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

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

Abstract

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

1. Introduction

Hochschild (aka derived Hochschild or Shukla) homology is in a precise sense the homology theory of associative algebras Reference 38, Section 3, and so plays an important role from a purely ring-theoretic perspective, classifying extensions and so on. However, Hochschild homology rose in prominence in the 1980s via its cyclic structure as explored by Connes and Tsygan and its subsequent connection to (rational) algebraic $K$-theory. To include torsion phenomena (and wider applications), Goodwillie and Waldhausen conjectured that the differential of algebraic $K$-theory should correspond to some form of Hochschild homology of algebras over the sphere spectrum (ring spectra). When Bökstedt succeeded in extending the definition of Hochschild homology to cover algebras over the sphere spectrum (“topological Hochschild homology”), he also managed to calculate its values at the prime fields, revealing the striking periodicity which has been fundamental to much of the subsequent development. Later, the first author and McCarthy confirmed Goodwillie and Waldhausen’s conjecture, leading to further advances and, ultimately, many calculations of algebraic $K$-theory.

In this paper, we try to emulate Bökstedt: we define Hochschild homology for algebras over the motivic sphere spectrum and calculate its values at prime fields. This is interesting for many reasons. Firstly, the motivic version over $\operatorname {Spec}(\mathbb{C})$ sheds light on the topological one, giving “reasons” for some of the relations from the classical case. Secondly, if we are to investigate the “number theory” of rings over the motivic sphere spectrum, we should access invariants of a $K$-theoretic nature with an ambitious goal of repeating the success in (equivariant) stable homotopy theory. We hope that this paper is a tiny step in the right direction. Our definition follows the interpretation of Hochschild homology as the homology of associative algebras in the spirit of Quillen. There are different options, but we do not pursue them in this paper.

Hochschild homology is a fundamental derived invariant for algebras and rings Reference 11. For a smooth algebra $A$ over a field $k$ of characteristic zero, such as the coordinate ring of a smooth affine variety, the Hochschild-Kostant-Rosenberg theorem identifies the Hochschild homology groups ${\mathbf{HH}}_{n}(A/k)$ with the Kähler differentials $\Omega _{A/k}^{n}$ of derivations. Using Hochschild homology, ${\mathbf{HH}}$, one constructs many other derived invariants such as Connes’ cyclic homology ${\mathbf{HC}}$. We refer to Reference 31 for background. Bökstedt-Hsiang-Madsen Reference 9 pioneered the refined theory of topological Hochschild homology ${\mathbf{THH}}$ and topological cyclic homology ${\mathbf{TC}}$; their trace methods remain of significant interest in algebraic $K$-theory, see Reference 13, Reference 20, and the modern viewpoint adapted to $\infty$-categories in Reference 36.

Motivic homotopy theory is an ${\mathbb{A}}^{1}$-invariant homotopy theory for algebraic varieties, originally developed by Morel and Voevodsky in the 1990s Reference 35, and motivated by the spectacular work of Voevodsky and Rost resolving the Milnor and Bloch-Kato conjectures relating Milnor $K$-theory with Galois cohomology Reference 52, Reference 54, and quadratic forms Reference 37, Reference 45. Since then, this framework has shown itself to be a valuable setting for studying algebro-geometric cohomology theories, with applications to algebraic geometry, number theory, and algebraic topology. See Reference 27 and Reference 30 for recent surveys.

In this paper, we study Hochschild homology in the motivic setting. Let $\mathscr{R}$ be a motivic ring spectrum such as algebraic cobordism, homotopy algebraic $K$-theory, or motivic cohomology Reference 49. Working in the stable motivic homotopy category $\mathbf{SH}(F)$ of a field $F$, we define the motivic Hochschild homology $\mathbf{MHH}(\mathscr{R})$ of $\mathscr{R}$ as the derived tensor product

$$\begin{equation} \mathscr{R}\wedge _{\mathscr{R}\wedge \mathscr{R}^{\text{op}}} \mathscr{R}. \cssId{texmlid5}{\tag{1}} \end{equation}$$

The primary purpose of this paper is to calculate the homotopy ring $\mathbf{MHH}_{\star }(\mathbb{F}_p)$ of motivic Hochschild homology of $\mathbf{M}\mathbb{F}_{p}$ over algebraically closed fields—the Suslin-Voevodsky motivic cohomology ring spectrum for $p$ any prime number. When the base field 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 in Reference 8 on topological Hochschild homology of the corresponding topological Eilenberg-MacLane spectrum $\mathbf{H}\mathbb{F}_p$. Additively, $\mathbf{THH}(\mathbb{F}_p)$ splits as sum of $\mathbf{H}\mathbb{F}_p$’s in the stable homotopy category. However, this is not the case for $\mathbf{MHH}(\mathbb{F}_p)$ and $\mathbf{M}\mathbb{F}_{p}$. The source of this extra layer of complexity is the abundance of $\tau$-torsion elements in the homotopy ring. 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 homotopy 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 Reference 23, Reference 53, and generating endofunctions $f\colon \mathbb{N}\acwgapcirclearrow$ with finite non-empty support containing some subset $S\subset \mathbb{N}$. It is the $\tau$-power in the equation $\mu _{i}^{p}=\tau ^{p-1}\mu _{i+1}$ that gives rise to the infinity of $\tau$-torsion classes $\mathsf{x}_{S,f}$ not witnessed topologically in $\mathbf{THH}_{\star }(\mathbb{F}_p)$.

Theorem 1.1.

Over an algebraically closed field of exponential characteristic $\neq p$, there is an algebra isomorphism

$$\begin{equation} \mathbf{MHH}_{\star }(\mathbb{F}_{p}) \cong \mathbb{F}_{p}[\tau ,\mu _{i},\mathsf{x}_{S,f}]_{i\in \mathbb{N}, (S\subset \operatorname {supp}f,f\colon \mathbb{N}\acwgapcirclearrow )}/\mathcal{I} \cssId{texmlid28}{\tag{2}} \end{equation}$$

with the ideal of relations

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

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

$$\begin{equation*} |\mathsf{x}_{S,f}|= (|S|+1)(-1,p-1)+p\sum _{j\in \operatorname {supp}f}f(j)(2p^{j},p^{j}-1). \end{equation*}$$

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 $\Omega ^2 S^3$ Reference 32. Behrens–Wilson showed that an analogous result is true $C_2$-equivariantly, with the base now $\Omega ^{2,1}S^{3,1}$ Reference 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 Reference 7. Equivariantly, classically and $C_2$-equivariantly, this splits as a wedge of smash powers of spheres. Putting this all together, we cannot have that 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 $\mathbf{MHH}(\mathbb{F}_{p})$, and then analyze how their homotopy classes conspire to describe the integral homotopy ring. We review some background and set our notation in Section 2. Remark 2.5 gives a Lefschetz Principle for the homotopy ring of $\mathbf{MHH}(\mathbb{F}_{p})$, which reduces our computation to the case of complex numbers. In Section 3, we divide the proof of Theorem 1.1 into the following steps.

Step 1

The dual motivic Steenrod algebra of our ground field $F$ at $p$, see Equation 9, contains classes $\tau _{i}$ for $i\geq 0$. Theorem 3.3 calculates the $\tau$-inverted or étale motivic Hochschild homology$$\begin{equation} \mathbf{MHH}_{\star }(\mathbb{F}_{p})[\tau ^{-1}] \cong \mathbb{F}_{p}[\tau ^{\pm 1},\mu _{i}]_{i\geq 0}/(\mu _{i}^{p}-\tau ^{p-1}\mu _{i+1}) \cong \mathbb{F}_{p}[\mu ,\tau ^{\pm 1}] \cong \mathbf{THH}_{\ast }(\mathbb{F}_{p})[\tau ^{\pm 1}]. \cssId{texmlid29}{\tag{3}} \end{equation}$$

Here the generator $\mu$ has bidegree $(2,0)$. The “homology suspension” classes $\mu _{i}\coloneq \sigma \tau _i$, see Section 2.2, generate the non-$\tau$-torsion part of $\mathbf{MHH}_{\star }(\mathbb{F}_{p})$ subject to the relation $\mu _{i}^{p}=\tau ^{p-1}\mu _{i+1}$.

Step 2

Theorem 3.6 calculates the mod-$\tau ^{p-1}$ motivic Hochschild homology$$\begin{equation} \mathbf{MHH}_{\star }(\mathbb{F}_p)/\tau ^{p-1} \coloneq \pi _{\star }(\mathbf{MHH}(\mathbb{F}_p)/\tau ^{p-1}) \cong (\bigotimes _{i\geq 0} \Gamma _{\mathbb{F}_{p}}(\bar{\mu }_{i}) \otimes \Lambda _{\mathbb{F}_{p}}(\bar{\lambda }_{i+1})) \otimes \mathbb{F}_{p}[\tau ]/\tau ^{p-1}. \cssId{texmlid3}{\tag{4}} \end{equation}$$

The bidegrees of the generators are $|\bar{\lambda }_{i}|=(2p^{i}-1,p^{i}-1)$, $|\bar{\mu }_{i}|=(2p^{i},p^{i}-1)$. The divided powers algebra generator $\bar{\mu }_{i}$ is the image of $\mu _{i}\in \mathbf{MHH}_{\star }(\mathbb{F}_p)$. It turns out that Equation 4 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.

Step 3

Lemma 3.9 shows that 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$$\begin{equation} \bar{\partial }\colon \mathbf{MHH}_{\ast +1,\ast }(\mathbb{F}_{p})/\tau ^{p-1} \xrightarrow {\partial } \mathbf{MHH}_{\ast ,\ast +p-1}(\mathbb{F}_{p}) \xrightarrow {q} \mathbf{MHH}_{\ast ,\ast +p-1}(\mathbb{F}_{p})/\tau ^{p-1}. \cssId{texmlid30}{\tag{5}} \end{equation}$$

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 $\mathbb{F}_{p,\tau } \coloneq \mathbb{F}_{p}[\tau ]/\tau ^{p-1}$ -algebra $\bigoplus _{i\geq 0} \Lambda _{\mathbb{F}_{p,\tau }} (\bar{\mu }_{i})$.

