(S^{1}\otimes\mathscr{R})\wedge_{\mathscr{R}}(S^{1}\otimes\mathscr{R})}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad\chi\colon
S^{1}\otimes\mathscr{R}\cong S^{1}\otimes\mathscr{R}$

on the “motivic Hochschild homology” — to implement this using finite
simplicial models of the circle, one must subdivide.

Hence, if $\mathbf{MHH}_{\star}(\mathscr{R})$ is flat over
$\mathscr{R}_{\star}$, which will turn out not to be true for
$\mathscr{R}=\mathbf{M}\mathbb{F}_{p}$, we get an $\mathscr{R}_{\star}$-Hopf
algebra structure on $\mathbf{MHH}_{\star}(\mathscr{R})$.

  * •

The tensor with spaces in the category of motivic spectra is $X{}\mapsto
X{}_{+}\wedge\mathscr{R}$ and the universal property defines a unique map of
motivic spectra

$\sigma^{+}\colon X{}_{+}\wedge\mathscr{R}\to X{}\otimes\mathscr{R}.$

If $X$ is a set considered as a motivic space, the inclusion of the points
$\\{x\\}\subseteq X$ induces the desired map
$X_{+}\wedge\mathscr{R}\cong\bigvee_{\\{x\\}\in X}\\{x\\}\otimes\mathscr{R}\to
X\otimes\mathscr{R}$. If $X{}$ is already pointed, the basepoint in $X{}$
makes $X{}\otimes\mathscr{R}$ an $\mathscr{R}$-algebra, giving rise to the
free extension to an $\mathscr{R}$-linear map

$\sigma\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.85138pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{R}\wedge
X_{+}\wedge\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
25.82365pt\raise 5.54723pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9639pt\hbox{$\scriptstyle{1\otimes\sigma^{+}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 48.85138pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.85138pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathscr{R}\wedge
X{}\otimes\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
98.4624pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\text{mult.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 121.53186pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
121.53186pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X{}\otimes\mathscr{R}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

  * •

In general, if $\mathcal{A}$ is a commutative $\mathscr{R}$-algebra then
$X\mapsto\mathcal{A}^{X}=\hom_{\mathscr{R}}(X_{+}\wedge\mathscr{R},\mathcal{A})$
is a cotensor (doesn’t depend on $\mathscr{R}$). The unit of adjunction

$\alpha^{\mathscr{R}}\colon\mathcal{A}\to(X\otimes^{\mathscr{R}}\mathcal{A})^{X}$

is a map of commutative $\mathscr{R}$-algebras. In the category of
$\mathscr{R}$-modules, the adjoint of $\alpha^{\mathscr{R}}$ takes the form

$\textstyle{\sigma^{\mathscr{R}}\colon(X_{+}\wedge\mathscr{R})\wedge_{\mathscr{R}}\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge\alpha^{\mathscr{R}}}$$\textstyle{(X_{+}\wedge\mathscr{R})\wedge_{\mathscr{R}}(X\otimes^{\mathscr{R}}\mathcal{A})^{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ev$\textstyle{(X\otimes^{\mathscr{R}}\mathcal{A}),}$

where ev is the evaluation.

Composition gives an $\mathscr{R}$-algebra map
$\nu\colon\mathscr{R}^{X}\wedge_{\mathscr{R}}(X\otimes^{\mathscr{R}}\mathcal{A})\to(X\otimes^{\mathscr{R}}\mathcal{A})^{X}$.
Assume that $X$ is a finite cell complex and that
$\pi_{\star}(X_{+}\wedge\mathscr{R})$ is a finitely generated
$\pi_{\star}\mathscr{R}$-module with basis $\mathcal{B}$. Then $\nu$ is an
equivalence and

$\nu^{-1}_{*}\alpha_{\star}\colon\pi_{\star}\mathcal{A}\to\hom_{\pi_{\star}\mathscr{R}}(\pi_{\star}(X_{+}\wedge\mathscr{R}),\pi_{\star}\mathscr{R})\otimes_{\pi_{\star}\mathscr{R}}\pi_{\star}(X\otimes\mathcal{A})$

satisfies

$\nu^{-1}_{\star}\alpha_{\star}(a)=\sum_{x\in\mathcal{B}}x^{*}\otimes\sigma^{\mathscr{R}}_{*}(x\otimes
a).$

Here $x^{*}$ is the basis element dual to $x$ and $x\otimes
a\in\pi_{\star}(X_{+}\wedge\mathscr{R})\otimes_{\pi_{\star}\mathscr{R}}\pi_{\star}\mathcal{A}=\pi_{\star}((X_{+}\wedge\mathscr{R})\wedge_{\mathscr{R}}\mathcal{A})$.
We’ll use this formula in Lemma 2.3 to get a relation in
$\mathbf{MHH}(\mathbb{F}_{p})$ (in the topological case, see [1, §5] for
$X=S^{1}$ using the circle action).

Note that
$X\otimes^{\mathscr{R}}(\mathscr{R}\wedge\mathscr{R})\cong\mathscr{R}\wedge(X\otimes\mathscr{R})$
is the tensor of $X$ with $\mathscr{R}\wedge\mathscr{R}$ in the category of
commutative $\mathscr{R}$-algebras, and there is a commutative diagram

$\textstyle{(X_{+}\wedge\mathscr{R})\wedge_{\mathscr{R}}(\mathscr{R}\wedge\mathscr{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\mathscr{R}}}$$\scriptstyle{\cong}$$\textstyle{X\otimes^{\mathscr{R}}(\mathscr{R}\wedge\mathscr{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathscr{R}\wedge
X_{+}\wedge\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge\sigma^{+}}$$\textstyle{\mathscr{R}\wedge
X\otimes\mathscr{R}}$

where the vertical isomorphisms are the associators.

###  2.1 Comparison of simplicial models

It will be convenient to make explicit some of the simplicial models and how
they interact. In this subsection, let $I=\Delta[1]$ be the simplicial
interval with boundary $S^{0}=\partial\Delta[1]$ and let
$S^{1}=I\coprod_{S^{0}}*$ be the simplicial circle. The subdivision of the
circle relevant for the comultiplication is $dS^{1}=(I\coprod
I)\coprod_{S^{0}\coprod S^{0}}S^{0}$ with weak equivalence $dS^{1}\to S^{1}$
given by sending the first interval to the base point. The pinch map
$\nabla\colon dS^{1}\to S^{1}\vee S^{1}$ identifies the endpoints. It is
sometimes convenient to write $dS^{1}$ as $*\coprod_{S^{0}}(I\times
S^{0})\coprod_{S^{0}}*$. Under the canonical isomorphism
$\mathscr{R}=*\otimes\mathscr{R}$ we get an identification
$S^{1}\otimes\mathscr{R}=(I\otimes^{\mathscr{R}}\mathscr{R})\wedge_{S^{0}\otimes\mathscr{R}}\mathscr{R}$
which is a concrete model for the derived smash
$\mathscr{R}\wedge^{L}_{\mathscr{R}\wedge\mathscr{R}}\mathscr{R}$ and
$dS^{1}\otimes\mathscr{R}=((I\coprod
I)\otimes\mathscr{R})\wedge_{(S^{0}\coprod
S^{0})\otimes\mathscr{R}}S^{0}\otimes\mathscr{R}\cong\mathscr{R}\wedge_{\mathscr{R}\wedge\mathscr{R}}I\otimes(\mathscr{R}\wedge\mathscr{R})\wedge_{\mathscr{R}\wedge\mathscr{R}}\mathscr{R}$.

Let $\mathscr{R}\to\mathcal{A}$ be a cofibration of cofibrant commutative
motivic ring spectra. Let $X\otimes^{\mathscr{R}}\mathcal{A}$ be the tensor in
the category of commutative $\mathscr{R}$-algebras of the space $X$ (all
smashes involved are over $\mathscr{R}$). If $\mathscr{M}$ and $\mathscr{N}$
are $\mathcal{A}$-modules, then the derived smash
$\mathscr{M}\wedge^{L}_{\mathcal{A}}\mathscr{N}$ is conveniently modeled as
$\mathscr{M}\wedge_{\mathcal{A}}(I\otimes^{\mathscr{R}}\mathcal{A})\wedge_{\mathcal{A}}\mathscr{N}$,
often referred to as the “two-sided bar construction over $\mathscr{R}$”. Note
that this does not depend on $\mathscr{R}$, in the sense that the map
$\mathscr{M}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{N}\to\mathscr{M}\wedge_{\mathcal{A}}(I\otimes^{\mathscr{R}}\mathcal{A})\wedge_{\mathcal{A}}\mathscr{N}$
is an equivalence. In the special case
$\mathcal{A}=S^{0}\otimes\mathscr{R}=\mathscr{R}\wedge\mathscr{R}$ we get an
identification between the tensor with the subdivided circle and the bar
construction
$S^{1}\otimes\mathscr{R}\cong\mathscr{R}\wedge_{\mathcal{A}}(I\otimes^{\mathscr{R}}\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}$
and
$dS^{1}\otimes\mathscr{R}\cong\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}$.
If one wishes to write the comultiplication

$\psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.32361pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S^{1}\otimes\mathscr{R}}$}}}}}}}{\hbox{\kern
39.32361pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
dS^{1}\otimes\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.9026pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 15.32362pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
76.78679pt\raise 5.68333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 99.17569pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
99.17569pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(S^{1}\vee
S^{1})\otimes\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
164.59506pt\raise 6.15pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 182.22006pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
182.22006pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(S^{1}\otimes\mathscr{R})\wedge_{\mathscr{R}}(S^{1}\otimes\mathscr{R})}$}}}}}}}\ignorespaces}}}}\ignorespaces$

in terms of the bar construction, a concrete way is to use the equivalence
$I\otimes\mathcal{A}\to\mathcal{A}$ and the augmentation
$I\otimes\mathcal{A}\to\mathscr{R}$ as in the diagram

$\textstyle{\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{(\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R})\wedge_{\mathscr{R}}(\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}\wedge_{\mathcal{A}}(I\otimes\mathcal{A})\wedge_{\mathcal{A}}\mathscr{R}.}$$\scriptstyle{\cong}$

This formula only uses the augmentation $\mathcal{A}\to\mathscr{R}$ and not
specifically that $\mathcal{A}=\mathscr{R}\wedge\mathscr{R}$. One may replace
the $\otimes$ by $\otimes^{\mathscr{R}}$ if convenient.

###  2.2 Some classes coming from the dual motivic Steenrod algebra

Let
$\mathcal{A}_{\star}=\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})$
be the dual motivic Steenrod algebra of our ground field $F$ at $p$,

$\mathcal{A}_{\star}=\begin{cases}\mathbb{M}_{\star}[\xi_{i},\tau_{i}]_{i\geq
0}/(\tau_{i}^{2}-\rho(\tau_{i+1}-\tau_{0}\xi_{i+1})-\tau\xi_{i+1})&p=2\\\
\mathbb{M}_{\star}[\xi_{i}]_{i\geq
0}\otimes_{\mathbb{M}_{\star}}\Lambda_{\mathbb{M}_{\star}}(\tau_{i})_{i\geq
0}&p\neq 2\end{cases}$ (9)

(where $\mathbb{M}_{\star}$ is the mod-$p$ motivic homology ring of $F$;
$\tau$ and $\rho$ are discussed below), whose Hopf algebroid structure is
given in [17, §5.1], [34, §12]. Our notation indicates that $\tau_{i}$ is an
exterior class when $p\neq 2$. By convention we set $\xi_{0}=1$. The bidegrees
of the generators in (9) are given by

$|\xi_{i}|=(2p^{i}-2,p^{i}-1),\qquad|\tau_{i}|=(2p^{i}-1,p^{i}-1)$

The coproducts of the generators are defined by

$\psi\xi_{i}=\sum_{j=0}^{i}\xi_{i-j}^{2^{j}}\otimes\xi_{j},\qquad\psi\tau_{i}=\tau_{i}\otimes
1+\sum_{j=0}^{i}\xi_{i-j}^{2^{j}}\otimes\tau_{j}.$ (10)

The left unit is the canonical inclusion. When $p=2$, the right unit is
determined by

$\eta_{R}(\rho)=\rho,\qquad\eta_{R}(\tau)=\tau+\rho\tau_{0}$

for the canonical classes $\tau\in\mathbb{M}_{0,-1}\cong\mu_{2}(F)$ and
$\rho\in\mathbb{M}_{-1,-1}\cong F^{\times}/(F^{\times})^{2}$. The mod $2$
Bockstein on $\tau$ equals $\rho$. While $\tau$ is always nontrivial — being
the class of $-1\in\mu_{2}(F)$ — we have $\rho=0$ if $\sqrt{-1}\in F$. The
graded mod-$2$ Milnor $K$-theory ring
$k_{\ast}^{M}\subseteq\mathbb{M}_{\star}$ of the base field $F$ is comprised
of primitive elements. The element $\tau$ is not primitive in general. If $F$
contains a primitive $p$th root of unity so that
$\mathbb{M}_{0,-1}\cong\mathbb{Z}/p\\{\tau\\}$, then $\mathbb{M}_{\star}\cong
k^{M}_{\ast}[\tau]$ by the norm residue isomorphism [33], [35]. We shall also
use the antipodal generators

$c(\tau_{i})=-\tau_{i}+\sum_{j=0}^{i-1}\xi_{i-j}^{p^{j}}c(\tau_{j}),\qquad
c(\xi_{i})=-\xi_{i}+\sum_{j=1}^{i-1}\xi_{i-j}^{p^{j}}c(\xi_{j})$ (11)

detailed in [17, §5]. For legibility, we will abuse notation by implicitly
using the antipodal classes (11) in our computations. Voevodsky defines in
[32, §3.1] the mod-$p$ rigid motivic Steenrod algebra

$\mathcal{A}_{\star}^{\textrm{rig}}:=\bigotimes_{i\geq
0}\text{S}_{\mathbb{F}_{p}}(\xi_{i+1})\otimes\Lambda_{\mathbb{F}_{p}}(\tau_{i}).$
(12)

The equation (10) gives the coproducts of the generators. For $p\neq 2$ this
is the dual topological Steenrod algebra at $p$.