Step 4

Lemma 3.13 shows that 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 $\bar{\partial }$.

Step 5

If $f\colon \mathbb{N}\acwgapcirclearrow$ has finite support and $S\subseteq \operatorname {supp}f$, we set$$\begin{equation*} \chi _{S,f} = \left(\prod _{m\in S}\bar{\lambda }_{m+1}\gamma _{pf(m)-p}\bar{\mu }_m\right) \left(\prod _{n\not \in S}\gamma _{pf(n)}\bar{\mu }_n\right) \in \mathbf{MHH}_{\star }(\mathbb{F}_{p})/\tau ^{p-1}. \end{equation*}$$

We define the $\tau$-torsion algebra generators in Theorem 1.1 by$$\begin{equation*} \mathsf{x}_{S,f} = \partial \chi _{S,f} \in \mathbf{MHH}_{\star }(\mathbb{F}_{p}). \end{equation*}$$

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 $\delta _n\colon \mathbb{N}\acwgapcirclearrow$ is zero except for $\delta _n(n)=1$. Applying the Bockstein operation $\partial$ to $\chi _{S,f}$ yields $\mathsf{x}_{S,f}=\sum _{n\in \operatorname {supp}(f)-S}\chi _{S\cup \{n\},f}$ 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.

Step 6

By combining the $\tau$-inverted and mod-$\tau ^{p-1}$ calculations we finally deduce Theorem 1.1. The power operations in the dual motivic Steenrod algebra give rise to the relation $\mu _{i}^{p}=\tau ^{p-1}\mu _{i+1}$. The Bockstein calculation $\mathsf{x}_{S,f}=\partial \chi _{S,f}$ implies 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 Definition 2.12 for the entity $\epsilon _{u}$.

For example, at the prime $p=2$, we obtain the relations$$\begin{equation*} \mathsf{x}_{\delta _0+\delta _1}\mathsf{x}_{\delta _2}+\mathsf{x}_{\delta _1+\delta _2}\mathsf{x}_{\delta _0}+\mathsf{x}_{\delta _2+\delta _0}\mathsf{x}_{\delta _1} = 0, \end{equation*}$$

$$\begin{equation*} \mathsf{x}_{\delta _0+\delta _1}\mathsf{x}_{\delta _1+\delta _2} = \mathsf{x}_{2\delta _1}\mathsf{x}_{\delta _0+\delta _2}. \end{equation*}$$

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 $\mathbb{F}_2[\tau ]$-algebras

$$\begin{equation*} \vcenter{\img[][175pt][55pt][{$\xymatrix{ \MHH_{\star}(\FF_2)\ar[r]\ar[d]&\FF_2[\tau,\mu_i]/(\mu_i^2-\tau\mu_{i+1})\ar[d]\\ \FF_2[\bmu_{i},\x_{S,f}]/\mathcal{I} \ar[r]&\FF_2[\bmu_i]/(\bmu_i^2)}$}]{Images/img37c048a5ada0c2dbae8cb18108883a08.svg}} \end{equation*}$$

where the ideal of relations is given by (see Section 2.5 for the definition of $t_{f+g}$)

$$\begin{equation*} \mathcal{I} = \left(\bar{\mu }_{i}^2,\mathsf{x}_{S,f}\cdot \mathsf{x}_{T,g}-\sum _{t_{f+g}\not =u\in \operatorname {supp}(f+g)-S\cup T} \epsilon _{u}\cdot \mathsf{x}_{S\cup T\cup \{u\},f+g}\right). \end{equation*}$$

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 $\mathsf{e}(F)\neq p$
$\mathbf{SH}(F)$	stable motivic homotopy category of $F$
$\mathbf{CAlg}(F)$	commutative motivic ring spectra of $F$
$\mathscr{R}$	motivic ring spectrum
$H^{\star }$, $h^{\star }$	(bigraded) integral, mod-$p$ motivic cohomology groups of $F$
$K^{M}_{\ast }$, $k^{M}_{\ast }$	(graded) 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 $\mathbb{F}_{p}[\tau ]/\tau ^{p-1}$
$\Gamma$, $\Lambda$	divided power and exterior 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

$$\begin{equation} \mathbf{MHH}(\mathscr{R};\mathscr{M}) \coloneq \mathscr{M}\wedge _{\mathscr{R}\wedge \mathscr{R}^{\text{op}}} \mathscr{R}\cssId{texmlid4}{\tag{6}} \end{equation}$$

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

When $\mathscr{R}=\mathscr{M}$, the derived tensor product Equation 6 specializes to $\mathbf{MHH}(\mathscr{R})$ in Equation 1. 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

$$\begin{equation} \mathbf{MHH}(\mathscr{R};\mathscr{M}) \simeq \mathscr{M}\wedge _{\mathscr{Q}\wedge \mathscr{R}^{\text{op}}} \mathscr{Q}. \cssId{texmlid11}{\tag{7}} \end{equation}$$

In the following, we assume that $\mathscr{R}$ is a cofibrant commutative motivic ring spectrum in any of the model categorical approaches to $\mathbf{SH}(F)$ as in Reference 14, Reference 21, Reference 24, Reference 28 (this assumption is superfluous in the $\infty$-category of motivic spectra Reference 41). Commutative motivic ring spectra are cotensored over motivic spaces via the free-forgetful adjunction $\mathcal{F}\dashv \mathcal{U}$ between $\mathbf{SH}(F)$ and commutative motivic ring spectra $\mathbf{CAlg}(F)$: if $\mathscr{X}$ is a motivic space, then $\mathscr{X}\otimes \mathscr{R}$ is the coequalizer of

$$\begin{equation*} \mathcal{F}(\mathscr{X}_{+}\wedge \mathcal{U}\mathscr{R})\leftleftarrows \mathcal{F}(\mathscr{X}_{+}\wedge \mathcal{U}\mathcal{F}\mathcal{U}\mathscr{R}). \end{equation*}$$

Here we use the canonical maps $\mathcal{U}\mathcal{F}\mathcal{U}\mathscr{R}\rightarrow \mathcal{U}\mathscr{R}$, $\mathscr{X}_{+}\wedge \mathcal{U}\mathcal{F}(\mathcal{U}\mathscr{R})\rightarrow \mathcal{U}\mathcal{F}(\mathscr{X}_{+}\wedge \mathcal{U}\mathscr{R})$, and $\mathcal{F}\mathcal{U}\mathcal{F}\rightarrow \mathcal{F}$. 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 ${\mathbb{A}}^{1}$-homotopy invariant and preserves coproducts (and so, in particular, sends pushouts to smashes).

•

$\ast \otimes \mathscr{R}\cong \mathscr{R}$, $S^0\otimes \mathscr{R}\cong \mathscr{R}\wedge \mathscr{R}$ and (since $S^1$ is the derived pushout of $\ast \gets S^0\to \ast$)$$\begin{equation*} \mathbf{MHH}(\mathscr{R};\mathscr{M})\simeq \mathscr{M}\wedge _{\mathscr{R}}(S^1\otimes \mathscr{R}). \end{equation*}$$

•

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

•

Choosing a point $\ast \to \mathscr{X}$ 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 automorphism $\{-1,1\}\to \{-1,1\}$ gives the anti-involution of the “dual Steenrod $\mathscr{R}$-Hopf algebroid” $S^0\otimes \mathscr{R}=\mathscr{R}\wedge \mathscr{R}$ (algebroid 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 suspensions of these maps give the pinch map$$\begin{equation*} S^1\cong [-1,1]\coprod _{\{-1,1\}}\ast \to [-1,1]\coprod _{\{-1,0,1\}}\ast \cong S^1\vee S^1 \end{equation*}$$

and the flip map $S^1\to S^1$, both of which are pointed maps, inducing the $\mathscr{R}$-Hopf algebroid structure$$\begin{equation*} \psi \colon S^1\otimes \mathscr{R}\longrightarrow (S^1\vee S^1)\otimes \mathscr{R}\cong (S^1\otimes \mathscr{R})\wedge _{\mathscr{R}}(S^1\otimes \mathscr{R}), \quad \chi \colon S^1\otimes \mathscr{R}\cong S^1\otimes \mathscr{R} \end{equation*}$$

on the “motivic Hochschild homology”—to implement this using finite simplicial models of the circle, one subdivides as in Reference 1.

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 $\mathscr{X}{}\mapsto \mathscr{X}{}_+\wedge \mathscr{R}$ and the universal property defines a unique map of motivic spectra$$\begin{equation*} \sigma ^+\colon \mathscr{X}{}_+\wedge \mathscr{R}\to \mathscr{X}{}\otimes \mathscr{R}. \end{equation*}$$

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$$\begin{equation} \sigma \colon \mathscr{R}\wedge X_+\wedge \mathscr{R}\xrightarrow {1\wedge \sigma ^+} \mathscr{R}\wedge X{}\otimes \mathscr{R}\xrightarrow {\mathrm{mult}} X{}\otimes \mathscr{R}. \cssId{texmlid9}{\tag{8}} \end{equation}$$

•

If $\mathcal{A}$ is a commutative $\mathscr{R}$-algebra, then the internal hom $\mathscr{X}\mapsto \mathcal{A}^\mathscr{X}=\hom _\mathscr{R}(\mathscr{X}_+\wedge \mathscr{R},\mathcal{A})$ is a cotensor (does not depend on $\mathscr{R}$). The unit of the adjunction$$\begin{equation*} \alpha ^\mathscr{R}\colon \mathcal{A}\to (\mathscr{X}\otimes ^\mathscr{R}\mathcal{A})^\mathscr{X} \end{equation*}$$

is a map of commutative $\mathscr{R}$-algebras. (Here $\mathscr{X}\otimes ^\mathscr{R}\mathcal{A}$ is the tensor in the category of commutative $\mathscr{R}$-algebras of the motivic space $\mathscr{X}$ with $\mathcal{A}$.) In the category of $\mathscr{R}$-modules, the adjoint of $\alpha ^\mathscr{R}$ takes the form$$\begin{equation*} \sigma ^\mathscr{R}\colon (\mathscr{X}_+\wedge \mathscr{R})\wedge _\mathscr{R}\mathcal{A}\xrightarrow {1\wedge \alpha ^\mathscr{R}} (\mathscr{X}_+\wedge \mathscr{R})\wedge _\mathscr{R}(\mathscr{X}\otimes ^\mathscr{R}\mathcal{A})^\mathscr{X}\xrightarrow {\mathrm{ev}} (\mathscr{X}\otimes ^\mathscr{R}\mathcal{A}), \end{equation*}$$

where $\mathrm{ev}$ is the evaluation. Composition gives an $\mathscr{R}$-algebra map $\nu \colon \mathscr{R}^\mathscr{X}\wedge _\mathscr{R}(\mathscr{X}\otimes ^\mathscr{R}\mathcal{A})\to (\mathscr{X}\otimes ^\mathscr{R}\mathcal{A})^\mathscr{X}$.

Assume that $\mathscr{X}=X$ is a finite cell complex and that $\pi _\star (X_+\wedge \mathscr{R})$ is a finitely generated free $\pi _\star \mathscr{R}$-module with basis $\mathcal{B}$. Then $\nu$ is an equivalence, and the composite (we identify $\pi _\star (\mathscr{R}^X\wedge _\mathscr{R}(X\otimes ^\mathscr{R}\mathcal{A}))$ with the given target)$$\begin{equation*} \nu ^{-1}_{\star }\alpha ^\mathscr{R}_\star \colon \pi _\star \mathcal{A}\xrightarrow {\alpha ^\mathscr{R}} \pi _\star (X\otimes ^\mathscr{R}\mathcal{A})^X \xrightarrow {\nu ^{-1}_{\star }} \hom _{\pi _\star \mathscr{R}}(\pi _\star (X_+\wedge \mathscr{R}),\pi _\star \mathscr{R})\otimes _{\pi _\star \mathscr{R}}\pi _\star (X\otimes ^\mathscr{R}\mathcal{A}) \end{equation*}$$

satisfies$$\begin{equation*} \nu ^{-1}_\star \alpha _\star (a)=\sum _{x\in \mathcal{B}}x^{\vee }\otimes \sigma ^\mathscr{R}_{\star }(x\otimes a). \end{equation*}$$

Here $x^{\vee }$ 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 will use this formula in Lemma 2.3 to get a relation in $\mathbf{MHH}(\mathbb{F}_p)$ (in the topological case, see Reference 1, §5 for $X=S^1$ using the circle action).

In the category of commutative $\mathscr{R}$-algebras, note that $\mathscr{X}\otimes ^\mathscr{R}(\mathscr{R}\wedge \mathscr{R})\cong \mathscr{R}\wedge (\mathscr{X}\otimes \mathscr{R})$ is the tensor of $\mathscr{X}$ with $S^{0}\otimes \mathscr{R}=\mathscr{R}\wedge \mathscr{R}$ with its left $\mathscr{R}$-algebra structure, and there is a commutative diagram$$\begin{equation*} \vcenter{\img[][196pt][57pt][{$\xymatrix{ (\XXX_+\smsh\RRR)\smsh_\RRR(\RRR\smsh\RRR)\ar[r]^-{\sigma^\RRR}\ar@{=}[d]^\cong&\XXX\otimes^\RRR(\RRR\smsh\RRR) \ar@{=}[d]^\cong\\ \RRR\smsh\XXX_+\smsh\RRR\ar[r]^-{1\smsh\sigma^+}&\RRR\smsh\XXX\otimes\RRR,}$}]{Images/img93ccfc441a3f18e3d06db88f1554c7a5.svg}} \end{equation*}$$

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 (see Reference 1 for a homological version in the topological setting). In this subsection, let $I=\Delta [1]$ be the simplicial interval with boundary $S^0=\partial \Delta [1]$ and let $S^1=I\coprod _{S^0}\ast$ be the simplicial circle. The subdivision of the circle relevant to 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 $\ast \coprod _{S^0}(I\times S^0)\coprod _{S^0}\ast$. Under the canonical isomorphism $\mathscr{R}=\ast \otimes \mathscr{R}$ 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

$$\begin{equation*} dS^1\otimes \mathscr{R}=((I\coprod I)\otimes \mathscr{R})\wedge _{(S^0\coprod S^0)\otimes \mathscr{R}}S^0\otimes \mathscr{R}\cong \mathscr{R}\wedge _{\mathscr{R}\wedge \mathscr{R}}I\otimes (\mathscr{R}\wedge \mathscr{R})\wedge _{\mathscr{R}\wedge \mathscr{R}}\mathscr{R}. \end{equation*}$$

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

$$\begin{equation*} \psi \colon S^1\otimes \mathscr{R}\xleftarrow {\sim } dS^1\otimes \mathscr{R}\xrightarrow {\nabla \otimes 1} (S^1\vee S^1)\otimes \mathscr{R}\xrightarrow {\cong } (S^1\otimes \mathscr{R})\wedge _\mathscr{R}(S^1\otimes \mathscr{R}) \end{equation*}$$

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

$$\begin{equation*} \vcenter{\img[][408pt][51pt][{$\xymatrix{ \RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR&\ar[l]_-{\sim} \RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA(I\otimes\AAA) \smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR\ar[d] \\ (\RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR)\smsh_\RRR(\RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR)&\ar[l]_-\cong\RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR\smsh_\AAA(I\otimes\AAA) \smsh_\AAA\RRR. }$}]{Images/img85e1e7aa7f1e1ce9d8bd15717f5ca0fb.svg}} \end{equation*}$$

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 $p$,

$$\begin{equation} \mathcal{A}_{\star } = \begin{cases} \mathbb{M}_{\star }[\xi _{i},\tau _{i}]_{i\geq 0}/(\tau _{i}^{2}-\rho (\tau _{i+1}-\tau _{0}\xi _{i+1})-\tau \xi _{i+1})& p=2 \\ \mathbb{M}_{\star }[\xi _{i}]_{i\geq 0}\otimes _{\mathbb{M}_{\star }} \Lambda _{\mathbb{M}_{\star }}(\tau _{i})_{i\geq 0} & p\ne 2 \end{cases} \cssId{texmlid2}{\tag{9}} \end{equation}$$

(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 Reference 23, §5.1, Reference 40, §5, Reference 53, §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 Equation 9 are given by

$$\begin{equation*} \vert \xi _{i}\vert =(2p^i-2,p^{i}-1),\qquad \vert \tau _{i}\vert =(2p^i-1,p^{i}-1). \end{equation*}$$

The coproducts of the generators are defined by

$$\begin{equation} \psi (\xi _{i}) =\sum _{j=0}^{i}\xi _{i-j}^{p^{j}}\otimes \xi _{j}, \qquad \psi (\tau _{i})=\tau _{i}\otimes 1+\sum _{j=0}^{i}\xi _{i-j}^{p^{j}}\otimes \tau _{j}. \cssId{texmlid8}{\tag{10}} \end{equation}$$

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

$$\begin{equation*} \eta _{R}(\rho )=\rho ,\qquad \eta _{R}(\tau )=\tau +\rho \tau _{0} \end{equation*}$$

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 Reference 52, Reference 54. We shall also use the antipodal generators

$$\begin{equation} c(\tau _i)=-\tau _i - \sum _{j=0}^{i-1} \xi _{i-j}^{p^j}c(\tau _j), \qquad c(\xi _i) = -\xi _i - \sum _{j=1}^{i-1} \xi _{i-j}^{p^j}c(\xi _j) \cssId{texmlid7}{\tag{11}} \end{equation}$$

detailed in Reference 23, §5. For legibility, we will abuse notation by implicitly using the antipodal classes Equation 11 in our computations. Voevodsky defines in Reference 51, §3.1 the mod-$p$ rigid motivic Steenrod algebra

$$\begin{equation} \mathcal{A}_{\star }^{\mathrm{rig}} \coloneq \bigotimes _{i\geq 0}\text{S}_{\mathbb{F}_{p}}(\xi _{i+1})\otimes \Lambda _{\mathbb{F}_{p}}(\tau _{i}). \cssId{texmlid31}{\tag{12}} \end{equation}$$

The equation 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

$$\begin{equation*} \overline{K} \leftarrow W(\overline{F}) \rightarrow \overline{F} \end{equation*}$$

induce isomorphisms on $\mathbb{M}_{\star }$ and $\mathcal{A}_{\star }$ according to Reference 55, §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 from Equation 8 the suspension operation

$$\begin{equation} \sigma \colon \mathbf{M}\mathbb{F}_p\wedge S^1_+ \wedge \mathbf{M}\mathbb{F}_p \longrightarrow \mathbf{M}\mathbb{F}_p\wedge \mathbf{MHH}(\mathbb{F}_p)\xrightarrow {\mathrm{mult}} \mathbf{MHH}(\mathbb{F}_p). \cssId{texmlid10}{\tag{13}} \end{equation}$$

We note the isomorphisms

$$\begin{align*} \pi _{\star } (\mathbf{M}\mathbb{F}_p\wedge S^1_+ \wedge \mathbf{M}\mathbb{F}_p) & \cong \pi _{\star } ((\mathbf{M}\mathbb{F}_p\wedge S^1_+) \wedge _{\mathbf{M}\mathbb{F}_p} (\mathbf{M}\mathbb{F}_p\wedge \mathbf{M}\mathbb{F}_p))\\ & \cong H_{\ast }(S^1_+;\mathbb{M}_{\star }) \otimes _{\mathbb{M}_{\star }} \mathcal{A}_{\star }\\ & \cong H_{\ast }(S^1_+;\mathbb{F}_p) \otimes \mathcal{A}_{\star }. \end{align*}$$

If $s_1\in H_1(S^1_+;\mathbb{F}_p)$ is the standard generator and $\zeta \in \mathcal{A}_{s,w}$, we let “$\sigma \zeta$” denote the “homology suspension” of $\zeta$, namely the image of $s_1\otimes \zeta$ in $\mathbf{MHH}_{s+1,w}(\mathbb{F}_p)$ under the composite in Equation 13 and also in $\pi _{s+1,w}(\mathbf{M}\mathbb{F}_p\wedge \mathbf{MHH}(\mathbb{F}_p))$ under the first map in Equation 13.