######  Remark 2.2.

Suppose $\overline{F}$ is an algebraically closed field of positive
characteristic $\neq p$. Its ring of Witt vectors $W(\overline{F})$ is a
henselian local ring with residue field $\overline{F}$. Let $\overline{K}$
denote an algebraic closure of the quotient field $K$ of $W(\overline{F})$. We
note that $\overline{K}$ has characteristic zero. The natural maps

$\overline{K}\leftarrow W(\overline{F})\rightarrow\overline{F}$

induce isomorphisms on $\mathbb{M}_{\star}$ and $\mathcal{A}_{\star}$
according in [36, §4,5,6]. These algebra isomorphisms preserve the classes
$\tau_{i}$ and $\xi_{i}$. Moreover, $\mathbb{M}_{\star}$ and
$\mathcal{A}_{\star}$ are invariant under extensions of algebraically closed
fields of characteristic zero.

The structure of the dual Steenrod algebra has some direct consequences for
motivic Hochschild homology. Recall the suspension operation

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{MHH}(\mathbb{F}_{p})}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Here, $s_{1}\in H_{1}(S^{1}_{+};\mathbb{F}_{p})$ is the standard generator and
if
$\zeta\in\pi_{s,w}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})$,
we let “$\sigma\zeta$” denote the “homology suspension” of $\zeta$, namely the
image of $s_{1}\zeta$ in $\mathbf{MHH}_{s+1,w}(\mathbb{F}_{p})$ and also in
$\pi_{s+1,w}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p}))$.

######  Lemma 2.3.

In the motivic Hochschild homology $\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ we
have the relations

$\tau^{p-1}\sigma\tau_{i+1}=(\sigma\tau_{i})^{p},\qquad\tau^{p-1}\sigma\xi_{i+1}=\rho\sigma\tau_{i+1}$

for all $i\geq 0$ (where $\rho:=0$ for odd primes $p$).

###### Proof.

Both relations are shown already in homology from which the homotopy versions
follow by the $\mathbf{M}\mathbb{F}_{p}$-algebra structure
$\text{mult}\colon\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p})\to\mathbf{MHH}(\mathbb{F}_{p})$
(splitting the inclusion of homotopy in homology).

We first show

$\tau^{p-1}(1\wedge\sigma^{+})_{*}(s_{1}\otimes\tau_{i+1})=[(1\wedge\sigma^{+})_{*}(s_{1}\otimes\tau_{i})]^{p}$

in the homology $\mathbb{M}_{\star}$-algebra
$\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p}))$. Let
$\mathscr{R}$ be a commutative $\mathbf{M}\mathbb{F}_{p}$-algebra and let
$\alpha\colon S^{s,w}\to\mathscr{R}$ represent a class in
$\pi_{s,w}\mathscr{R}$. Commutative motivic ring spectra are
$E_{\infty}$-algebras and the induced composite

$\textstyle{\mathbf{M}\mathbb{F}_{p}\wedge((E\Sigma_{p})_{+}\wedge_{\Sigma_{p}}(S^{s,w})^{\wedge
p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(E\Sigma_{p})_{+}\wedge_{\Sigma_{p}}\mathscr{R}^{\wedge_{\mathbf{M}\mathbb{F}_{p}}p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$mult$\textstyle{\mathscr{R}}$

(where $E\Sigma_{p}$ is the nerve of the translation category of the symmetric
group $\Sigma_{p}$ on $p$ letters) gives us a “power operation”

$P(\alpha)\colon\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge((E\Sigma_{p})_{+}\wedge_{\Sigma_{p}}(S^{s,w})^{\wedge
p}))\to\pi_{\star}\mathscr{R}.$

The image under $P(\alpha)$ of generators may be called Dyer-Lashof operations
on $\alpha$. Precomposing with

$H_{*}(B\Sigma_{p};\mathbb{F}_{p})\otimes
H_{*}(S^{s,w};\mathbf{M}\mathbb{F}_{p})\to\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge((E\Sigma_{p})_{+}\wedge_{\Sigma_{p}}(S^{s,w})^{\wedge
p}))$

defined on the chain level as the diagonal

$C_{*}(E\Sigma_{p},\mathbb{F}_{p})/\Sigma_{p}\otimes
C_{*}(S^{s,w};\mathbf{M}\mathbb{F}_{p})\to
C_{*}(E\Sigma_{p},\mathbb{F}_{p})\otimes_{\Sigma_{p}}C_{*}(S^{s,w};\mathbf{M}\mathbb{F}_{p})^{\otimes_{\mathbb{F}[\tau]}p}$

and evaluation at the classical choice of generator of
$H_{i}(BC_{p};\mathbb{F}_{p})\subseteq H_{i}(\Sigma_{p},\mathbb{F}_{p})$ gives
us the (topological) Dyer-Lashof operation
$Q_{i}(\alpha)\in\pi_{i+ps,pw}\mathscr{R}$ on $\alpha$. We do the usual shift
to upper indexing with
$Q^{r}(\alpha)\overset{\cdot}{=}Q_{(2r-s)(p-1)}(\alpha)$ (for $p$ odd;
$Q^{r}(\alpha)=Q_{r-s}(\alpha)$ for $p=2$) so that $Q^{0}(\alpha)=\alpha$ and
$Q^{s}(\alpha)=\alpha^{p}$ when $2r=s$ (for $p$ odd; $r=s$ for $p=2$).

If in Section 2.1 we set $\mathscr{R}=\mathbf{M}\mathbb{F}_{p}$ and
$\mathcal{A}=\mathscr{R}\wedge\mathscr{R}$ and let $X$ be any space with
finite basis $\\{x\\}$ for the homology $H_{*}(X;\mathbb{F}_{p})$ and recall
that
$\alpha^{\mathscr{R}}\colon\mathcal{A}\to(X\otimes^{\mathscr{R}}\mathcal{A})^{X}$
and $\nu\colon\mathscr{R}^{X}\wedge_{X}(X\otimes^{\mathscr{R}}\mathcal{A})\to
X\otimes^{\mathscr{R}}\mathcal{A}$ were homomorphisms of commutative
$\mathscr{R}$-algebras we get that

$Q^{s}\nu^{-1}_{*}\alpha^{\mathscr{R}}_{*}=\nu^{-1}_{*}\alpha^{\mathscr{R}}_{*}Q^{s}$

and so

$\sum_{x}\sum_{a+b=s}Q^{a}x^{*}\otimes Q^{b}\sigma^{\mathscr{R}}(x\otimes
a)=\sum_{x}x^{*}\otimes\sigma^{\mathscr{R}}(x\otimes Q^{s}a).$

When $X=S^{1}$ the Dyer-Lashof operations $Q^{a}(x^{*})$ are trivial for
$a\neq 0$ and so we get

$Q^{s}\sigma^{\mathscr{R}}(x\otimes a)=\sigma^{\mathscr{R}}(x\otimes Q^{s}a).$

Restricting to the generator $x=s_{1}\in H_{1}(S^{1};\mathbb{F}_{p})$ and
multiplying down to homotopy, we get the crucial formula

$Q^{s}\sigma=\sigma
Q^{s}\colon\pi_{s,w}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})\to\mathbf{MHH}_{s+1+2s(p-1),pw}(\mathbb{F}_{p}).$

By construction, the power operations are preserved under base change. Over
any algebraically closed field, we claim there is a relation

$\tau^{p-1}\tau_{i+1}=Q^{p^{i}}\tau_{i}\in\pi_{2p^{i+1}-1,p^{i+1}-p}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p}).$
(13)

By Remark 2.2 the relation holds over $\bar{F}$ if and only if it does so over
$\bar{K}$. Consequently, by rigidity is suffices to know that the relation
holds over the complex numbers, which follows by Betti realization to the
topological situation (the motivic correction factor $\tau^{p-1}$ ensures the
weights agree). Thus, for the antipodal classes (11), we obtain the formula

$\tau^{p-1}\sigma\tau_{i+1}=\sigma
Q^{p^{i}}\tau_{i}=Q^{p^{i}}\sigma\tau_{i}=(\sigma\tau_{i})^{p}.$ (14)

We use this result to prove the vanishing of
$(1\wedge\sigma^{+})_{*}(s_{1}\otimes\tau^{p-1}\xi_{i+1})$. Let
$\beta\colon\mathbf{M}\mathbb{F}_{p}\to S^{1}\wedge\mathbf{M}\mathbb{F}_{p}$
be the $p$-Bockstein, i.e., the $\mathbf{M}\mathbb{F}_{p}$-linear boundary map
in the fiber sequence of Eilenberg-MacLane spectra associated with the short
exact sequence $p\mathbb{Z}/p^{2}\subseteq\mathbb{Z}/p^{2}\to\mathbb{Z}/p$.
For any commutative ring spectrum $\mathscr{R}$ the map $(\beta\wedge
1)_{*}\colon\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge\mathscr{R})\to\pi_{*}(S^{1}\wedge\mathbf{M}\mathbb{F}_{p}\wedge\mathscr{R})$
is a derivation (since $p\mathbb{Z}/p^{2}\subseteq\mathbb{Z}/p^{2}$ is a
square zero ideal) and as usual we allow ourselves the shorthand $\beta$ for
$(\beta\wedge 1)_{*}$. By construction of the motivic Steenrod algebra, see
[17, §5], [34, §9], the generators in the dual motivic Steenrod algebra
$\mathcal{A}_{\star}=\pi_{\star}(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})$
are connected via

$\xi_{i+1}=\beta\tau_{i+1}.$

Obviously, the diagram

$\textstyle{\mathbf{M}\mathbb{F}_{p}\wedge
S^{1}_{+}\wedge\mathbf{M}\mathbb{F}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge\sigma^{+}}$$\scriptstyle{\beta\wedge
1}$$\textstyle{\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\wedge
1}$$\textstyle{S^{1}\wedge\mathbf{M}\mathbb{F}_{p}\wedge
S^{1}_{+}\wedge\mathbf{M}\mathbb{F}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge\sigma^{+}}$$\textstyle{S^{1}\wedge\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{MHH}(\mathbb{F}_{p})}$

commutes, and since the power operations commute with
$\sigma^{\mathbf{M}\mathbb{F}_{p}}=1\wedge\sigma^{+}$ we get for $p$ odd
(where $\beta\tau=0$) that

$\displaystyle 0$
$\displaystyle=\text{mult.}\beta[\sigma^{\mathbf{M}\mathbb{F}_{p}}_{*}(s_{1}\otimes\tau_{i})]^{p}=\text{mult.}\beta\sigma^{\mathbf{M}\mathbb{F}_{p}}_{*}(s_{1}\otimes\tau^{p-1}\tau_{i+1})$
$\displaystyle=\sigma\beta\tau^{p-1}\tau_{i+1}=\sigma\tau^{p-1}\xi_{i+1}=\tau^{p-1}\sigma\xi_{i+1}.$

For $p=2$ we’ll see that we can easily read the last formula directly from the
$d^{1}$-differentials in the Tor-spectral sequence, but we may also use the
Bockstein and compute

$\displaystyle 0$
$\displaystyle=\text{mult.}\beta[\sigma^{\mathbf{M}\mathbb{F}_{2}}_{*}(s_{1}\otimes\tau_{i})]^{2}=\text{mult.}\beta\sigma^{\mathbf{M}\mathbb{F}_{2}}_{*}(s_{1}\otimes\tau\tau_{i+1})$
$\displaystyle=\sigma(\beta(\tau\tau_{i+1}))=\sigma(\beta(\tau)\tau_{i+1}+\tau\beta(\tau_{i+1}))=\sigma(\rho\tau_{i+1}+\tau\xi_{i+1})=\rho\sigma\tau_{i+1}+\tau\sigma\xi_{i+1}.$

∎

###  2.3 Tor spectral sequence for motivic Hochschild homology

A motivic spectrum is cellular if it belongs to the smallest full subcategory
of the stable motivic homotopy category, which is closed under homotopy
colimits and contains the motivic spheres $S^{p,q}$ for all
$p,q\in\mathbb{Z}$, see [9, §2.8]. The cellularity assumption is central in
motivic homotopy theory, see, e.g., [28, §2.3]. It is, moreover, needed for
running the motivic Tor spectral sequence.