We now show two relations useful in the forthcoming spectral sequence calculations. Remark 2.7 generalizes our second formula to all base fields when $p=2$.

Lemma 2.3.

In the motivic Hochschild homology $\mathbf{MHH}_\star (\mathbb{F}_p)$ of an algebraically closed field of exponential characteristic $\neq p$, we have the relations

$$\begin{equation*} \tau ^{p-1}\sigma \tau _{i+1}=(\sigma \tau _{i})^p,\qquad \tau ^{p-1}\sigma \xi _{i+1}=0 \end{equation*}$$

for all $i\geq 0$.

Proof.

Suppose that $\mathscr{R}$ is a commutative $\mathbf{M}\mathbb{F}_p$-algebra and let $\alpha \colon S^{s,w}\to \mathscr{R}$ represent a class in $\pi _{s,w}\mathscr{R}$. We write $E\Sigma _p$ for the nerve of the translation category of the symmetric group $\Sigma _p$ on $p$ letters.

The “power operation”

$$\begin{equation*} P(\alpha )\colon \pi _\star (\mathbf{M}\mathbb{F}_p\wedge ((E\Sigma _p)_+\wedge _{\Sigma _p} (S^{s,w})^{\wedge p}))\to \pi _\star \mathscr{R} \end{equation*}$$

is the homotopy of the $\mathbf{M}\mathbb{F}_p$-module map $\mathbf{M}\mathbb{F}_p\wedge ((E\Sigma _p)_+\wedge _{\Sigma _p} (S^{s,w})^{\wedge p})\to \mathscr{R}$ adjoint to the composite

$$\begin{equation*} E\Sigma _p\wedge _{\Sigma _p} (S^{s,w})^{\wedge p}\xrightarrow {\mathrm{id}\wedge \alpha ^{p}} E\Sigma _p\wedge _{\Sigma _p} \mathscr{R}^{\wedge p}\xrightarrow {E\Sigma _p\to *}\mathscr{R}^{\wedge p}/\Sigma _p\xrightarrow {\mathrm{mult}} \mathscr{R}. \end{equation*}$$

Precomposing with the map

$$\begin{equation*} H_{\ast }(B\Sigma _p;\mathbb{F}_p)\otimes \pi _\star (\mathbf{M}\mathbb{F}_p\wedge S^{s,w}) \to \pi _\star (\mathbf{M}\mathbb{F}_p\wedge ((E\Sigma _p)_+\wedge _{\Sigma _p} (S^{s,w})^{\wedge p})) \end{equation*}$$

defined on the chain level (via the monoidal Quillen equivalence between $\mathbf{M}\mathbb{F}_p$-modules and motives with mod-$p$ coefficients, see Reference 16, Reference 23, Reference 42, Reference 43) as the diagonal for the Suslin-Voevodsky motivic complex Reference 33, Reference 50

$$\begin{equation*} C_{\ast }(E\Sigma _p,\mathbb{F}_p)/\Sigma _p\otimes \mathbb{Z}/p(w)[s]\to C_{\ast }(E\Sigma _p,\mathbb{F}_p)\otimes _{\Sigma _p} \mathbb{Z}/p(w)[s]^{\otimes p} \end{equation*}$$

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 $p=2$). We refer to Reference 29, §1.5 for a survey of Dyer-Lashof operations.

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 $H_{\ast }(X;\mathbb{F}_p)$. Recall that $\alpha ^\mathscr{R}\colon \mathcal{A}\to (X\otimes ^\mathscr{R}\mathcal{A})^X$ and $\nu \colon \mathscr{R}^X\wedge _\mathscr{R}(X\otimes ^\mathscr{R}\mathcal{A})\to X\otimes ^\mathscr{R}\mathcal{A}$ are maps of commutative $\mathscr{R}$-algebras. For appropriate $r$ and $s$, we obtain

$$\begin{equation*} Q^r\nu ^{-1}_{\star }\alpha ^\mathscr{R}_{\star }=\nu ^{-1}_{\star }\alpha ^\mathscr{R}_{\star }Q^s \end{equation*}$$

and so

$$\begin{equation*} \sum _x\sum _{a+b=s}Q^ax^{\vee }\otimes Q^b\sigma ^\mathscr{R}(x\otimes a)=\sum _x x^{\vee }\otimes \sigma ^\mathscr{R}(x\otimes Q^s a). \end{equation*}$$

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

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

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

$$\begin{equation*} Q^s\sigma =\sigma Q^s\colon \pi _{s,w}(\mathbf{M}\mathbb{F}_p\wedge \mathbf{M}\mathbb{F}_p)\to \mathbf{MHH}_{s+1+2s(p-1),pw}(\mathbb{F}_p) \end{equation*}$$

for $p$ odd, and

$$\begin{equation*} Q^s\sigma =\sigma Q^s\colon \pi _{s,w}(\mathbf{M}\mathbb{F}_2\wedge \mathbf{M}\mathbb{F}_2)\to \mathbf{MHH}_{1+2s,2w}(\mathbb{F}_2). \end{equation*}$$

By construction, the power operations are preserved under base change. Over any algebraically closed field, we claim there is a relation (due to Steinberger Reference 10, III.2 in the topological setting)

$$\begin{equation} \tau ^{p-1}\tau _{i+1}=Q^{p^i}\tau _i\in \pi _{2p^{i+1}-1,p^{i+1}-p}(\mathbf{M}\mathbb{F}_p\wedge \mathbf{M}\mathbb{F}_p). \cssId{texmlid32}{\tag{14}} \end{equation}$$

By Remark 2.2 and rigidity, it 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 Equation 11, we obtain the first formula

$$\begin{equation} \tau ^{p-1}\sigma \tau _{i+1}=\sigma Q^{p^i}\tau _i=Q^{p^i}\sigma \tau _i= (\sigma \tau _{i})^p. \cssId{texmlid33}{\tag{15}} \end{equation}$$

We use this result to prove the vanishing of $(1\wedge \sigma ^+)_{\star }(s_1\otimes \tau ^{p-1}\xi _{i+1})$ for the map $1\wedge \sigma ^+\colon \mathbf{M}\mathbb{F}_p\wedge S^1_+\wedge \mathbf{M}\mathbb{F}_p\to \mathbf{M}\mathbb{F}_p\wedge \mathbf{MHH}(\mathbb{F}_p)$. This relation will be shown in homology from which the homotopy versions follow by the $\mathbf{M}\mathbb{F}_p$-algebra structure $\mathrm{mult}\colon \mathbf{M}\mathbb{F}_p\wedge \mathbf{MHH}(\mathbb{F}_p)\to \mathbf{MHH}(\mathbb{F}_p)$ (splitting the inclusion of homotopy in homology). 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 $(\beta \wedge 1)_{\star }\colon \pi _\star (\mathbf{M}\mathbb{F}_p\wedge \mathscr{R})\to \pi _{\star }(S^1\wedge \mathbf{M}\mathbb{F}_p\wedge \mathscr{R})$ 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 $(\beta \wedge 1)_{\ast }$. By construction of the motivic Steenrod algebra, see Reference 23, §5, Reference 53, §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

$$\begin{equation*} \xi _{i+1}=\beta \tau _{i+1}. \end{equation*}$$

The diagram

$$\begin{equation*} \vcenter{\img[][221pt][58pt][{$\xymatrix{\M\FF_p\smsh S^1_+\smsh\M\FF_p\ar[r]^{1\smsh\sigma^+}\ar[d]^{\beta\smsh1}&\M\FF_p\smsh\MHH(\FF_p)\ar[d]^{\beta\smsh1}\\ S^1\smsh\M\FF_p\smsh S^1_+\smsh\M\FF_p\ar[r]^{1\smsh\sigma^+}&S^1\smsh\M\FF_p\smsh\MHH(\FF_p) }$}]{Images/imgebe1bea129579e6e2ec39b2a4a0cc33b.svg}} \end{equation*}$$

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

$$\begin{align*} 0&=\mathrm{mult}\beta [\sigma ^{\mathbf{M}\mathbb{F}_p}_{\star }(s_1\otimes \tau _i)]^p =\mathrm{mult}\beta \sigma ^{\mathbf{M}\mathbb{F}_p}_{\star }(s_1\otimes \tau ^{p-1}\tau _{i+1})\\ &=\sigma \beta \tau ^{p-1}\tau _{i+1}=\sigma \tau ^{p-1}\xi _{i+1} =\tau ^{p-1}\sigma \xi _{i+1}. \end{align*}$$

For $p=2$, we will see that the last formula follows directly from the $d^1$-differentials in the Tor-spectral sequence, but we may also use the Bockstein and compute ($\rho =0$ since the base field is algebraically closed)

$$\begin{align*} 0&=\mathrm{mult}\beta [\sigma ^{\mathbf{M}\mathbb{F}_2}_{\star }(s_1\otimes \tau _i)]^2 =\mathrm{mult}\beta \sigma ^{\mathbf{M}\mathbb{F}_2}_{\star }(s_1\otimes \tau \tau _{i+1})\\ &=\sigma (\beta (\tau \tau _{i+1}))= \sigma (\beta (\tau )\tau _{i+1}+\tau \beta (\tau _{i+1}))= \sigma (\rho \tau _{i+1}+\tau \xi _{i+1})=\tau \sigma \xi _{i+1}. \end{align*}$$

This finishes the proof.

■

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 Reference 12, §2.8. The cellularity assumption is central in motivic homotopy theory, see, e.g., Reference 46, §2.3. It is, moreover, needed for running the motivic Tor spectral sequence (we refer to Reference 18 for the topological setting).