We begin by relating the integral Tor spectral sequence to the bar
construction. Our setup is a map of motivic ring spectra
$\mathscr{R}\rightarrow\mathscr{Q}$ and an $\mathscr{Q}$-$\mathscr{R}$
bimodule $\mathscr{M}$. We assume that $\mathscr{R}$ is a commutative motivic
ring spectrum and $\mathcal{A}=\mathscr{Q}\wedge\mathscr{R}$ is a cofibrant
$\mathscr{R}$-algebra. Then the derived smash product
$\mathscr{M}\wedge_{\mathcal{A}}\mathscr{Q}$ in (8) is the realization of the
simplicial object

$\\{[s]\mapsto\mathscr{M}\wedge_{\mathscr{R}}\mathcal{A}^{\wedge_{\mathscr{R}}^{s}}\wedge_{\mathscr{R}}\mathscr{Q}\\}.$

The skeletal filtration yields the $E^{1}$ page of the Tor spectral sequence,
which — if $\pi_{\star}\mathcal{A}$ is flat over $\pi_{\star}\mathscr{R}$ —
takes the form

$E^{1}_{s,\star}=B_{s}(\pi_{\star}\mathscr{M},\pi_{\star}\mathcal{A},\pi_{\star}\mathscr{Q})=\pi_{\star}\mathscr{M}\otimes_{\pi_{\star}\mathscr{R}}\pi_{\star}\mathcal{A}^{\otimes_{\pi_{\star}\mathscr{R}}^{s}}\otimes_{\pi_{\star}\mathscr{R}}\pi_{\star}\mathscr{Q}.$

It is conventional to denote the generators of the bar complex by
$[m_{0}|a_{1}|\dots|a_{s}|n_{s+1}]$. When
$\mathscr{M}=\mathscr{Q}=\mathscr{R}$ we abbreviate $[1|a_{1}|\dots|a_{s}|1]$
to $[a_{1}|\dots|a_{s}]$. The homology of $E^{1}_{s,\star}$ computes the
$E^{2}$ page of the Tor spectral sequence (15). We recall the $d^{1}$
differential is given by the alternating sum of the face maps

$d_{i}[m_{0}|a_{1}|\dots|a_{s}|n_{s+1}]=\begin{cases}[m_{0}\cdot
a_{1}|a_{2}|\dots|a_{s}|n_{s+1}]&i=0\\\ [m_{0}|a_{1}|\dots|a_{i}\cdot
a_{i+1}|\dots|a_{s}|n_{s+1}]&0<i<s\\\ [m_{0}|a_{1}|\dots|a_{s-1}|a_{s}\cdot
n_{s+1}]&i=s.\end{cases}$

If $\mathcal{A}$ is commutative and the modules $\mathscr{M}$ and
$\mathscr{Q}$ are commutative $\mathcal{A}$-algebras, then the skeletal
filtration of the simplicial object
$\\{[s]\mapsto\mathscr{M}\wedge_{\mathscr{R}}\mathcal{A}^{\wedge_{\mathscr{R}}^{s}}\wedge_{\mathscr{R}}\mathscr{Q}\\}$
is isomorphic to
$(\mathscr{M}\wedge_{\mathscr{R}}\mathscr{Q})\otimes_{\mathcal{A}}(S^{1}\otimes\mathcal{A})$
_in the category of commutative $\mathscr{R}$-algebras_. This is isomorphic to
the more Hochschild-homology-looking
$\\{[s]\mapsto(\mathscr{M}\wedge_{\mathscr{R}}\mathscr{Q})\wedge\mathscr{Q}^{\wedge
s}\\}$, and the filtration arises from the simplicial circle. Hence the Tor
spectral sequence (15) is a $\pi_{\star}\mathscr{R}$-algebra spectral sequence
with the multiplicative structure on the $E^{1}$ page given by the shuffle
product introduced by Eilenberg-Mac Lane [11].

######  Proposition 2.4.

Suppose $\mathscr{M}$, $\mathscr{R}$, and $\mathscr{S}$ are cellular motivic
spectra. Then the skeletal filtration of the simplicial circle gives rise to a
strongly convergent trigraded Tor spectral sequence

$E^{2}_{h,t,w}=\mathbf{Tor}^{\pi_{\star}(\mathscr{S}\wedge\mathscr{R}^{\text{op}})}_{h,t,w}(\pi_{\star}(\mathscr{M}),\pi_{\star}(\mathscr{S}))\Rightarrow\mathbf{MHH}_{h+t,w}(\mathscr{R};\mathscr{M}).$
(15)

Here, $h$ is the homological grading on the torsion product and $(t,w)$ is the
internal grading for the bigraded motivic homotopy groups in topological
degree $t$ and weight $w$. The differentials are of the form

$d^{r}\colon E^{r}_{\ast,\ast,\ast}\rightarrow E^{r}_{\ast-r,\ast+r-1,\ast}.$

If $\mathscr{R}$ is commutative and $\mathscr{S}$ and $\mathscr{M}$ are
commutative $\mathscr{R}$-algebras, then the Tor spectral sequence is a
spectral sequence of $\mathscr{R}$-algebras. The pinch map on the circle
induces the Hopf-algebra structure on the torsion groups on the $E^{2}$ page.
If $E^{2}$,…,$E^{r}$ are all flat over $\mathscr{R}_{\star}$ for $2\leq r$,
then the $E^{r}$ page inherits an $\mathscr{R}$-Hopf algebra structure; in
particular, the $r$-th differential $d^{r}\colon E^{r}\to E^{r}$ satisfies the
“co-Leibniz” rule in the sense that it commutes with the coproduct $\psi\colon
E^{r}\to E^{r}\otimes E^{r}$.

###### Proof.

This follows from (8), [1, §4], [9, Proposition 7.7], [25, §2]. ∎

The suspension map $\sigma\colon\mathscr{R}\wedge S^{1}\wedge\mathscr{R}\to
S^{1}\otimes\mathscr{R}$ has a simple interpretation under the isomorphism

$S^{1}\otimes\mathscr{R}\cong\\{[s]\mapsto(\mathscr{R}\wedge_{\mathscr{R}}\mathscr{R})\wedge(\mathscr{R}\wedge\mathscr{R})^{\wedge_{\mathscr{R}}s}\\}$

of the Tor-interpretation: it is the map from
$S^{1}\wedge(\mathscr{R}\wedge\mathscr{R})=\\{[s]\mapsto\bigvee_{\\{1,\dots,s\\}}(\mathscr{R}\wedge\mathscr{R})\\}$
sending the $i$th summand to the inclusion on the $i$th factor (and units
elsewhere). In particular, if $x\in\pi_{d,w}(\mathscr{R}\wedge\mathscr{R})$,
then $\sigma_{*}x\in\pi_{d+1,w}(S^{1}\otimes\mathscr{R})$ is the class
represented by $[x]\in E^{1}_{1,d,w}$.

The Hopkins-Morel equivalence shown by Hoyois [16, Proposition 8.1] implies
the cellularity assumption in Proposition 2.4 holds for
$\mathscr{M}=\mathscr{R}=\mathscr{S}=\mathbf{M}\mathbb{F}_{p}$ since the base
scheme $F$ is a field of exponential characteristic $\mathsf{e}(F)\neq p$. In
this case, we have the Tor spectral sequence

$E^{2}_{h,t,w}=\mathbf{Tor}^{\mathcal{A}_{\star}}_{h,t,w}(\mathbb{M}_{\star},\mathbb{M}_{\star})\Rightarrow\mathbf{MHH}_{h+t,w}(\mathbf{M}\mathbb{F}_{p}).$
(16)

######  Remark 2.5.

By Remark 2.5 and (16) it follows that, for algebraically closed fields,
$\mathbf{MHH}_{\star}(\mathbf{M}\mathbb{F}_{p})$ is independent of the
exponential characteristic $\neq p$.

###  2.4 Torsion products

For reference we record some basic facts about the structure of
$\operatorname{Tor}$ in simple situations. If $(\mathsf{x}_{i})$ is a basis
for an $\mathbb{F}_{p}$-vector space, we denote the corresponding divided
power, exterior, and symmetric algebras by
$\Gamma_{\mathbb{F}_{p}}(\mathsf{x}_{i})$,
$\Lambda_{\mathbb{F}_{p}}(\mathsf{x}_{i})$, and
$\text{S}_{\mathbb{F}_{p}}(\mathsf{x}_{i})$, respectively. Let
$\gamma_{n}\mathsf{x}$ denote the $n$th divided power of a class $\mathsf{x}$
in degree $d$. Then the graded divided power algebra
$\Gamma_{\mathbb{F}_{p}}(\mathsf{x})$ is generated by elements
$\gamma_{p^{j}}\mathsf{x}$ in degree $dp^{j}$ subject to the relations
$\gamma_{0}\mathsf{x}=1$, $\gamma_{1}\mathsf{x}=\mathsf{x}$, and

$\gamma_{m}\mathsf{x}\cdot\gamma_{n}\mathsf{x}=\left(\begin{array}[]{c}m+n\\\\[4.30554pt]
m\end{array}\right)\gamma_{m+n}\mathsf{x}.$

The symmetric algebra functor $\text{S}_{\mathbb{F}_{p}}(-)$ is left adjoint
to the forgetful functor from $\mathbb{F}_{p}$-algebras to
$\mathbb{F}_{p}$-modules, and the symmetric algebra
$\text{S}_{\mathbb{F}_{p}}(\mathsf{x}_{i})$ is canonically isomorphic to a
polynomial ring. We let $\text{P}^{h}_{\mathbb{F}_{p}}(\mathsf{x})$ denote the
height $h$ truncated polynomial ring
$\text{P}_{\mathbb{F}_{p}}(\mathsf{x})/\mathsf{x}^{h}$. With these
definitions, an exercise in binomial coefficients shows there is an
isomorphism of algebras

$\Gamma_{\mathbb{F}_{p}}(\mathsf{x})=\mathbb{F}_{p}\\{\gamma_{p^{j}}\mathsf{x}|j\geq
0\\}\cong\bigotimes_{j\geq
0}\text{P}^{p}_{\mathbb{F}_{p}}(\gamma_{p^{j}}\mathsf{x}).$ (17)

We shall make repeatedly use of the following torsion product computations,
see [25, §6].

######  Lemma 2.6.

  1. (i)

For the symmetric algebra $\text{S}_{\mathbb{F}_{p}}(\mathsf{x})$ on a
generator $\mathsf{x}$ in even degree $d$, there is an
$\mathbb{F}_{p}$-bialgebra isomorphism

$\mathbf{Tor}^{\text{S}_{\mathbb{F}_{p}}(\mathsf{x})}_{\ast}(\mathbb{F}_{p},\mathbb{F}_{p})\cong\Lambda_{\mathbb{F}_{p}}(\sigma\mathsf{x}).$

Here, $\sigma\mathsf{x}$ in degree $(1,d)$ is a coalgebra primitive
represented in the bar complex by $[\mathsf{x}]$.