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. If $\mathcal{A}$ is commutative and the modules $\mathscr{M}$ and $\mathscr{Q}$ are commutative $\mathcal{A}$-algebras, then the homotopy colimit 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. Moreover, the derived smash product $\mathscr{M}\wedge _{\mathcal{A}}\mathscr{Q}$ in Equation 7 is the homotopy colimit of the diagram over the opposite simplex category $\Delta ^{\mathrm{op}}$ given by

$$\begin{equation*} [s]\mapsto \mathscr{M}\wedge _{\mathscr{R}} \mathcal{A}^{\wedge _{\mathscr{R}}^{s}}\wedge _{\mathscr{R}} \mathscr{Q}. \end{equation*}$$

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

$$\begin{equation*} E^1_{s,\star } = B_s(\pi _{\star }\mathscr{M},\pi _{\star }\mathcal{A},\pi _{\star }\mathscr{Q}) = \pi _{\star }\mathscr{M}\otimes _{\pi _{\star }\mathscr{R}}\pi _{\star }\mathcal{A}^{\otimes _{\pi _{\star }\mathscr{R}}^{s}} \otimes _{\pi _{\star }\mathscr{R}} \pi _{\star }\mathscr{Q}. \end{equation*}$$

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 Equation 16. We recall the $d^{1}$ differential is given by the alternating sum of the face maps

$$\begin{equation*} d_{i}[m_0|a_1|\dots |a_s|n_{s+1}] = \begin{cases}[m_0\cdot a_1|a_2|\dots |a_s|n_{s+1}] &i=0\\ [m_0|a_1|\dots |a_i\cdot a_{i+1}|\dots |a_s|n_{s+1}]&0<i<s\\ [m_0|a_1|\dots |a_{s-1}|a_s\cdot n_{s+1}]&i=s. \end{cases} \end{equation*}$$

The isomorphim $\mathscr{M}\wedge _\mathscr{R}(\mathscr{Q}\wedge \mathscr{R})^{\wedge _{\mathscr{R}}^{s}}\wedge _\mathscr{R}\mathscr{Q}\cong (\mathscr{M}\wedge _\mathscr{R}\mathscr{Q})\wedge \mathscr{Q}^{\wedge s}$ given by multiplication relates our description of the derived smash product to the Hochschild homology style description

$$\begin{equation*} [s]\mapsto (\mathscr{M}\wedge _\mathscr{R}\mathscr{Q})\wedge \mathscr{Q}^{\wedge s}. \end{equation*}$$

Proposition 2.4.

Let $\mathscr{R}\rightarrow \mathscr{Q}$ be a map of cellular motivic ring spectra, where $\mathscr{R}$ is commutative and $\mathcal{A}=\mathscr{Q}\wedge \mathscr{R}$ is a cofibrant $\mathscr{R}$-algebra, and let $\mathscr{M}$ be an $\mathscr{Q}$-$\mathscr{R}$ bimodule. Then the skeletal filtration on the simplicial circle gives rise to a strongly convergent trigraded Tor spectral sequence

$$\begin{equation} E^{2}_{h,t,w} = \mathbf{Tor}^{\pi _{\star }(\mathcal{A})}_{h,t,w}(\pi _{\star }\mathscr{M},\pi _{\star }\mathscr{S}) \Rightarrow \mathbf{MHH}_{h+t,w}(\mathscr{R};\mathscr{M}). \cssId{texmlid12}{\tag{16}} \end{equation}$$

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

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

If $\mathscr{R}$ is commutative and $\mathscr{Q}$ and $\mathscr{M}$ are commutative $\mathscr{R}$-algebras, then the Tor spectral sequence is a spectral sequence of $\mathscr{R}_\star$-algebras with the multiplicative structure on the $E^{1}$ page given by the shuffle product introduced by Eilenberg-MacLane Reference 15. The pinch map on the circle induces the Hopf-algebra structure on the torsion groups on the $E^2$ page. If $E^2,\dots ,E^r$ are all flat over $\mathscr{R}_\star$ for $2\leq r$, then the $E^r$ page inherits an $\mathscr{R}_\star$-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 Equation 7, Reference 1, §4, Reference 12, Proposition 7.7, Reference 39, §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

$$\begin{equation*} S^1\otimes \mathscr{R}\cong \{[s]\mapsto (\mathscr{R}\wedge _{\mathscr{R}}\mathscr{R})\wedge (\mathscr{R}\wedge \mathscr{R})^{\wedge _{\mathscr{R}}s}\}. \end{equation*}$$

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 $\sigma _{\star }x\in \pi _{d+1,w}(S^1\otimes \mathscr{R})$ is the class represented by $[x]\in E^1_{1,d,w}$.

The Hopkins-Morel equivalence shown by Hoyois Reference 22, 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

$$\begin{equation} E^{2}_{h,t,w} = \mathbf{Tor}^{\mathcal{A}_{\star }}_{h,t,w}(\mathbb{M}_{\star },\mathbb{M}_{\star }) \Rightarrow \mathbf{MHH}_{h+t,w}(\mathbf{M}\mathbb{F}_{p}). \cssId{texmlid13}{\tag{17}} \end{equation}$$

Remark 2.5.

By Remark 2.2 and Equation 17 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

We shall repeatedly make use of some torsion product computations, see Reference 39, §6, and Section 1.1 for our notation.

Lemma 2.6.

(i)

For the polynomial algebra $\mathbb{F}_{p}[\mathsf{x}]$ on a generator $\mathsf{x}$ in even degree $d$, there is an $\mathbb{F}_{p}$-bialgebra isomorphism$$\begin{equation*} \mathbf{Tor}^{\mathbb{F}_{p}[\mathsf{x}]}_{\ast }(\mathbb{F}_{p},\mathbb{F}_{p}) \cong \Lambda _{\mathbb{F}_{p}}(\sigma \mathsf{x}). \end{equation*}$$

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

(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$$\begin{equation*} \mathbf{Tor}^{\Lambda _{\mathbb{F}_{p}}(\mathsf{x})}_{\ast }(\mathbb{F}_{p},\mathbb{F}_{p}) \cong \Gamma _{\mathbb{F}_{p}}(\sigma \mathsf{x}). \end{equation*}$$

Here, $\gamma _j\sigma \mathsf{x}$ in degree $(j,dj)$ is represented in the bar complex by $[\mathsf{x}\vert \mathsf{x}\vert \dots \vert \mathsf{x}]$ and has coproduct$$\begin{equation*} \psi (\gamma _{k}\sigma \mathsf{x}) = \underset{i+j=k}{\sum } \gamma _{i}\sigma \mathsf{x}\otimes \gamma _{j}\sigma \mathsf{x}. \end{equation*}$$

Remark 2.7.

As an example, let us give a direct proof of the relation $\tau \sigma \xi _{i+1}=\rho \sigma \tau _{i+1}\in \mathbf{MHH}_{\star }(\mathbb{F}_2)$ shown for algebraically closed fields in Lemma 2.3. Consider the $E^1$ page

$$\begin{equation*} E^1_{s,\star }=\mathcal{A}_\star ^{\otimes _{\mathbb{M}_\star }s}\cong \mathbb{M}_\star \otimes _{\mathbb{M}_\star }\mathcal{A}_\star ^{\otimes _{\mathbb{M}_\star }s}\otimes _{\mathbb{M}_\star }\mathbb{M}_{\star } \end{equation*}$$

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

$$\begin{equation*} d^1[\tau _i|\tau _i]=[\tau ^2_i]=\tau [\xi _{i+1}]+\rho [\tau _{i+1}]+\rho [\tau _0\xi _{i+1}] \end{equation*}$$

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

$$\begin{align*} d^1\gamma _{j+2}\sigma \tau _i&=\sum _{a=1}^{j-1}[\tau _i|\dots |\tau _i^2|\dots |\tau _i]\\ & = [\tau _i^2]\cdot [\tau _i |\dots |\tau _i]\\ & =(\tau [\xi _{i+1}]+\rho [\tau _{i+1}]+\rho [\tau _0\xi _{i+1}])\cdot [\tau _i | \dots |\tau _i]\\ &=(\tau \sigma \xi _{i+1}+\rho \sigma \tau _{i+1}+\rho \sigma (\tau _0\xi _{i+1}))\cdot \gamma _j\sigma \tau _i. \end{align*}$$

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

$$\begin{equation*} d^1\gamma _{j+2}\sigma \tau _i=\tau \sigma \xi _{i+1}\cdot \gamma _j\sigma \tau _i. \end{equation*}$$

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

$$\begin{equation*} E^2_{\ast ,\star }=\mathbb{M}_\star \otimes _{\mathbb{F}_p} \bigotimes _{i\geq 0}\Lambda _{\mathbb{F}_{p}}(\sigma \xi _{i+1})\otimes \Gamma _{\mathbb{F}_{p}}(\sigma \tau _i). \end{equation*}$$

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 relation (described up to a unit in $\mathbb{F}_p$)

$$\begin{equation*} d^{p-1}\gamma _p\sigma \tau _i\overset{\cdot }{=}\tau ^{p-1}\sigma \xi _{i+1}. \end{equation*}$$

2.5. A Bockstein type complex

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

$$\begin{equation*} C=\bigotimes _{i\geq 0}\Gamma _{\mathbb{F}_{p}}(\bar{\mu }_{i}) \otimes \Lambda _{\mathbb{F}_{p}}(\bar{\lambda }_{i+1}) \end{equation*}$$

and $\partial \colon C\to C$ is the derivation generated by $\partial (\gamma _{j+p}\bar{\mu }_i)=\bar{\lambda }_{i+1}\gamma _j\bar{\mu }_i$ and $\partial (\bar{\lambda }_{i+1})=0$. Set $B^\partial =\operatorname {im}\partial$, $Z^\partial =\ker \partial$ and $H^\partial =Z^\partial /B^\partial$—this subsection aims to calculate these. In our application $C$ will be the mod-$\tau ^{p-1}$ motivic Hochschild homology of $\mathbb{F}_p$ (the reader may recognize it as $\operatorname {Tor}^{\mathcal{A}^{\mathrm{rig}}_\star }_{\ast }(\mathbb{F}_p,\mathbb{F}_p)$) and $\partial$ 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 many others exist. 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 $f\colon \mathbb{N}\acwgapcirclearrow$ has finite non-empty support, $\operatorname {supp}f=\{n\in \mathbb{N}\mid f(n)\neq 0\}$, we write $t_f=t_{\operatorname {supp}f}$. For every $j\in \mathbb{N}$, let $\delta _j\colon \mathbb{N}\acwgapcirclearrow$ be the function with $\operatorname {supp}\delta _j=\{j\}$ and $\delta _j(j)=1$.