  2. (ii)

For the exterior algebra $\Lambda_{\mathbb{F}_{p}}(\mathsf{x})$ on a generator
$\mathsf{x}$ in odd degree $d$, there is an $\mathbb{F}_{p}$-bialgebra
isomorphism

$\mathbf{Tor}^{\Lambda_{\mathbb{F}_{p}}(\mathsf{x})}_{\ast}(\mathbb{F}_{p},\mathbb{F}_{p})\cong\Gamma_{\mathbb{F}_{p}}(\sigma\mathsf{x}).$

Here, $\gamma_{j}\sigma\mathsf{x}$ in degree $(j,dj)$ is represented in the
bar complex by $[\mathsf{x}|\mathsf{x}|\dots|\mathsf{x}]$ and has coproduct

$\psi\gamma_{k}\sigma\mathsf{x}=\underset{i+j=k}{\sum}\gamma_{i}\sigma\mathsf{x}\otimes\gamma_{j}\sigma\mathsf{x}.$

######  Remark 2.7.

As an example, let us reconstruct a direct proof of the equation
$\tau\sigma\xi_{i+1}=\rho\sigma\tau_{i+1}\in\mathbf{MHH}_{\star}(\mathbb{F}_{2})$
of Lemma 2.3. Consider the $E^{1}$ page

$E^{1}_{s,\star}=\mathcal{A}_{\star}^{\otimes_{\mathbb{M}_{\star}}s}\cong\mathbb{M}_{\star}\otimes_{\mathbb{M}_{\star}}\mathcal{A}_{\star}^{\otimes_{\mathbb{M}_{\star}}s}\otimes_{\mathbb{M}_{\star}}\mathbb{M}_{\star}$

of the spectral sequence for $\mathbf{MHH}(\mathbb{F}_{2})$, where
$\mathcal{A}_{\star}=\pi_{\star}(\mathbf{M}\mathbb{F}_{2}\wedge\mathbf{M}\mathbb{F}_{2})$
is the dual Steenrod algebra. Then

$d^{1}[\tau_{i}|\tau_{i}]=[\tau^{2}_{i}]=\tau[\xi_{i+1}]+\rho[\tau_{i+1}]+\rho[\tau_{0}\xi_{i+1}]$

and $d^{1}[\tau_{0}|\xi_{i+1}]=[\tau_{0}\xi_{i+1}]$, so that
$\tau[\xi_{i+1}]+\rho[\tau_{i+1}]$ is a boundary. Hence
$\tau\sigma\xi_{i+1}=\rho\sigma\tau_{i+1}$.

With the notation $\gamma_{j}\sigma\tau_{i}=[\tau_{i}|\dots|\tau_{i}]\in
E^{1}_{j,\star}$ and $\sigma x=[x]$ the shuffle product yields an explicit
formula for the $d^{1}$ differentials

$\displaystyle d^{1}\gamma_{j+2}\sigma\tau_{i}$
$\displaystyle=\sum_{a=1}^{j-1}[\tau_{i}|\dots|\tau_{i}^{2}|\dots|\tau_{i}]=[\tau_{i}^{2}]\cdot[\tau_{i}\dots|\tau_{i}]=(\tau[\xi_{i+1}]+\rho[\tau_{i+1}]+\rho[\tau_{0}\xi_{i+1}])\cdot[\tau_{i}\dots|\tau_{i}]$
$\displaystyle=(\tau\sigma\xi_{i+1}+\rho\sigma\tau_{i+1}+\rho\sigma(\tau_{0}\xi_{i+1}))\cdot\gamma_{j}\sigma\tau_{i}.$

When the ground field contains a square root of $-1$, so that $\rho=0$, we get
the formula

$d^{1}\gamma_{j+2}\sigma\tau_{i}=\tau\sigma\xi_{i+1}\cdot\gamma_{j}\sigma\tau_{i}.$

Conversely, for odd primes $p$ we can use Lemma 2.3 to _deduce_ differentials
by a simple weight count — simplifying the corresponding topological argument.
Lemma 2.6 tells us that

$E^{2}=\mathbb{M}_{\star}\otimes_{\mathbb{F}_{p}}\bigotimes_{i\geq}\Lambda_{\mathbb{F}_{p}}(\sigma\xi_{i+1})\otimes\Gamma_{\mathbb{F}_{p}}(\sigma\tau_{i}).$

We know that $\tau^{p-1}\sigma\xi_{i+1}$ has to be hit by a differential. When
the ground field is algebraically closed,
$\mathbb{M}_{\star}=\mathbb{F}_{p}[\tau]$ with $\tau\in\mathbb{M}_{0,-1}$. In
this case the source of the differential hitting $\tau^{p-1}\sigma\xi_{i+1}$
must come from linear combinations of monomials in $\sigma\xi_{i}$s and
$\gamma_{j}\sigma\tau_{i}$s of total degree $2p^{i+1}$ and weight at least
$p^{i+1}-p$. A quick count shows that the only monomial with sufficient weight
is $\gamma_{p}\sigma\tau_{i}$ and so we have the equation (described up to a
unit in $\mathbb{F}_{p}$)

$d^{p-1}\gamma_{p}\sigma\tau_{i}\overset{\cdot}{=}\tau^{p-1}\sigma\xi_{i+1}.$

###  2.5 A Bockstein type complex

We end the section by doing an entirely algebraic exercise which will be
needed later on. Let $p$ be any prime and consider the commutative
differential graded $\mathbb{F}_{p}$-algebra $(C,D)$, where

$C=\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\bar{\mu}_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1})$

and $D\colon C\to C$ is the derivation generated by
$D(\gamma_{j+p}\bar{\mu}_{i})=\bar{\lambda}_{i+1}\gamma_{j}\bar{\mu}_{i}$. As
before, $B^{D}=\mathrm{im}D$, $Z^{D}=\ker D$ and $H^{D}=Z^{D}/B^{D}$ — the aim
of this subsection is to calculate these. In the application $C$ will be the
mod-$\tau^{p-1}$ motivic Hochschild homology of $\mathbb{F}_{p}$ (the reader
may recognize it as
$\operatorname{Tor}^{\mathcal{A}^{\mathrm{rig}}_{\star}}_{*}(\mathbb{F}_{p},\mathbb{F}_{p})$)
and $D$ will be derived from a Bockstein.

We first fix some notation. For each non-empty finite set of natural numbers
$S\subseteq\mathbb{N}$, we choose an element $t_{S}\in S$ with the property
that $t_{S\cup T}\in\\{t_{S},t_{T}\\}$. The minimum, $t_{S}=\min S$, is a good
choice, but there are many others. Down the road, such a choice amounts to a
particular choice of basis, and there is no reason to prefer one over the
other, except that in concrete examples, some can be more convenient. If the
function $f\colon\mathbb{N}\circlearrowleft$ has finite non-empty support, we
write $t_{f}=t_{\mathrm{supp}f}$. For every $j\in\mathbb{N}$, let
$\delta_{j}\colon\mathbb{N}\circlearrowleft$ be the function with
$\mathrm{supp}\,\delta_{j}=\\{j\\}$ and $\delta_{j}(j)=1$.

######  Definition 2.8.

Let $J$ denote the set of pairs $(S,f)$, where the function
$f\colon\mathbb{N}\circlearrowleft$ has finite support and
$S\subseteq\mathrm{supp}f$. The subset $K\subseteq J$ consists of the pairs
$(S,f)$, where the support of $f$ is non-empty and $S$ does not contain
$t_{f}$.

######  Definition 2.9.

For $(S,f)\in J$, we set

$\chi_{S,f}:=\left(\prod_{m\in
S}\bar{\lambda}_{m+1}\gamma_{pf(m)-p}\bar{\mu}_{m}\right)\left(\prod_{n\not\in
S}\gamma_{pf(n)}\bar{\mu}_{n}\right)\in C.$ (18)

In particular, $\chi_{\emptyset,0}=1$,
$\chi_{\emptyset,p^{j}\delta_{n}}=\gamma_{p^{j+1}}\bar{\mu}_{n}$ and
$\chi_{\\{m\\},\delta_{m}}=\lambda_{m+1}$.

We note that

$D\chi_{S,f}=\sum_{n\in\mathrm{supp}(f)-S}\chi_{S\cup\\{n\\},f}$

since $D\gamma_{n}\bar{\mu}_{i}=\bar{\lambda}_{i+1}\gamma_{n-p}\bar{\mu}_{i}$.
Next we construct sub-complexes of $(C,D)$.

######  Definition 2.10.

If $f\colon\mathbb{N}\circlearrowleft$ has finite support, the _associated
$f$-cube_ is the sub-complex

$(C^{f},D):=(\bigoplus_{S\subseteq\mathrm{supp}f}\mathbb{F}_{p}\\{\chi_{S,f}\\},D)\subseteq(C,D).$

If $f=0$, then $C^{f}=\mathbb{F}_{p}$ . Furthermore, let $Z^{f}=\ker D\cap
C^{f}$, $B^{D}=\mathrm{im}D\cap C^{f}$ and $H^{f}=Z^{f}/B^{f}$.

Note that if $f=0$, then $Z^{f}=H^{f}=\mathbb{F}_{p}$. Recall the number
$t_{f}\in\mathrm{supp}f$ chosen once and for all (whenever $\mathrm{supp}f$ is
non-empty) just before Definition 2.8.

######  Lemma 2.11.

If $f\colon\mathbb{N}\circlearrowleft$ has finite nonempty support, then
$(C^{f},D)$ is contractible so that $H^{f}=0$. Furthermore, $B^{f}=Z^{f}$ is
generated by the $D\chi_{S,f}$ with $t_{f}\notin S\subseteq\mathrm{supp}f$.

###### Proof.

For $N=|\mathrm{supp}f|$ and $k=0,\dots,N$, let $C^{f}_{k}\subseteq C^{f}$ be
the span of the $\chi_{S,f}$ with $k=|S|$. From the formula
$D\chi_{S,f}=\sum_{n\in\mathrm{supp}(f)-S}\chi_{S\cup\\{n\\},f}$ we see that
the differential restricts to a chain complex

$(C^{f},{D})=\left\\{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.60347pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C^{f}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.60855pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{D}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.60347pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.60347pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{f}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.88875pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{D}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.81041pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.81041pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
93.31549pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{D}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 111.31041pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
111.31041pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{f}_{N-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
134.52243pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{D}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 152.51735pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
152.51735pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{f}_{N}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right\\}.$

Here the $\mathbb{F}_{p}$-vector space $C^{f}_{k}$ is of dimension
$\binom{N}{k}$ with basis elements $\chi_{S,f}$, where $|S|=k$. Note that the
set

$\\{{D}\chi_{S,f}\mid k=|S|,t_{f}\notin S\\}$

is linearly independent because only ${D}\chi_{S,f}$ has a nontrivial
$\chi_{\\{t_{f}\\}\cup S,f}$-coefficient. Hence the rank of ${D}\colon
C^{f}_{k}\to C^{f}_{k+1}$ is at least $\binom{N-1}{k}$, and we deduce that

$\displaystyle\dim H_{k}(C^{f},{D})$ $\displaystyle=\dim\ker\\{{D}\colon
C^{f}_{k}\to C^{f}_{k+1}\\}-\dim\mathrm{im}\\{{D}\colon C^{f}_{k-1}\to
C^{f}_{k}\\}$
$\displaystyle\leq\binom{N}{k}-\binom{N-1}{k}-\binom{N-1}{k-1}=0$

and so $B^{f}=Z^{f}$ is generated by the $D\chi_{S,f}$ with $t_{f}\notin
S\subseteq\mathrm{supp}f$, as claimed. The calculation works when $k=0$ or
$k=N$ (but not for $N=0$ since then we cannot choose $t_{f}$). ∎

We analyze the multiplicative structure.

######  Definition 2.12.

For functions $f,g\colon\mathbb{N}\circlearrowleft$ with finite support and
non-empty finite sets $S,T\subseteq\mathbb{N}$ define
$K_{S,T,f,g}\in\mathbb{F}_{p}$ by

$K_{S,T,f,g}=\left(\prod_{s\in S}\binom{fs-1+gs}{fs-1}\right)\left(\prod_{t\in
T}\binom{ft+gt-1}{ft}\right)\left(\prod_{c\notin S\cup
T}\binom{fc+gc}{fc}\right)$

if $(S,f),(T,g)\in J$ and $S\cap T=\emptyset$, and set $K_{S,T,f,g}=0$
otherwise. Moreover, we define

$\epsilon_{u,S,T,f,g}=K_{S\cup\\{u\\},T\cup\\{t_{f+g}\\},f+g}+K_{S\cup\\{t_{f+g}\\},T\cup\\{u\\},f+g}.$

Note that when $(S,f),(T,g)\in J$ and $S\cap T=\emptyset$, each factor in the
formula $K_{S,T,f,g}$ is $1$ unless the index is in
$\mathrm{supp}\,f\cap\mathrm{supp}\,g=\mathrm{supp}(f\cdot g)$, and so we can
restrict to these factors to simplify the calculation. We will need
$\epsilon_{u,S,T,f,g}$ only in the case when $u\in\mathrm{supp}(f+g)$,
$u\not=t_{f+g}$ and $u\notin S\cup T$.

The following lemma, a consequence of the defining relations among divided
power generators of $C$, explains the relevance of these numbers.

######  Lemma 2.13.

For $(S,f),(T,g)\in J$ we have

$\chi_{S,f}\chi_{T,g}=K_{S,T,f,g}\cdot\chi_{S\cup T,f+g}$

and if $(S,f),(T,g)\in K$, then

$D\chi_{S,f}\cdot D\chi_{T,g}=\sum_{t_{f+g}\not=u\in\mathrm{supp}(f+g)-S\cup
T}\epsilon_{u}\cdot D\chi_{S\cup T\cup\\{u\\},f+g}.$

∎

######  Lemma 2.14.

The multiplication gives an extra grading indexed by the generators of the
commutative differential graded sub-algebra $\bigoplus_{f}C^{f}\subseteq C$.
In particular, if $f,g\colon\mathbb{N}\circlearrowleft$ there is a commutative
diagram

$\textstyle{C^{f}\otimes
C^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{f+g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\,C.}$

Here the rows are given by the evident inclusion and the columns by
multiplication. The resulting algebra inclusions

$\bigoplus_{f\colon\mathbb{N}\circlearrowleft}C^{f}\subseteq C$

and

${\mathbb{F}_{p}}[\bar{\mu}_{i}]/\bar{\mu}_{i}^{p}\subseteq C$

induce isomorphisms of graded commutative $\mathbb{F}_{p}$-algebras

$\bigoplus_{f\colon\mathbb{N}\circlearrowleft}C^{f}\cong\bigotimes_{i\geq
0}\left(\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1})\otimes\bigotimes_{j>0}{\mathbb{F}_{p}}[\gamma_{p^{j}}\bar{\mu}_{i}]/(\gamma_{p^{j}}\bar{\mu}_{i})^{p}\right)$

and

$C\cong\left(\bigotimes_{i\geq
0}{\mathbb{F}_{p}}[\bar{\mu}_{i}]/\bar{\mu}_{i}^{p}\right)\otimes\left(\bigoplus_{f\colon\mathbb{N}\circlearrowleft}C^{f}\right).$

###### Proof.

The multiplicative structure follows from Lemma 2.13, and the last two
isomorphisms follow from the fact that a monomial in $C$ does not have any
$\bar{\mu}_{i}$-factors of the form $\chi_{S,f}$. ∎

######  Corollary 2.15.

As an $\mathbb{F}_{p}$-algebra,

$H^{D}\cong\bigotimes_{i\geq
0}\mathbb{F}_{p}[\bar{\mu}_{i}]/\bar{\mu}_{i}^{p}$

and $Z^{D}$ is the subalgebra of $C$ generated by the $\bar{\mu}_{i}$ with
$i\geq 0$ and the $D\chi_{(S,f)}$ with $t_{f}\notin S\subseteq\mathrm{supp}f$.
More explicitly, and writing $\mathsf{x}_{S,f}=D\chi_{(S,f)}$, the relation
expressed in Lemma 2.13 gives an isomorphism

$Z^{D}\cong\mathbb{F}_{p}[\bar{\mu}_{i},\mathsf{x}_{S,f}]_{i\in\mathbb{N},(S,f)\in
K}/(\bar{\mu}_{i}^{p},\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\sum_{u}\epsilon_{u}\cdot\mathsf{x}_{S\cup
T\cup\\{u\\},f+g}).$

Here, $u\in\mathrm{supp}(f+g)-S\cup T$ and $u\neq t_{f+g}$.

##  3 Motivic Hochschild homology over algebraically closed fields

In this section we work over an algebraically closed field $F$ of exponential
characteristic $\mathsf{e}(F)\neq p$. Then $\rho=0$ since every unit is a
square, and

$\mathbb{M}_{\star}\cong k^{M}_{\ast}[\tau]\cong\mathbb{F}_{p}[\tau]$ (19)

by [30, Corollary 4.3, p.254], where $|\tau|=(0,-1)$. From (9) and (19) we
deduce

$\mathcal{A}_{\star}=\begin{cases}\mathbb{F}_{p}[\tau,\xi_{i+1},\tau_{i}]_{i\geq
0}/(\tau_{i}^{2}-\tau\xi_{i+1})&p=2\\\ \mathbb{F}_{p}[\tau,\xi_{i+1}]_{i\geq
0}\otimes\Lambda_{\mathbb{F}_{p}}(\tau_{i})_{i\geq 0}&p\neq 2.\end{cases}$
(20)

If $p=2$ and we invert $\tau$ in $\mathcal{A}_{\star}$, then $\xi_{i}$ is no
longer needed as a generator because $\xi_{i+1}=\tau^{-1}\tau_{i}^{2}$:

$\pi_{\star}((\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})[\tau^{-1}])\cong\mathcal{A}_{\star}[\tau^{-1}]=\begin{cases}\mathbb{F}_{p}[\tau^{\pm
1},\tau_{i}]_{i\geq 0}&p=2\\\ \mathbb{F}_{p}[\tau^{\pm 1},\xi_{i+1}]_{i\geq
0}\otimes\Lambda_{\mathbb{F}_{p}}(\tau_{i})_{i\geq 0}&p\neq 2.\end{cases}$
(21)

Likewise, since $\mathcal{A}_{\star}$ is free as an
$\mathbb{F}_{p}[\tau]$-module, taking the quotient by $\tau^{p-1}$ (for any
prime $p$) gives an isomorphism of Hopf algebras

$\pi_{\star}((\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})/\tau^{p-1})\cong\mathcal{A}_{\star}/\tau^{p-1}=(\bigotimes_{i\geq
0}\mathbb{F}_{p}[\xi_{i+1}]\otimes\Lambda_{\mathbb{F}_{p}}(\tau_{i}))\otimes\mathbb{F}_{p,\tau}.$
(22)

Here $\mathbb{F}_{p,\tau}$ is shorthand for $\mathbb{F}_{p}[\tau]/\tau^{p-1}$.
In Section 3.2, we use (22) to compute the coefficients of the mod
$\tau^{p-1}$ reduction of $\mathbf{MHH}(\mathbb{F}_{p})$.

###  3.1 Étale motivic Hochschild homology

We refer to [3], [12] for $\tau$-self maps and applications towards étale
hyperdescent for motivic spectra. Suppose $\mathscr{R}/p$ is a motivic
$E_{\infty}$ ring spectrum defined over an algebraically closed field. Then
the canonical map

$\mathscr{R}/p\to\mathscr{R}/p[\tau^{-1}]$ (23)

exhibits the $\tau$-periodization as a motivic $E_{\infty}$ ring spectrum
under $\mathscr{R}/p$; see [4, §12], [12, §8] for recent expositions. If
$\mathscr{R}$ happens to be cellular, then so is $\mathscr{R}/p[\tau^{-1}]$.
Owing to [3, Theorem 1.2], (23) is an étale localization (the
$\rho$-completion in [3] is obsolete over algebraically closed fields, and for
$p\neq 2$ the étale localization involves only the “+”-part of
$\mathscr{R}/p$). We note that (23) induces an isomorphism on $\tau$-inverted
homotopy groups.

At all primes, the $\tau$-periodic mod-$p$ motivic Steenrod algebra agrees
with the tensor product of the topological mod-$p$ Steenrod algebra with the
Laurent polynomial ring $\mathbb{F}_{p}[\tau^{\pm 1}]$. This observation
implies that after $p$-completion the $\tau$-periodic motivic stable homotopy
groups are isomorphic to the classical stable homotopy groups with $\tau^{\pm
1}$ adjoined [19], [21, §4]. In Section 3.1, we prove a similar statement for
motivic and topological Hochschild homology.

We calculate
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})[\tau^{-1}]\cong\pi_{\star}(\mathbf{MHH}(\mathbb{F}_{p})[\tau^{-1}])$
directly by the Tor spectral sequence, using the relations and differentials
from Lemma 2.3 and Remark 2.7 and by appealing to Equation 21 and the
naturally induced equivalence of motivic spectra

$\mathbf{MHH}(\mathbb{F}_{p})[\tau^{-1}]=(\mathbf{M}\mathbb{F}_{p}\wedge_{\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p}}\mathbf{M}\mathbb{F}_{p})[\tau^{-1}]\xrightarrow{\simeq}\mathbf{M}\mathbb{F}_{p}[\tau^{-1}]\wedge_{(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})[\tau^{-1}]}\mathbf{M}\mathbb{F}_{p}[\tau^{-1}].$
(24)

As before we set $\mu_{i}:=\sigma\tau_{i}$ and $\lambda_{i}:=\sigma\xi_{i}$.

######  Lemma 3.1.

The Tor spectral sequence of $\mathbf{MHH}(\mathbb{F}_{p})[\tau^{-1}]$
collapses at the $E^{p}$ page and

$E^{\infty}=(\bigotimes_{i\geq 0}\Lambda_{\mathbb{F}_{p}}(\mu_{i}))[\tau^{\pm
1}].$

For $p$ odd the only nonzero differentials $d^{r}$ for $r>1$ are generated by

$d^{p-1}(\gamma_{j}\mu_{i})\overset{\cdot}{=}\tau^{p-1}\lambda_{i+1}\gamma_{j-p}\mu_{i}$
(25)

for all $i\geq 0$, $j\geq p$.

###### Proof.

Lemma 2.6, Equation 21, and Equation 24 yield the $E^{2}$ page. When $p=2$, we
have

$E^{2}=(\bigotimes_{i\geq 0}\Lambda_{\mathbb{F}_{2}}(\mu_{i}))[\tau^{\pm 1}].$
(26)

Since all the $\mu_{i}$s have filtration degree $1$, there are no non-trivial
differentials and we conclude that $E^{\infty}=E^{2}$. When $p$ is odd, the
$E^{2}$ page takes the form

$E^{2}=(\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\mu_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\lambda_{i+1}))[\tau^{\pm
1}].$ (27)

The Tor spectral sequence starts out as an augmented unital
$\mathbb{M}_{\star}[\tau^{-1}]$-Hopf algebra since (27) is flat over
$\mathbb{M}_{\star}[\tau^{-1}]$. Arguing as in [1, §4], [2, §5], [14, §1.2],
[24], [25], we’ll see that the non-trivial differentials are as claimed. More
precisely, since the shortest differential in the lowest total degree must go
from an algebra generator (these lie in filtration powers of $p$) to a
coalgebra primitive (these lie in filtration $1$), the differentials $d^{r}$
for $1<r<p-1$ are all zero. Recall from Remark 2.7 that we established the
said differential for $j=p$ integrally:
$d^{p-1}(\gamma_{j}\mu_{i})\overset{\cdot}{=}\tau^{p-1}\lambda_{i+1}$ and we
move from there by induction on $j\geq p$ and the coalgebra structure in Lemma
2.6; this is, for $k\geq 0$, the calculation