Definition 2.8.

Let $J$ denote the set of pairs $(S,f)$, where the function $f\colon \mathbb{N}\acwgapcirclearrow$ has finite support and $S\subseteq \operatorname {supp}f$. 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

$$\begin{equation} \chi _{S,f} \coloneq \left(\prod _{m\in S}\bar{\lambda }_{m+1}\gamma _{pf(m)-p}\bar{\mu }_m\right) \left(\prod _{n\not \in S}\gamma _{pf(n)}\bar{\mu }_n\right) \in C. \cssId{texmlid34}{\tag{18}} \end{equation}$$

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

$$\begin{equation*} \partial \chi _{S,f}=\sum _{n\in \operatorname {supp}(f)-S}\chi _{S\cup \{n\},f} \end{equation*}$$

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

Definition 2.10.

If $f\colon \mathbb{N}\acwgapcirclearrow$ has finite support, the associated $f$-cube is the sub-complex

$$\begin{equation*} (C^f,\partial ) \coloneq \left(\bigoplus _{S\subseteq \operatorname {supp}f} \mathbb{F}_p\{\chi _{S,f}\},\partial \right) \subseteq (C,\partial ). \end{equation*}$$

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

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

Lemma 2.11.

If $f\colon \mathbb{N}\acwgapcirclearrow$ has finite non-empty support, then $(C^f,\partial )$ is contractible so that $H^f=0$. Furthermore, $B^f=Z^f$ is generated by the $\partial \chi _{S,f}$ with $t_f\notin S\subseteq \operatorname {supp}f$.

Proof.

For $N=|\operatorname {supp}f|$ 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 $\partial \chi _{S,f}=\sum _{n\in \operatorname {supp}(f)-S}\chi _{S\cup \{n\},f}$ we see that the differential restricts to a chain complex

$$\begin{equation*} (C^f,{\partial })=\left\{ C^f_{0}\xrightarrow [\qquad ]{\partial } C^f_{1}\xrightarrow [\qquad ]{\partial } \dots \xrightarrow [\qquad ]{\partial } C^f_{N-1}\xrightarrow [\qquad ]{\partial } C^f_{N}\right\}. \end{equation*}$$

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

$$\begin{equation*} \{{\partial }\chi _{S,f}\mid k=|S|, t_f\notin S\} \end{equation*}$$

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

$$\begin{align*} \dim H_k(C^f,{\partial }) & = \dim \ker \{{\partial }\colon C^f_{k}\to C^f_{k+1}\}-\dim \operatorname {im}\{{\partial }\colon C^f_{k-1}\to C^f_{k}\}\\ & \leq \binom{N}{k}-\binom{N-1}{k}-\binom{N-1}{k-1}=0 \end{align*}$$

and so $B^f=Z^f$ is generated by the $\partial \chi _{S,f}$ with $t_f\notin S\subseteq \operatorname {supp}f$, 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 $f,g\colon \mathbb{N}\acwgapcirclearrow$ with finite support and non-empty finite sets $S,T\subseteq \mathbb{N}$ define $K_{S,T,f,g}\in \mathbb{F}_{p}$ by

$$\begin{equation*} K_{S,T,f,g} = \left(\prod _{s\in S}\binom{f(s)-1+g(s)}{f(s)-1}\right) \left(\prod _{t\in T}\binom{f(t)+g(t)-1}{f(t)}\right) \left(\prod _{c\notin S\cup T}\binom{f(c)+g(c)}{f(c)}\right) \end{equation*}$$

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

$$\begin{equation*} \epsilon _{u,S,T,f,g} = K_{S\cup \{u\},T\cup \{t_{f+g}\},f+g} + K_{S\cup \{t_{f+g}\},T\cup \{u\},f+g}. \end{equation*}$$

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 $\operatorname {supp}f\cap \operatorname {supp}g=\operatorname {supp}(f\cdot g)$, 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 $u\in \operatorname {supp}(f+g)$, $u\not =t_{f+g}$ and $u\notin S\cup T$.

Lemma 2.13, 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

$$\begin{equation*} \chi _{S,f}\chi _{T,g}=K_{S,T,f,g}\cdot \chi _{S\cup T,f+g} \end{equation*}$$

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

$$\begin{equation*} \partial \chi _{S,f}\cdot \partial \chi _{T,g} = \sum _{t_{f+g}\not =u\in \operatorname {supp}(f+g)-S\cup T}\epsilon _u\cdot \partial \chi _{S\cup T\cup \{u\},f+g}. \end{equation*}$$

Lemma 2.14.

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

$$\begin{equation*} \vcenter{\img[][94pt][49pt][{$\xymatrix{ C^f\otimes C^g \ar[r] \ar[d] & C\otimes C \ar[d] \\ C^{f+g}\ar[r] & C. }$}]{Images/img7457842005bd110d4c98fcfc20fffc00.svg}} \end{equation*}$$

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

$$\begin{equation*} \bigoplus _{f\colon \mathbb{N}\acwgapcirclearrow }C^f\subseteq C \end{equation*}$$

and

$$\begin{equation*} {\mathbb{F}_{p}}[\bar{\mu }_{i}]/\bar{\mu }_i^p\subseteq C \end{equation*}$$

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

$$\begin{equation*} \bigoplus _{f\colon \mathbb{N}\acwgapcirclearrow } C^f \cong \bigotimes _{i\geq 0} \left(\Lambda _{\mathbb{F}_{p}}(\bar{\lambda }_{i+1}) \otimes \bigotimes _{j> 0}{\mathbb{F}_{p}}[\gamma _{p^{j}}\bar{\mu }_{i}]/(\gamma _{p^{j}}\bar{\mu }_{i})^p\right) \end{equation*}$$

and

$$\begin{equation*} C \cong \left(\bigotimes _{i\geq 0}{\mathbb{F}_{p}}[\bar{\mu }_{i}]/\bar{\mu }_{i}^p\right) \otimes \left(\bigoplus _{f\colon \mathbb{N}\acwgapcirclearrow }C^f\right). \end{equation*}$$

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,

$$\begin{equation*} H^\partial \cong \bigotimes _{i\geq 0}\mathbb{F}_{p}[\bar{\mu }_{i}]/\bar{\mu }_i^p \end{equation*}$$

and $Z^\partial$ is the subalgebra of $C$ generated by the $\bar{\mu }_i$ with $i\geq 0$ and the $\partial \chi _{S,f}$ with $t_f\notin S\subseteq \operatorname {supp}f$. More explicitly, and writing $\mathsf{x}_{S,f}=\partial \chi _{S,f}$, the relation expressed in Lemma 2.13 gives an isomorphism

$$\begin{equation*} Z^\partial \cong \mathbb{F}_p[\bar{\mu }_{i},\mathsf{x}_{S,f}]_{i\in \mathbb{N}, (S,f)\in K}/(\bar{\mu }_{i}^p,\mathsf{x}_{S,f} \cdot \mathsf{x}_{T,g}-\sum _{u}\epsilon _u\cdot \mathsf{x}_{S\cup T\cup \{u\},f+g}). \end{equation*}$$

Here, $u\in \operatorname {supp}(f+g)-S\cup T$ 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

$$\begin{equation} \mathbb{M}_{\star } \cong k^{M}_{\ast }[\tau ] \cong \mathbb{F}_{p}[\tau ] \cssId{texmlid14}{\tag{19}} \end{equation}$$

by Reference 48, Corollary 4.3, p.254, where $\vert \tau \vert =(0,-1)$. From Equation 9 and Equation 19 we deduce

$$\begin{equation} \mathcal{A}_{\star } = \begin{cases} \mathbb{F}_{p}[\tau ,\xi _{i+1},\tau _{i}]_{i\geq 0}/(\tau _{i}^{2}-\tau \xi _{i+1}) & p=2 \\ \mathbb{F}_{p}[\tau ,\xi _{i+1}]_{i\geq 0}\otimes \Lambda _{\mathbb{F}_{p}}(\tau _{i})_{i\geq 0} & p\neq 2. \end{cases} \cssId{texmlid35}{\tag{20}} \end{equation}$$

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

$$\begin{equation} \pi _\star ((\mathbf{M}\mathbb{F}_p\wedge \mathbf{M}\mathbb{F}_p)[\tau ^{-1}])\cong \mathcal{A}_{\star }[\tau ^{-1}] = \begin{cases} \mathbb{F}_{p}[\tau ^{\pm 1},\tau _{i}]_{i\geq 0} & p=2 \\ \mathbb{F}_{p}[\tau ^{\pm 1},\xi _{i+1}]_{i\geq 0}\otimes \Lambda _{\mathbb{F}_{p}}(\tau _{i})_{i\geq 0} & p\neq 2. \end{cases} \cssId{texmlid17}{\tag{21}} \end{equation}$$

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

$$\begin{equation} \pi _\star ((\mathbf{M}\mathbb{F}_p\wedge \mathbf{M}\mathbb{F}_p)/\tau ^{p-1})\cong \mathcal{A}_{\star }/\tau ^{p-1} = (\bigotimes _{i\geq 0}\mathbb{F}_{p}[\xi _{i+1}]\otimes \Lambda _{\mathbb{F}_{p}}(\tau _{i})) \otimes \mathbb{F}_{p,\tau }. \cssId{texmlid15}{\tag{22}} \end{equation}$$

Here $\mathbb{F}_{p,\tau }$ is shorthand for $\mathbb{F}_{p}[\tau ]/\tau ^{p-1}$. In Section 3.2, we use Equation 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 Reference 3, Reference 17 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

$$\begin{equation} \mathscr{R}/p\to \mathscr{R}/p[\tau ^{-1}] \cssId{texmlid16}{\tag{23}} \end{equation}$$

exhibits the $\tau$-periodization as a motivic $E_{\infty }$ ring spectrum under $\mathscr{R}/p$; see Reference 4, §12, Reference 17, §8 for recent expositions. If $\mathscr{R}$ happens to be cellular, then so is $\mathscr{R}/p[\tau ^{-1}]$. Owing to Reference 3, Theorem 1.2, Equation 23 is an étale localization (the $\rho$-completion in Reference 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 Equation 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 Reference 25, Reference 27, §4. In this section, 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

$$\begin{equation} \mathbf{MHH}(\mathbb{F}_{p})[\tau ^{-1}] = (\mathbf{M}\mathbb{F}_{p}\wedge _{\mathbf{M}\mathbb{F}_{p}\wedge \mathbf{M}\mathbb{F}_{p}}\mathbf{M}\mathbb{F}_{p})[\tau ^{-1}] \xrightarrow {\simeq } \mathbf{M}\mathbb{F}_{p}[\tau ^{-1}]\wedge _{(\mathbf{M}\mathbb{F}_{p}\wedge \mathbf{M}\mathbb{F}_{p})[\tau ^{-1}]}\mathbf{M}\mathbb{F}_{p}[\tau ^{-1}]. \cssId{texmlid18}{\tag{24}} \end{equation}$$

Our calculation uses the classes

$$\begin{equation*} \mu _{i}\coloneq \sigma \tau _i, \lambda _{i}\coloneq \sigma \xi _{i}. \end{equation*}$$

Lemma 3.1.