$\displaystyle\psi
d^{p-1}(\gamma_{p+k}\mu_{i})-\psi(\tau^{p-1}\lambda_{i+1}\gamma_{k}\mu_{i})$
$\displaystyle=(d^{p-1}\otimes 1+1\otimes d^{p-1})\psi(\gamma_{p+k}\mu_{i})$
$\displaystyle\phantom{=}~{}-\tau^{p-1}(\lambda_{i+1}\otimes
1+1\otimes\lambda_{i+1})(\Sigma_{a+b=k}\gamma_{a}\mu_{i}\otimes\gamma_{b}\mu_{i})$
$\displaystyle=(d^{p-1}\otimes 1)(\gamma_{p+k}\mu_{i}\otimes 1)+(1\otimes
d^{p-1})(1\otimes\gamma_{p+k}\mu_{i})$
$\displaystyle\phantom{=}~{}+\tau^{p-1}\underset{a+b=p+k;a,b>0}{\sum}(\lambda_{i+1}\gamma_{a-p}\mu_{i}\otimes\gamma_{b}\mu_{i}+\gamma_{a}\mu_{i}\otimes\lambda_{i+1}\gamma_{b-p}\mu_{i})$
$\displaystyle\phantom{=}~{}-\tau^{p-1}\underset{a+b=k}{\sum}(\lambda_{i+1}\gamma_{a}\mu_{i}\otimes\gamma_{b}\mu_{i}+\gamma_{a}\mu_{i}\otimes\lambda_{i+1}\gamma_{b}\mu_{i})$
$\displaystyle=(d^{p-1}\otimes 1)(\gamma_{p+k}(\mu_{i})\otimes 1)+(1\otimes
d^{p-1})(1\otimes\gamma_{p+k}(\mu_{i}))$

shows the difference
$d^{p-1}(\gamma_{p+k}\mu_{i})-\tau^{p-1}\lambda_{i+1}\gamma_{k}\mu_{i}$ is a
coalgebra primitive; however, $0$ is the only such element in the given
degree. The remaining algebra generators on the $E^{p}$ page are in filtration
degree $\leq 1$, and hence $E^{\infty}=E^{p}$. ∎

######  Remark 3.2.

Alternatively, an appeal to rigidity for extensions of algebraically closed
fields as in Remark 2.2 or [27] (in characteristic zero) reduces to
considering the complex numbers. Over $\mathbb{C}$, the differential (25) is
forced by Bökstedt’s differential
$d^{p-1}(\gamma_{j}\mu_{i})=\lambda_{i+1}\gamma_{j-p}\mu_{i}$ in the Tor
spectral sequence for $\mathbf{THH}_{\ast}(\mathbb{F}_{p})$. In the motivic
case, the correction term $\tau^{p-1}$ ensures agreement of the weights.

######  Theorem 3.3.

There are isomorphisms

$\mathbf{MHH}_{\star}(\mathbb{F}_{p})[\tau^{-1}]\cong\mathbb{F}_{p}[\tau^{\pm
1},\mu_{i}]_{i\geq
0}/(\mu_{i}^{p}-\tau^{p-1}\mu_{i+1})\cong\mathbb{F}_{p}[\mu,\tau^{\pm
1}]\cong\mathbf{THH}_{\ast}(\mathbb{F}_{p})[\tau^{\pm 1}].$

The generator $\mu$ has bidegree $(2,0)$.

###### Proof.

Equation 24 shows the $E^{\infty}$ page for
$\mathbf{MHH}(\mathbb{F}_{p})[\tau^{-1}]$ is the Laurent polynomials in $\tau$
of the $E^{\infty}$ page for $\mathbf{THH}(\mathbb{F}_{p})$. The result now
follow from Lemma 3.1 and the multiplicative extension

$\mu_{i}^{p}=\tau^{p-1}\mu_{i+1}.$ (28)

of Lemma 2.3. ∎

Hence all the classes $\mu_{i}\in\mathbf{MHH}(\mathbb{F}_{p})$ are nontrivial
and we may identify the $\tau$-free part in
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ with

$\mathbb{F}_{p}[\tau,\mu_{i}]_{i\geq 0}/(\mu_{i}^{p}-\tau^{p-1}\mu_{i+1}).$
(29)

This is depicted graphically for $p=2$ and $p=3$ in Figure 1 and Figure 2,
respectively.

Figure 1: The étale motivic Hochschild homology of $\mathbb{F}_{2}$. The
vertical lines indicate $\tau$-multiplication, while the horizontal and
diagonal lines depict powers of $\mu_{i}$, $i=0,1,2,3$. Figure 2: The étale
motivic Hochschild homology of $\mathbb{F}_{3}$ depicted in the same graphical
style as Figure 1.

###  3.2 Reduced motivic Hochschild homology

To proceed to the next step in our strategy for calculating
$\mathbf{MHH}(\mathbb{F}_{p})$ over an algebraically closed field $F$ with
$\mathsf{e}(F)\neq p$, we form the cofiber of $\tau^{n}$ (for our
calculations, it suffices to consider $n=p-1$)

$\Sigma^{0,n}\mathbf{M}\mathbb{F}_{p}\overset{\tau^{n}}{\rightarrow}\mathbf{M}\mathbb{F}_{p}\rightarrow\mathbf{M}\mathbb{F}_{p}/\tau^{n}.$
(30)

We thank Markus Spitzweck for informing us that
$\mathbf{M}\mathbb{F}_{p}/\tau^{n}$ is a motivic $E_{\infty}$ ring spectrum
for all $n\geq 1$. His argument goes as follows: $\mathbf{M}\mathbb{Z}$ is
strongly periodizable by [29, Corollary C.3]. Thus the mod-$p$ coefficient
ring $\mathbb{M}_{\star}$ is a motivic differential graded algebra, i.e., a
graded $E_{\infty}$ ring spectrum in complexes of $\mathbb{F}_{p}$-vector
spaces. In fact, $\mathbb{M}_{\star}$ is formal so that
$\mathbb{M}_{\star}/\tau^{n}$ is $E_{\infty}$ over $\mathbb{M}_{\star}$ for
all $n\geq 1$. This implies the corresponding claim for
$\mathbf{M}\mathbb{F}_{p}/\tau^{n}$. In effect, let $\mathcal{E}$ be the free
$E_{\infty}$ algebra in graded complexes on a generator $\tau$ in degree
$(0,1)$. Its $0$-truncation, with respect to the natural $t$-structure on the
derived category of graded abelian groups, is the formal model
$\mathbb{F}_{p}[\tau]$. Thus $\mathbb{M}_{\star}$ and $\mathbb{F}_{p}[\tau]$
are equivalent since the natural map
$\mathcal{E}\rightarrow\mathbb{M}_{\star}$ is the $0$-truncation. When $n=1$,
we also refer to Gheorghe [13] for the fact that
$\mathbf{M}\mathbb{F}_{2}\rightarrow\mathbf{M}\mathbb{F}_{2}/\tau$ is a map of
motivic $E_{\infty}$ ring spectra.

Inserting $\mathbf{M}\mathbb{F}_{p}/\tau^{n}$ into (1) yields the derived
smash product

$\mathbf{MHH}(\mathbb{F}_{p})/\tau^{n}\simeq\mathbf{M}\mathbb{F}_{p}/\tau^{n}\wedge_{(\mathbf{M}\mathbb{F}_{p}\wedge\mathbf{M}\mathbb{F}_{p})/\tau^{n}}\mathbf{M}\mathbb{F}_{p}/\tau^{n}.$
(31)

Owing to (30) and cellularity of $\mathbf{M}\mathbb{F}_{p}$, see §2.3, it
follows that $\mathbf{M}\mathbb{F}_{p}/\tau^{n}$ is cellular. Thus (31) gives
rise to the Tor spectral sequence

$E^{2}_{h,t,w}=\mathbf{Tor}^{\mathcal{A}_{\star}/\tau^{n}}_{h,t,w}(\mathbb{M}_{\star}/\tau^{n},\mathbb{M}_{\star}/\tau^{n})\Rightarrow\mathbf{MHH}_{h+t,w}(\mathbb{F}_{p})/\tau^{n}.$
(32)

Recall that $\mathbb{F}_{p,\tau}$ is shorthand for
$\mathbb{F}_{p}[\tau]/\tau^{p-1}$. Lemma 2.6 and (22) implies the Tor spectral
sequence (15) for $\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ takes the
form

$\bar{E}^{2}_{\ast,\star}\cong\left(\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\sigma\tau_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\sigma\xi_{i+1})\right)\otimes\mathbb{F}_{p,\tau}\Rightarrow\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}.$
(33)

This is a first quadrant spectral sequence; the horizontal direction is the
“filtration”, the vertical direction is the “degree”, and every term is graded
by “weight.” Recall that if $x$ has filtration $f_{x}$, degree $d_{x}$ and
weight $w_{x}$, we write $|x|=(f_{x},d_{x};w_{x})$ so that the differentials
take the form

$d^{r}\colon\bar{E}^{r}_{f,d;w}\to\bar{E}^{r}_{f-r,d+r-1;w}.$

In (33), we set $\bar{\mu}_{i}:=\sigma\tau_{i}$ and
$\bar{\lambda}_{i+1}:=\sigma\xi_{i+1}$. The bar is meant to signify that the
generators are mod-$\tau^{p-1}$ classes and should not be confused with the
conjugate classes. For these classes, we note the degrees

  1. 1.

$|\bar{\lambda}_{i+1}|=(1,2p^{i+1}-2;p^{i+1}-1)$,

  2. 2.

$|\gamma_{p^{j}}\bar{\mu}_{i}|=(p^{j},2p^{i+j}-p^{j};p^{i+j}-p^{j})$.

Thus for $x=\bar{\lambda}_{i+1}$ and $x=\gamma_{p^{j}}\bar{\mu}_{i}$ we have
the congruence $w_{x}\equiv 0\bmod p-1$. Hence if
$x\in\bar{E}^{2}_{\ast,\star}$ in (33) has weight $w_{x}=-n+(p-1)m$, $0\leq
n\leq p-1$, then $n$ equals $x$’s $\tau$-multiplicity. Another helpful
bookkeeping device for our calculation is the Chow degree of $x$, see [5,
Definition 3.1] and [20, Definition 2.1.10] for related terminology, defined
by

$c(x)=f_{x}+2w_{x}-d_{x}$

In particular, we have

  1. 1.

$c(\bar{\lambda}_{i+1})=1+2(p^{i+1}-1)-(2p^{i+1}-2)=1$

  2. 2.

$c(\gamma_{p^{j}}\bar{\mu}_{i})=p^{j}+2(p^{i+j}-p^{j})-(2p^{i+j}-p^{j})=0$

Every homogeneous class $x\in\bar{E}^{2}_{\ast,\star}$ in (33) is a monomial
in the generators $\bar{\lambda}_{i+1}$ and $\gamma_{p^{j}}\bar{\mu}_{i}$. The
Chow degree $c(x)$ records the number of $\lambda_{i+1}$ classes in $x$, and
the equality $0\leq c(x)\leq f_{x}$ follows from the definition.

######  Lemma 3.4.

The Tor spectral sequence (33) for
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ collapses at its $E^{2}$
page.

###### Proof.

For $r\geq 2$ and $x\in E^{r}_{\ast,\star}$ we note the equality of weights
$w_{x}=w_{d^{r}x}$. If $x=\tau$, then $d^{r}\tau=0$ since (33) is an
$\mathbb{F}_{p}[\tau]$-algebra spectral sequence. If $x=\bar{\lambda}_{i+1}$
or $x=\gamma_{p^{j}}\bar{\mu}_{i}$, the congruence $w_{d^{r}x}\equiv 0\bmod
p-1$ shows the monomials in $d^{r}x$ are not $\tau$-divisible. Hence,
$d^{r}x=0$, and we are done, or $c(d^{r}x)\geq 0$. It remains to note that
$c(d^{r}x)=c(x)-2r+1<0$. ∎

######  Lemma 3.5.

There are no multiplicative extensions in the mod-$\tau^{p-1}$ Tor spectral
sequence (33).

###### Proof.

The Chow degree of $x$ equals $c(x)=2f_{x}+2w_{x}-(d_{x}+f_{x})$. To find a
hidden extension for $g=(\gamma_{p^{j}}\bar{\mu}_{i})^{p}=0$, we search among
the $x$’s that satisfy

  1. 1.

$d_{x}+f_{x}=d_{g}+f_{g}=2p^{i+j+1}$,

  2. 2.

$w_{x}=w_{g}=p(p^{i+j}-p^{j})$,

  3. 3.

$0<f_{x}<f_{g}=p^{j+1}$.