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

$$\begin{equation*} E^{\infty } =(\bigotimes _{i\geq 0}\mathbb{F}_p[\mu _{i}]/\mu _i^p)[\tau ^{\pm 1}]. \end{equation*}$$

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

$$\begin{equation} d^{p-1}(\gamma _{j}\mu _{i}) \overset{\cdot }{=} \tau ^{p-1}\lambda _{i+1}\gamma _{j-p}\mu _{i} \cssId{texmlid20}{\tag{25}} \end{equation}$$

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

Proof.

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

$$\begin{equation} E^{2} = (\bigotimes _{i\geq 0}\Lambda _{\mathbb{F}_{2}}(\mu _{i}))[\tau ^{\pm 1}]. \cssId{texmlid36}{\tag{26}} \end{equation}$$

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

$$\begin{equation} E^{2} = (\bigotimes _{i\geq 0}\Gamma _{\mathbb{F}_{p}}(\mu _{i}) \otimes \Lambda _{\mathbb{F}_{p}}(\lambda _{i+1})) [\tau ^{\pm 1}]. \cssId{texmlid19}{\tag{27}} \end{equation}$$

The Tor spectral sequence starts as an augmented unital $\mathbb{M}_\star [\tau ^{-1}]$-Hopf algebra since Equation 27 is flat over $\mathbb{M}_\star [\tau ^{-1}]$. Arguing as in Reference 1, §4, Reference 2, §5, Reference 20, §1.2, Reference 34, Reference 39, we’ll see that the nontrivial 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

$$\begin{align*} & \psi (d^{p-1}(\gamma _{p+k}\mu _{i})) - \psi (\tau ^{p-1}\lambda _{i+1}\gamma _{k}\mu _{i})\\ & \qquad = (d^{p-1}\otimes 1+1\otimes d^{p-1})\psi (\gamma _{p+k}\mu _{i}) \\ & \qquad \quad - \tau ^{p-1} (\lambda _{i+1}\otimes 1+1\otimes \lambda _{i+1}) (\Sigma _{a+b=k}\gamma _{a}\mu _{i}\otimes \gamma _{b}\mu _{i}) \\ & \qquad = (d^{p-1}\otimes 1)(\gamma _{p+k}\mu _{i}\otimes 1) + (1\otimes d^{p-1})(1\otimes \gamma _{p+k}\mu _{i}) \\ & \qquad \quad + \tau ^{p-1} \underset{a+b=p+k; a,b>0}{\sum } ( \lambda _{i+1}\gamma _{a-p}\mu _{i}\otimes \gamma _{b}\mu _{i} + \gamma _{a}\mu _{i}\otimes \lambda _{i+1}\gamma _{b-p}\mu _{i} ) \\ & \qquad \quad - \tau ^{p-1} \underset{a+b=k}{\sum }(\lambda _{i+1}\gamma _{a}\mu _{i}\otimes \gamma _{b}\mu _{i} +\gamma _{a}\mu _{i}\otimes \lambda _{i+1}\gamma _{b}\mu _{i}) \\ & \qquad = (d^{p-1}\otimes 1)(\gamma _{p+k}\mu _{i}\otimes 1) + (1\otimes d^{p-1})(1\otimes \gamma _{p+k}\mu _{i}) \end{align*}$$

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 Reference 44 (in characteristic zero) reduces to considering complex numbers. Over $\mathbb{C}$, the differential Equation 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

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

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 follows from Lemma 3.1 and the multiplicative extension

$$\begin{equation} \mu _{i}^{p}= \tau ^{p-1}\mu _{i+1} \cssId{texmlid37}{\tag{28}} \end{equation}$$

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

$$\begin{equation} \mathbb{F}_{p}[\tau ,\mu _{i}]_{i\geq 0}/(\mu _{i}^{p}-\tau ^{p-1}\mu _{i+1}). \cssId{texmlid38}{\tag{29}} \end{equation}$$

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

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 $n=p-1$)

$$\begin{equation} \Sigma ^{0,n}\mathbf{M}\mathbb{F}_{p} \overset{\tau ^{n}}{\rightarrow } \mathbf{M}\mathbb{F}_{p} \rightarrow \mathbf{M}\mathbb{F}_{p}/\tau ^{n}. \cssId{texmlid22}{\tag{30}} \end{equation}$$

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 and thus the mod-$p$ homology ring $\mathbb{M}_{\star }$ is the homology of an $E_{\infty }$ ring spectrum $C_{\star }\mathbb{F}_{p}$ in graded complexes of $\mathbb{F}_{p}$-vector spaces, see Reference 47, Appendix C, Corollary C.3 for details. As noted below, $C_{\star }\mathbb{F}_{p}$ is formal, i.e., equivalent as an $E_{\infty }$ ring spectrum in graded complexes of $\mathbb{F}_{p}$-vector spaces to the (bigraded) homology $\mathbb{M}_{\star }$ equipped with trivial differentials, $C_{\star }\mathbb{F}_{p}/\tau ^{n}$ is $E_{\infty }$ over $\mathbb{M}_{\star }$ for all $n\geq 1$. This implies the corresponding claim for $\mathbf{M}\mathbb{F}_{p}/\tau ^{n}$. To show formality, consider the free $E_{\infty }$ algebra $\mathcal{E}$ in graded complexes of $\mathbb{F}_{p}$-vector spaces on a generator $\tau$ in bidegree $(0,1)$. 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 $C_{\star }\mathbb{F}_{p}$ and $\mathbb{F}_{p}[\tau ]$ are equivalent since the natural map $\mathcal{E}\rightarrow C_{\star }\mathbb{F}_{p}$ is the $0$-truncation. When $n=1$, we also refer to Gheorghe Reference 19 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 Equation 1 yields the derived smash product

$$\begin{equation} \mathbf{MHH}(\mathbb{F}_{p})/\tau ^{n} \simeq \mathbf{M}\mathbb{F}_{p}/\tau ^{n} \wedge _{(\mathbf{M}\mathbb{F}_{p}\wedge \mathbf{M}\mathbb{F}_{p})/\tau ^{n}} \mathbf{M}\mathbb{F}_{p}/\tau ^{n}. \cssId{texmlid23}{\tag{31}} \end{equation}$$

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

$$\begin{equation} E^{2}_{h,t,w} = \mathbf{Tor}^{\mathcal{A}_{\star }/\tau ^{n}}_{h,t,w}(\mathbb{M}_{\star }/\tau ^{n},\mathbb{M}_{\star }/\tau ^{n}) \Rightarrow \mathbf{MHH}_{h+t,w}(\mathbb{F}_{p})/\tau ^{n}. \cssId{texmlid39}{\tag{32}} \end{equation}$$

Recall that $\mathbb{F}_{p,\tau }$ is shorthand for $\mathbb{F}_{p}[\tau ]/\tau ^{p-1}$. Lemma 2.6 and Equation 22 imply the Tor spectral sequence Equation 16 for $\mathbf{MHH}_{\star }(\mathbb{F}_{p})/\tau ^{p-1}$ takes the form

$$\begin{equation} \bar{E}^{2}_{\ast ,\star } \cong \left(\bigotimes _{i\geq 0} \Gamma _{\mathbb{F}_{p}}(\sigma \tau _{i}) \otimes \Lambda _{\mathbb{F}_{p}}(\sigma \xi _{i+1})\right) \otimes \mathbb{F}_{p,\tau } \Rightarrow \mathbf{MHH}_{\star }(\mathbb{F}_{p})/\tau ^{p-1}. \cssId{texmlid24}{\tag{33}} \end{equation}$$

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

$$\begin{equation*} d^r\colon \bar{E}^r_{f,d;w}\to \bar{E}^r_{f-r,d+r-1;w}. \end{equation*}$$

In Equation 33, we set $\bar{\mu }_{i}\coloneq \sigma \tau _i$ and $\bar{\lambda }_{i+1}\coloneq \sigma \xi _{i+1}$. The bar signifies 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): $|\bar{\lambda }_{i+1}|=(1,2p^{i+1}-2;p^{i+1}-1)$,
(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 Equation 33 has weight $w_{x}=-n+(p-1)m$, $0\leq n<p-1$, then $n$ equals $x$’s $\tau$-multiplicity. Another helpful bookkeeping device for our calculation is the Chow degree of $x$, see Reference 5, Definition 3.1 and Reference 26, Definition 2.1.10 for related terminology, defined by

$$\begin{equation*} c(x)=f_{x}+2w_{x}-d_{x}. \end{equation*}$$

In particular, we have

(1): $c(\bar{\lambda }_{i+1})=1+2(p^{i+1}-1)-(2p^{i+1}-2)=1$
(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 Equation 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 Equation 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 Equation 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 Equation 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): $d_{x}+f_{x}=d_g+f_g=2p^{i+j+1}$,
(2): $w_{x}=w_g=p(p^{i+j}-p^{j})$,
(3): $0<f_{x}<f_g=p^{j+1}$.