This rules out the existence of multiplicative extensions, since for the Chow
degree, we have

$c(x)=2f_{x}+2w_{x}-(d_{x}+f_{x})=2f_{x}+2p(p^{i+j}-p^{j})-2p^{j+i+1}=2(f_{x}-p^{j+1})<0.$

Likewise, a hidden extension for $\bar{\lambda}_{i+1}^{2}=0$ would be a class
$x$ with $|x|=(1,4(p^{i+1}-1),2p^{i+1}-2)$; by inspection, no such class
exists since all possible $x$ of filtration $1$ have weight $p^{j}-1$, $j\geq
0$. ∎

######  Theorem 3.6.

There is an isomorphism of graded commutative $\mathbb{F}_{p,\tau}$-algebras

$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}\cong(\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\bar{\mu}_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1}))\otimes\mathbb{F}_{p,\tau}.$

The bidegrees of the generators are $|\bar{\mu}_{i}|=(2p^{i},p^{i}-1)$ and
$|\bar{\lambda}_{i+1}|=(2p^{i+1}-1,p^{i+1}-1)$.

######  Remark 3.7.

The reader may recognize the answer as
$\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}\cong
C\otimes\mathbb{F}_{p,\tau}$ where
$C=\mathbf{Tor}^{\mathcal{A}_{\star}^{\textrm{rig}}}_{\ast,\star}(\mathbb{F}_{p},\mathbb{F}_{p})$
appeared in Section 2.5.

###  3.3 Integral motivic Hochschild homology

We now turn to the integral case of the Tor spectral sequence

$E^{r}_{h,t,w}\Rightarrow\mathbf{MHH}_{h+t,w}(\mathbf{M}\mathbb{F}_{p}).$ (34)

There is a natural comparison map $q\colon
E^{r}_{\star;*}\rightarrow\bar{E}^{r}_{\star;*}$ to the mod-$\tau^{p-1}$ Tor
spectral sequence analyzed in §3.2. Due to Theorem 3.6 we have the following
non-trivial mod-$\tau^{p-1}$ classes and their representatives in the bar
complex:

  1. 1.

$\bar{\lambda}_{i+1}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ is the
class of the permanent cycle
$[\bar{\xi}_{i+1}]\in\bar{E}^{1}_{1,2p^{i+1}-2;p^{i+1}-1}$,

  2. 2.

$\gamma_{j}\bar{\mu}_{i}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$ is
the class of the permanent cycle
$[\bar{\tau}_{i}|\dots|\bar{\tau}_{i}]\in\bar{E}^{1}_{j,j(2p^{i}-1);j(p^{i}-1)}$.

As before, to aid the bookkeeping we also set

$\lambda_{i+1}=[\xi_{i+1}]\in E^{1}_{1,2p^{i+1}-2;p^{i+1}-1}$

and

$\gamma_{j}\mu_{i}=[\tau_{i}|\dots|\tau_{i}]\in
E^{1}_{j,j(2p^{i}-1);j(p^{i}-1)},$

even though the $\gamma_{j}\mu_{i}$s turn out to be permanent cycles for $j<p$
only.

As already noted, when $p$ is an odd prime
$E^{2}=\mathbb{F}_{p}[\tau]\otimes\bigotimes_{i\geq
0}\Lambda(\lambda_{i+1})\otimes\Gamma(\mu_{i})$.

######  Lemma 3.8.

Let $p$ be a prime.

  * •

For $0<r<p$ the étale localization

$L^{r}_{\text{{\'{e}}t}}\colon E^{r}\to E^{r}[\tau^{-1}]$

is an injection.

  * •

For $1<r<p-1$, the differentials $d^{r}:E^{r}\to E^{r}$ are all zero.

  * •

For all $p$

$d^{p-1}\gamma_{j+p}\mu_{i}\overset{\cdot}{=}\tau^{p-1}\lambda_{i+1}\gamma_{j}\mu_{i}$

for $i,j\geq 0$ and for odd $p$, this generates the $d^{p-1}$-differential
multiplicatively.

###### Proof.

Since the dual Steenrod algebra has no $\tau$-torsion we have that
$L^{1}_{\text{{\'{e}}t}}$ is an injection, and from the Tor-calculations we
get that for odd primes $p$ also $L^{2}_{\text{{\'{e}}t}}$ is an injection.
Assume that for given $0<r<p$ $L^{r}_{\text{{\'{e}}t}}$ is injective. For
$1<r<p-1$ we have established that the differential on $E^{r}[\tau^{-1}]$ is
trivial, and so the differential on $E^{r}$ is trivial too. Hence
$L^{r+1}_{\text{{\'{e}}t}}$ is injective, showing that (for odd primes $p$)
$E^{2}=E^{3}=\dots=E^{p-1}$.

Finally, since for all primes $p$ we now have $L^{p-1}_{\text{{\'{e}}t}}$ is
an injection, the formula
$d^{p-1}\gamma_{j+p}\mu_{i}\overset{\cdot}{=}\tau^{p-1}\lambda_{i+1}\gamma_{j}\mu_{i}$
follows from the same formula in $E^{p-1}[\tau^{-1}]$. ∎

The case for odd and even primes $p$ takes slightly different paths from here
on. The case $p=2$ is in many ways the simplest one but requires more care in
that it turns out to be neither practical nor necessary to muddle through with
the integral spectral sequence calculation: everything emanates from the
torsion and $\tau$-inverted $\mathbf{MHH}$s together with minimal information
about the integral $E^{1}$-page and an analysis of the Bockstein homology
(called “a Bockstein type complex” in Section 2.5 since it also appears in the
odd primary case in a slightly different guise) giving the answer — with all
multiplicative extensions — without more ado.

####  3.3.1 The even case

Let $p=2$. Since $\tau$ is a non-zero divisor in $\mathcal{A}_{\star}$,
multiplication by $\tau$ gives the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{1}_{f,d;w+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{E^{1}_{f,d;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\bar{E}^{1}_{f,d;w}0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We recall that the mod-$\tau$ spectral sequence collapses at $\bar{E}^{2}$ and
has no multiplicative extensions:
$\mathbf{MHH}_{\star}(\mathbb{F}_{2})/\tau\cong\bar{E}^{2}$. Moving on to the
abutment, the $\tau$-Bockstein on $\mathbf{MHH}_{\star}(\mathbb{F}_{2})$ is
the composite

$\bar{\partial}\colon\mathbf{MHH}_{\ast+1,\ast}(\mathbb{F}_{2})/\tau\xrightarrow{\partial}\mathbf{MHH}_{\ast,\ast+1}(\mathbb{F}_{2})\xrightarrow{q}\mathbf{MHH}_{\ast,\ast+1}(\mathbb{F}_{p})/\tau.$
(35)

Since (35) is a derivation we only need to know its value on the generators.
These are obtained from the integral $d^{1}$-differentials analyzed in Remark
2.7 as follows. Since $\bar{\lambda}_{i+1}$ is hit by the $d^{1}$-boundary
$\lambda_{i+1}=[\xi_{i+1}]\in E^{1}_{1,2p^{i+1}-2;p^{i+1}-1}$ we get
$\bar{\partial}\bar{\lambda}_{i+1}=0$, and since $\gamma_{j+2}\bar{\mu}_{i}$
is hit by $\gamma_{j+2}\mu_{i}=[\tau_{i}|\dots|\tau_{i}]\in
E^{1}_{j,j(2p^{i}-1);j(p^{i}-1)}$ and
$d^{1}\gamma_{j+2}\mu_{i}=\tau\lambda_{i+1}\gamma_{j}\mu_{i}$ we deduce the
following lemma.

######  Lemma 3.9.

The nontrivial $\tau$-Bocksteins on $\mathbf{MHH}_{\star}(\mathbb{F}_{2})$ are
generated by

$\bar{\partial}\gamma_{j+2}\bar{\mu}_{i}=\bar{\lambda}_{i+1}\gamma_{j}\bar{\mu}_{i}$

for all $i,j\geq 0$, i.e.,
$(\mathbf{MHH}_{\star}(\mathbb{F}_{2})/\tau,\bar{\partial})=(C,D)$, where
$(C,D)$ is the commutative differential graded algebra of Section 2.5.

Combined with Lemma 2.14, and using that the $\tau$-free element
$\mu_{i}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})$ maps to
$\bar{\mu}_{i}\in\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$, we deduce
the following result.

######  Corollary 3.10.

The Bockstein homology of $\mathbf{MHH}_{\star}(\mathbb{F}_{2})/\tau$ is
isomorphic to the graded commutative $\mathbb{F}_{2}$-algebra
$\bigoplus_{i\geq 0}\Lambda(\bar{\mu}_{i})$.

Corollary 3.10 lets us conclude that the $\tau$-torsion classes in
$\mathbf{MHH}_{\star}(\mathbb{F}_{2})$ are not $\tau$-divisible. The
$\tau$-torsion in $\mathbf{MHH}_{\star}(\mathbb{F}_{2})$ agrees with the image
of
$\partial\colon\mathbf{MHH}_{\star}(\mathbb{F}_{2})/\tau\to\mathbf{MHH}_{\star}(\mathbb{F}_{2})$
and maps injectively via
$q\colon\mathbf{MHH}_{\star}(\mathbb{F}_{2})\to\mathbf{MHH}_{\star}(\mathbb{F}_{2})/\tau$.

There is a naturally induced commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tau\text{-torsion})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\cong}$$\textstyle{\mathbf{MHH}_{\star}(\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathbb{F}_{2}[\tau,\mu_{0},\mu_{1},\dots]/(\mu_{i}^{2}-\tau\mu_{i+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{im}\bar{\partial}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigotimes_{i\geq
0}\Gamma(\bar{\mu}_{i})\otimes\Lambda(\bar{\lambda}_{i+1}).}$

More elegantly, using Corollary 2.15, we have a pullback diagram of
commutative $\mathbb{F}_{2}[\tau]$-algebras

$\textstyle{\mathbf{MHH}_{\star}(\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}[\tau,\mu_{i}]/\mu_{i}^{2}-\tau\mu_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{2}[\tau,\bar{\mu}_{i},\mathsf{x}_{S,f}]/\mathcal{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{2}[\tau,\bar{\mu}_{i}]/(\bar{\mu}_{i}^{2},\tau)}$

with indexation $i\in\mathbb{N}$, $(S,f)\in K$ (see 2.8), and

$\mathcal{I}=(\tau,\bar{\mu}_{i}^{2},\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\sum_{t_{f+g}\not=u\in\mathrm{supp}(f+g)-S\cup
T}\epsilon_{u}\cdot\mathsf{x}_{S\cup T\cup\\{u\\},f+g}).$

Here $\mu_{i}$ maps to $\bar{\mu}_{i}$ and $\mathsf{x}_{S,f}$ maps to zero.
When we are done with the odd case, we’ll see that by replacing $2$ with $p$,
we have the general formula.

####  3.3.2 The odd case

Let $p$ be an odd prime. The first task is using our knowledge of $d^{p-1}$ to
calculate $E^{p}$. Consider short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{p-1}_{f,d;w+p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{p-1}}$$\textstyle{E^{p-1}_{f,d;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\bar{E}^{p-1}_{f,d;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and the injection

$L^{p-1}_{\text{{\'{e}}t}}\colon E^{p-1}_{f,d;w}\to
E^{p-1}_{f,d;w}[\tau^{-1}].$

######  Definition 3.11.

For $p\leq r$, let $P(r)$ be the conjunction of the propositions $P(r)_{1}$,
$P(r)_{2}$, and $P(r)_{3}$ defined as follows:

  1. $P(r)_{1}$:

$\textstyle{E^{r}_{f,d;w+p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{p-1}}$$\textstyle{E^{r}_{f,d;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\bar{E}^{r}_{f,d;w}}$
is exact,

  2. $P(r)_{2}$:

in $E^{r}_{f,d;w}$ we have $\ker L^{r}_{\text{{\'{e}}t}}=\ker\tau^{p-1}$, and

  3. $P(r)_{3}$:

for $p\leq j<r$ the $jth$ differential $d^{j}$ is trivial (so that
$E^{p}=E^{r}$).

To simplify notation, consider the $\mathbb{F}_{p}$-algebra
$C=\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\bar{\mu}_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1})$
(with the above isomorphism $\bar{E}^{r}\cong C[\tau]/\tau^{p-1}$ for $r\geq
2$) and the derivation $D\colon C\to C$ generated by
$D(\gamma_{j+p}\bar{\mu}_{i})=\bar{\lambda}_{i+1}\gamma_{j}\bar{\mu}_{i}$. Let
$B^{D}=\mathrm{im}D$, $Z^{D}=\ker D$ and $H^{D}=Z^{D}/B^{D}$.

######  Lemma 3.12.

The proposition $P(p)$ is true.

Then $E^{p}$ is isomorphic to $Z^{D}[\tau]/\tau^{p-1}B^{D}[\tau]$ and under
this isomorphism $E^{p}/\ker L^{p}_{\text{{\'{e}}t}}$ is isomorphic to
$H^{D}[\tau]$.

Furthermore, the map $q\colon E^{p}\to\bar{E}^{p}\cong C[\tau]/\tau^{p-1}$
factors over $Z^{D}[\tau]/\tau^{p-1}\subseteq C[\tau]/\tau^{p-1}$ and the map

$\ker\\{\tau^{p-1}\colon E^{p}\to E^{p}\\}\subseteq E^{p}\to\bar{E}^{p}\cong
C[\tau]/\tau^{p-1}$

is an injection factoring as an isomorphism $\ker\\{\tau^{p-1}\colon E^{p}\to
E^{p}\\}\cong B^{D}[\tau]/\tau^{p-1}$ followed by the injection
$B^{D}[\tau]/\tau^{p-1}\subseteq C[\tau]/\tau^{p-1}$. Summing up, the
resulting diagram of commutative $\mathbb{F}_{p}[\tau]$-algebras

$\textstyle{E^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{D}[\tau]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{D}[\tau]/\tau^{p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{D}[\tau]/\tau^{p-1}}$

is a pullback.

###### Proof.

For odd $p$, the first thing to notice is that $E^{p-1}$ is a free
$\mathbb{F}_{p}[\tau^{p-1}]$-module and that the differential factors
$d^{p-1}=\tau^{p-1}{\bar{\partial}}$ where ${\bar{\partial}}$ (aka the
Bockstein) is homogeneous with respect to the $\tau^{p-1}$-grading on
$E^{p-1}$ and ${\bar{\partial}}^{2}=0$. Let $Q$ be the degree zero part of
$E^{p-1}$ (so that $E^{p-1}=Q[\tau^{p-1}]$ and $Q\subseteq E^{p-1}\to
E^{p-1}/\tau^{p-1}\cong\bar{E}^{p-1}=\bar{E}^{p}$ is an isomorphism). If
$Z^{\bar{\partial}}=\ker\\{{\bar{\partial}}\colon Q\to Q\\}$, then $\ker
d^{p-1}=Z^{\bar{\partial}}[\tau^{p-1}]$, whereas if
$B^{\bar{\partial}}=\mathrm{im}\\{{\bar{\partial}}\colon Q\to Q\\}$, then
$\mathrm{im}\,d^{p-1}=\tau^{p-1}B^{\bar{\partial}}[\tau^{p-1}]$, and if
$H^{\bar{\partial}}=Z^{\bar{\partial}}/B^{\bar{\partial}}$, then (as an
$\mathbb{F}_{p}[\tau^{p-1}]$-module)

$E^{p}=Z^{\bar{\partial}}\oplus\tau^{p-1}H^{\bar{\partial}}[\tau^{p-1}],$

and $q\colon E^{p}\to\bar{E}^{p}$ may be identified with the composite

$Z^{\bar{\partial}}\oplus\tau^{p-1}H^{\bar{\partial}}[\tau^{p-1}]\to
Z^{\bar{\partial}}\subseteq Q$

of the projection to the degree zero part followed by the inclusion. Hence

$\ker
q=\tau^{p-1}H^{\bar{\partial}}[\tau^{p-1}]=\mathrm{im}\\{\tau^{p-1}\colon
E^{p}\to E^{p}\\},$ $\ker\\{\tau^{p-1}\colon E^{p}\to
E^{p}\\}=B^{\bar{\partial}}=\ker L^{p}_{\text{{\'{e}}t}},$

and

$E^{p}/\ker\tau^{p-1}\cong H^{\bar{\partial}}[\tau^{p-1}]\subseteq
H^{\bar{\partial}}[\tau^{\pm(p-1)}]\cong E^{p}[\tau^{-1}]$

Since $P(r)_{3}$ is vacuous in this case, we have proven $P(p)$.

The formulation with the pullback follows when writing the above out as
$\mathbb{F}_{p}[\tau]$-algebras, so that $\ker d^{p-1}=Z^{D}[\tau]$ and
$\mathrm{im}\,d^{p-1}=\tau^{p-1}B^{D}[\tau]$ and remembering that
$\bar{E}^{p-1}\cong C[\tau]/\tau^{p-1}$. ∎

######  Lemma 3.13.

For all $r\geq p$ the proposition $P(r)$ is true. Hence,

  1. 1.

$E^{\infty}=E^{p}$,

  2. 2.

the algebra map from the $\mathbb{F}_{p}[\tau^{p-1}]$-free part to the
$\tau^{p-1}$-localization

$L_{\text{{\'{e}}t}}\colon\left[\mathbf{MHH}_{\star}(\mathbb{F}_{p})\right]/\ker\tau^{p-1}\to\mathbf{MHH}_{\star}(\mathbb{F}_{p})[\tau^{-1}]\cong\mathbb{F}_{p}[\tau^{\pm
1},\mu_{i}]_{i\geq 0}/(\mu_{i}^{p}-\tau^{p-1}\mu_{i+1})$

is injective so that
$\left[\mathbf{MHH}_{\star}(\mathbb{F}_{p})\right]/\ker\tau^{p-1}\cong\mathbb{F}_{p}[\tau,\mu_{i}]_{i\geq
0}/(\mu_{i}^{p}-\tau^{p-1}\mu_{i+1})$,

  3. 3.

the algebra map induced by
$q\colon\mathbf{MHH}(\mathbb{F}_{p})\to\mathbf{MHH}(\mathbb{F}_{p})/\tau^{p-1}$

$q\colon\left[\mathbf{MHH}_{\star}(\mathbb{F}_{p})\right]/\mathrm{im}\,\tau^{p-1}\to\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}\cong(\bigotimes_{i\geq
0}\Gamma_{\mathbb{F}_{p}}(\bar{\mu}_{i})\otimes\Lambda_{\mathbb{F}_{p}}(\bar{\lambda}_{i+1}))\otimes\mathbb{F}_{p,\tau}$

is injective, and

  4. 4.

the composite
$\ker\tau^{p-1}\subseteq\mathbf{MHH}_{\star}(\mathbb{F}_{p})\to\left[\mathbf{MHH}_{\star}(\mathbb{F}_{p})\right]/\mathrm{im}\,\tau^{p-1}$
is injective.

###### Proof.

By Lemma 3.12 we have $P(p)$ so we only need to show that $P(r)$ implies
$P(r+1)$ for all $r\geq p$. Note that if $P(r)$ and $P(r+1)_{3}$ are true,
then $P(r+1)$ is true. Recall from Lemma 3.1 and Lemma 3.4 that the $r$th
differentials in both the localized and reduced $\operatorname{Tor}$-spectral
sequences are trivial.

Assume $P(r)$ and consider $x\in E^{r}_{f,d;w}$. From the fact that
$d^{r}\colon\bar{E}^{r}\to\bar{E}^{r}$ is trivial so that $0=d^{r}qx=qd^{r}x$
we get that there is a $y\in E^{r}_{f-r,d+r-1;w+p-1}$ so that $P(r)_{1}$
implies that $d^{r}x=\tau^{p-1}y$. Since $d^{r}\colon
E^{r}[\tau^{-1}]\to\bar{E}^{r}[\tau^{-1}]$ is trivial we get that
$0=d^{r}L^{r}_{\text{{\'{e}}t}}x=L^{r}_{\text{{\'{e}}t}}d^{r}x=L^{r}_{\text{{\'{e}}t}}\tau^{p-1}y=\tau^{p-1}L^{r}_{\text{{\'{e}}t}}y$
so that $0=L^{r}_{\text{{\'{e}}t}}y$ and $P(r)_{2}$ implies that
$0=\tau^{p-1}y=d^{r}x$.

The other points then follow directly, where in the last point we have used
that $\ker\tau^{p-1}=\ker L_{\text{{\'{e}}t}}$ gives that
$\ker\tau^{p-1}\cap\mathrm{im}\,\tau^{p-1}=0$. ∎

Summing up in the language of Lemma 3.12, we have achieved a pullback of
commutative $\mathbb{F}_{p}[\tau]$-algebras

$\textstyle{E^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{D}[\tau]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{D}[\tau]/\tau^{p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{D}[\tau]/\tau^{p-1}.}$
(36)

Moreover, the pullback survives to the abutment in the sense that the maps out
of $E^{\infty}$ are the associated graded versions of maps induced from maps
of commutative ring spectra.

We now set out analyzing $Z^{D}$ and $H^{D}$.

###  3.4 Multiplicative extensions

From Lemma 3.13 we deduced the pullback (36) of commutative
$\mathbb{F}_{p}[\tau]$-algebras, which, given the information of Corollary
2.15, takes the form

$\textstyle{E^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}[\tau,\mu_{i}]/\mu_{i}^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}[\tau,\bar{\mu}_{i},\mathsf{x}_{S,f}]/\overline{\mathcal{I}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}[\tau,\bar{\mu}_{i}]/(\bar{\mu}_{i}^{p},\tau^{p-1})}$

with indexation $i\in\mathbb{N}$, $(S,f)\in K$, and

$\overline{\mathcal{I}}=(\tau^{p-1},\bar{\mu}_{i}^{p},\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\sum_{t_{f+g}\not=u\in\mathrm{supp}(f+g)-S\cup
T}\epsilon_{u}\cdot\mathsf{x}_{S\cup T\cup\\{u\\},f+g}).$

Here $\mu_{i}$ maps to $\bar{\mu}_{i}$ and $\mathsf{x}_{S,f}$ maps to zero.
Moreover, the pullback survives to the abutment in the sense that the maps out
of $E^{\infty}$ are the associated graded versions of maps induced from maps
of commutative ring spectra.

In the abutment, we know that $\mu_{i}^{p}=\tau^{p-1}\mu_{i+1}$, but can there
be further extensions? Since $\mu_{i}$ maps to $\bar{\mu}_{i}$, such an
extension must be witnessed when passing from the associated graded
$\mathbb{F}_{p}[\tau,\bar{\mu}_{i},\mathsf{x}_{S,f}]/\overline{\mathcal{I}}$
to $\mathbf{MHH}_{\star}(\mathbb{F}_{p})/\tau^{p-1}$, but this we have seen in
the mod $\tau^{p-1}$-calculation is not the case. In conclusion, we have shown
the following result.

######  Theorem 3.14.

There is an isomorphism of graded commutative $\mathbb{F}_{p}[\tau]$-algebras

$\mathbf{MHH}_{*}(\mathbb{F}_{p})\cong\mathbb{F}_{p}[\tau,\mu_{i},\mathsf{x}_{S,f}]_{i\in\mathbb{N},(S,f)\in
K}/\mathcal{I}$

where the indexing set $K$ is given in Definition 2.8 and $\mathcal{I}$ is the
ideal generated by

  * •

$\mu_{i}^{p}-\tau^{p-1}\mu_{i+1}$,

  * •

$\tau^{p-1}\mathsf{x}_{S,f}$, and

  * •

$\mathsf{x}_{S,f}\cdot\mathsf{x}_{T,g}-\sum_{u}\epsilon_{u,S,T,f,g}\cdot\mathsf{x}_{S\cup
T\cup\\{u\\},f+g}$ where the sum runs over all elements $u\notin S\cup T$ so
that $(f+g,S\cup T\cup\\{u\\})\in K$, and the coefficient
$\epsilon_{u,S,T,f,g}\in\mathbb{F}_{p}$ is given in Definition 2.12.

##### Acknowledgments

We thank Markus Spitzweck for his help with an argument in Section 3.2. The
authors acknowledge the support of the Centre for Advanced Study at the
Norwegian Academy of Science and Letters in Oslo, Norway, which funded and
hosted our research project “Motivic Geometry” during the 2020/21 academic
year. This research was supported by grants from the RCN Frontier Research
Group Project no. 250399 “Motivic Hopf Equations” and no. 312472 “Equations in
Motivic Homotopy.”

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Department of Mathematics, University of Bergen, Norway

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Department of Mathematics, University of California, Los Angeles, USA

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Department of Mathematics,, Reed College, Portland, USA

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Department of Mathematics F. Enriques, University of Milan, Italy

Department of Mathematics, University of Oslo, Norway

email<EMAIL_ADDRESS><EMAIL_ADDRESS>