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

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

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

$$\begin{equation*} \mathbf{MHH}_{\star }(\mathbb{F}_p)/\tau ^{p-1} \cong (\bigotimes _{i\geq 0} \Gamma _{\mathbb{F}_{p}}(\bar{\mu }_{i}) \otimes \Lambda _{\mathbb{F}_{p}}(\bar{\lambda }_{i+1})) \otimes \mathbb{F}_{p,\tau }. \end{equation*}$$

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 $C=\mathbf{Tor}^{\mathcal{A}_{\star }^{\mathrm{rig}}}_{\ast ,\star }(\mathbb{F}_{p},\mathbb{F}_{p})$ appeared in Section 2.5.

3.3. Integral motivic Hochschild homology

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

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

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

(1): $\bar{\lambda }_{i+1}\in \mathbf{MHH}_{\star }(\mathbb{F}_{p})/\tau ^{p-1}$ is the class of the permanent cycle$$\begin{equation*} [\bar{\xi }_{i+1}]\in \bar{E}^{1}_{1,2p^{i+1}-2;p^{i+1}-1}, \end{equation*}$$
(2): $\gamma _j\bar{\mu }_{i}\in \mathbf{MHH}_{\star }(\mathbb{F}_{p})/\tau ^{p-1}$ is the class of the permanent cycle$$\begin{equation*} [\bar{\tau }_i|\dots |\bar{\tau }_i]\in \bar{E}^{1}_{j,j(2p^{i}-1);j(p^{i}-1)}. \end{equation*}$$

As before, to aid the bookkeeping, we also set

$$\begin{equation*} \lambda _{i+1}=[\xi _{i+1}]\in E^{1}_{1,2p^{i+1}-2;p^{i+1}-1} \end{equation*}$$

and

$$\begin{equation*} \gamma _j\mu _{i}=[\tau _i|\dots |\tau _i]\in E^{1}_{j,j(2p^{i}-1);j(p^{i}-1)}, \end{equation*}$$

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.

(1)

For $0<r<p$ the étale localization$$\begin{equation*} L^r_{\mathrm{\acute{e}t}}\colon E^r\to E^r[\tau ^{-1}] \end{equation*}$$

is an injection.

(2)

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

(3)

For all $p$$$\begin{equation*} d^{p-1}\gamma _{j+p}\mu _i\overset{\cdot }{=}\tau ^{p-1}\lambda _{i+1}\gamma _j\mu _i \end{equation*}$$

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 $L^1_{\mathrm{\acute{e}t}}$ is an injection, and from the Tor-calculations we get that for odd primes $p$ also $L^2_{\mathrm{\acute{e}t}}$ is an injection. Assume that for given $0<r<p$ $L^r_{\mathrm{\acute{e}t}}$ 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 $L^{r+1}_{\mathrm{\acute{e}t}}$ 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 $L^{p-1}_{\mathrm{\acute{e}t}}$ is an injection, the formula

$$\begin{equation*} d^{p-1}\gamma _{j+p}\mu _i\overset{\cdot }{=}\tau ^{p-1}\lambda _{i+1}\gamma _j\mu _i \end{equation*}$$

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

$$\begin{equation*} 0\longrightarrow E^1_{f,d;w+1} \xrightarrow [\quad ]{\tau } E^1_{f,d;w} \xrightarrow [\quad ]{q} \bar{E}^1_{f,d;w} \longrightarrow 0. \end{equation*}$$

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

$$\begin{equation} \bar{\partial }\colon \mathbf{MHH}_{\ast +1,\ast }(\mathbb{F}_{2})/\tau \xrightarrow {\partial } \mathbf{MHH}_{\ast ,\ast +1}(\mathbb{F}_{2}) \xrightarrow {q} \mathbf{MHH}_{\ast ,\ast +1}(\mathbb{F}_{2})/\tau . \cssId{texmlid26}{\tag{35}} \end{equation}$$

Since Equation 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 $\lambda _{i+1}=[\xi _{i+1}]\in E^{1}_{1,2^{i+2}-2;2^{i+1}-1}$ we get $\bar{\partial }\bar{\lambda }_{i+1}=0$, and since $\gamma _{j+2}\bar{\mu }_i$ is hit by $\gamma _{j+2}\mu _i=[\tau _i|\dots |\tau _i]\in E^{1}_{j,j(2^{i+1}-1);j(2^{i}-1)}$ and $d^1\gamma _{j+2}\mu _{i}=\tau \lambda _{i+1}\gamma _{j}\mu _{i}$ we deduce Lemma 3.9.

Lemma 3.9.

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

$$\begin{equation*} \bar{\partial }\gamma _{j+2}\bar{\mu }_{i}=\bar{\lambda }_{i+1}\gamma _{j}\bar{\mu }_{i} \end{equation*}$$

for all $i,j\geq 0$, i.e., $(\mathbf{MHH}_\star (\mathbb{F}_2)/\tau ,\bar{\partial })=(C,\partial )$, where $(C,\partial )$ is the commutative differential graded algebra of Section 2.5.

Combined with Corollary 2.15, and using that the $\tau$-free element $\mu _{i}\in \mathbf{MHH}_{\star }(\mathbb{F}_{2})$ maps to $\bar{\mu }_{i}\in \mathbf{MHH}_{\star }(\mathbb{F}_{2})/\tau$, 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

$$\begin{equation*} \vcenter{\img[][346pt][56pt][{$\xymatrix@C=1pc{ 0\ar[r]&(\tau\mathrm{-torsion})\ar[r]\ar[d]^q_\cong& \MHH_{\star}(\FF_2)\ar[r]\ar[d]^q & \FF_2[\tau,\mu_0,\mu_1,\dots]/(\mu_{i}^2-\tau\mu_{i+1})\ar[r] & 0\\ 0\ar[r] & \mathrm{im}\bar\partial\ar[r] & \bigotimes_{i\geq0}\GAMMA(\bmu_{i})\otimes\LAMBDA(\blambda_{i+1}). & & }$}]{Images/img0aed209b7100b9c757c4c056b3bd5c2d.svg}} \end{equation*}$$

More elegantly, using Corollary 2.15, we have a pullback diagram of commutative $\mathbb{F}_2[\tau ]$-algebras

$$\begin{equation*} \vcenter{\img[][176pt][55pt][{$\xymatrix{ \MHH_{\star}(\FF_2)\ar[r]\ar[d]&\FF_2[\tau,\mu_i] /\mu_i^2-\tau\mu_{i+1}\ar[d]\\ \FF_2[\tau,\bmu_{i},\x_{S,f}] /\mathcal{I} \ar[r]&\FF_2[\tau,\bmu_i] /(\bmu_i^2,\tau)}$}]{Images/imgdd3e496f83ff46938bca3746e0639a0c.svg}} \end{equation*}$$

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

$$\begin{equation*} \mathcal{I}= (\tau ,\bar{\mu }_{i}^2,\mathsf{x}_{S,f}\cdot \mathsf{x}_{T,g}-\sum _{t_{f+g}\not =u\in \operatorname {supp}(f+g)-S\cup T}\epsilon _{u}\cdot \mathsf{x}_{S\cup T\cup \{u\},f+g}). \end{equation*}$$

Here $\mu _i$ maps to $\bar{\mu }_i$ and $\mathsf{x}_{S,f}$ maps to zero. When we finish 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 uses our knowledge of $d^{p-1}$ to calculate $E^p$. Consider the short exact sequence

$$\begin{equation*} 0 \longrightarrow E^{p-1}_{f,d;w+p-1} \xrightarrow [\quad ]{\tau ^{p-1}} E^{p-1}_{f,d;w} \xrightarrow [\quad ]{q} \bar{E}^{p-1}_{f,d;w} \longrightarrow 0 \end{equation*}$$

and the injection

$$\begin{equation*} L^{p-1}_{\mathrm{\acute{e}t}}\colon E^{p-1}_{f,d;w}\to E^{p-1}_{f,d;w}[\tau ^{-1}]. \end{equation*}$$

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:

$P(r)_{1}$:: $E^{r}_{f,d;w+p-1} \xrightarrow {\tau ^{p-1}} E^{r}_{f,d;w} \xrightarrow [\quad ]{q} \bar{E}^{r}_{f,d;w}$ is exact,
$P(r)_{2}$:: in $E^{r}_{f,d;w}$ we have $\ker L^{r}_{\mathrm{\acute{e}t}} =\ker \tau ^{p-1}$, and
$P(r)_{3}$:: 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 $\partial \colon C\to C$ generated by $\partial (\gamma _{j+p}\bar{\mu }_i)=\bar{\lambda }_{i+1}\gamma _j\bar{\mu }_i$. Let $B^\partial =\operatorname {im}\partial$, $Z^\partial =\ker \partial$ and $H^\partial =Z^\partial /B^\partial$.

Lemma 3.12.

The proposition $P(p)$ is true. Hence, $E^p$ is isomorphic to $Z^\partial [\tau ]/\tau ^{p-1}B^\partial [\tau ]$ and under this isomorphism $E^p/\ker L^{p}_{\mathrm{\acute{e}t}}$ is isomorphic to $H^\partial [\tau ]$.

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

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

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

$$\begin{equation*} \vcenter{\img[][122pt][52pt][{$\xymatrix{E^p\ar[r]\ar[d]&H^\partial[\tau]\ar[d]\\ Z^\partial[\tau]/\tau^{p-1}\ar[r]&H^\partial[\tau]/\tau^{p-1}}$}]{Images/img2414fb07dffff058f06a3ff231689d82.svg}} \end{equation*}$$

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 $B^{\bar{\partial }}=\operatorname {im}\{{\bar{\partial }}\colon Q\to Q\}$, then $\operatorname {im}d^{p-1}=\tau ^{p-1}B^{\bar{\partial }}[\tau ^{p-1}]$, and if $H^{\bar{\partial }}=Z^{\bar{\partial }}/B^{\bar{\partial }}$, then (as an $\mathbb{F}_p[\tau ^{p-1}]$-module)

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

and