2.1 Some exact functors

Let A be an algebra over the ground field $\Bbbk $ of characteristic $p>0$. We equip A with the trivial (p-)differential graded structure by declaring that $d_0\equiv 0$, $\partial _0\equiv 0$ and A sits in degree zero. In this subsection, we study a functor relating the usual homotopy category $\mathcal {C}(A,d_0)$ of A with its p-DG homotopy category $\mathcal {C}(A,\partial _0)$.

To do this, recall that a chain complex of A-modules consists of a collection of A-modules and homomorphisms $d_M:M_i{\longrightarrow } M_{i-1}$ called boundary maps

satisfying $d_{M}^2=0$ for all

. A null-homotopic map is a sequence of A-module maps $f_i:M_i{\longrightarrow } N_i$,

, of A-modules, as depicted in the diagram below:

which satisfy $f_{i}=d_{N}\circ h_i+h_{i-1}\circ d_M$ for all

. The homotopy category $\mathcal {C}(A,d_0)$, by construction, is the quotient of the category of chain complexes over A by the ideal of null-homotopic morphisms.

For ease of notation, we will use bullet points to stand for a general (p)-chain complex index in what follows. Similarly, a p-chain complex of A-modules consists of a collection of A-modules and homomorphisms $\partial _M :M_i {\longrightarrow } M_{i-1}$ called p-boundary maps

satisfying $\partial _M^p \equiv 0$. A p-chain complex can also be regarded as a graded module over the tensor product algebra $A\otimes H_0$, where $H_0=\Bbbk [\partial _0]/(\partial _0^p)$ is a graded Hopf algebra, where $\mathrm {deg}(\partial _0)=-1$ and $\mathrm {deg}(A)=0$.

We introduce some special notation for some specific indecomposable p-chain complexes over $\Bbbk $, by setting

(2.1) $$ \begin{align} U_i:=H_0/(\partial_0^{i+1}) ,\quad \quad 0\leq i \leq p-1. \end{align} $$

In particular, $U_i$ has dimension $i+1$. We will also use these modules with degree shifted up by an integer , which we denote by $U_i\{a\}$. Then $U_i\{a\}$ is concentrated in degrees $a, a-1,\dots , a-i$:

A map of p-complexes $f \colon M_{\bullet } {\longrightarrow } N_{\bullet }$ of A-modules is said to be null-homotopic if there exists

$$\begin{align*}h: M_{\bullet}{\longrightarrow} N_{\bullet+p-1}\end{align*}$$

such that

(2.2) $$ \begin{align} f = \sum_{i=0}^{p-1} \partial^{i}_N \circ h \circ \partial^{p-1-i}_M. \end{align} $$

The p-homotopy category, $\mathcal {C}(A,\partial _0)$, is then the quotient of p-chain complexes of A-modules by the ideal of null-homotopic morphisms. It is a triangulated category, whose homological shift functor $[1]_{\partial }$ is defined by

(2.3) $$ \begin{align} M[1]_{\partial}:=M\otimes U_{p-2}\{p-1\} \end{align} $$

for any p-complex of A-modules. The inverse functor $[-1]_{\partial }$ is given by

(2.4) $$ \begin{align} M[-1]_{\partial}:=M\otimes U_{p-2}\{-1\}, \end{align} $$

which is a consequence of the fact that $U_{p-2}\otimes U_{p-2}$ decomposes into a direct sum of $\Bbbk \{2-p\}$ and copies of free $H_0$-modules.

Slash homology

As an analogue of the usual homology functor, we have the notion of slash homology groups [Reference Khovanov and QiKQ15] of a p-complex. To recall its definition, let us set $A=\Bbbk $. For each $0\le k \le p-2$ form the graded vector space

$$\begin{align*}\mathrm{H}^{/k}(M) = \dfrac{\mathrm{Ker}(\partial^{k+1}_M)}{\mathrm{Im} (\partial^{p-k-1}_M)+\mathrm{Ker} (\partial^{k}_M)}.\end{align*}$$

The original -grading on M gives a decomposition

The differential $\partial _M$ induces a map, also denoted $\partial _M$, which takes $\mathrm {H}_{i}^{/ k}(U)$ to $\mathrm {H}_{ i-1}^{/k-1}(U)$. Define the slash homology of M as

(2.5) $$ \begin{align} \mathrm{H}_{\bullet }^{/}(M) = \bigoplus_{k=0}^{p-2} \mathrm{H}_{\bullet }^{/k}(M). \end{align} $$

Also, let

$$\begin{align*}\mathrm{H}_{i}^/(M) := \bigoplus_{k=0}^{p-2}\mathrm{H}^{/k}_{i}(M).\end{align*}$$

We have the decompositions

(2.6)

$\mathrm {H}^/_{\bullet }(M)$ is a bigraded $\Bbbk $-vector space, equipped with an operator $\partial _M$ of bidegree $(-1,-1)$, $\partial _M{\ :\ }\mathrm {H}^{/k}_{i}{\longrightarrow } \mathrm {H}^{/k-1}_{i-1}$.

Forgetting the k-grading gives us a graded vector space $\mathrm {H}_{\bullet }^/(M)$ with differential $\partial _M$, which we can view as a graded $H_0$-module. $\mathrm {H}^/_{\bullet }(M)$ is isomorphic to M in the homotopy category of p-complexes $\mathcal {C}(\Bbbk ,\partial _0)$, and we can decompose

(2.7) $$ \begin{align} M\cong \mathrm{H}^/_{\bullet}(M)\oplus P(M) \end{align} $$

in the abelian category of H-modules, where $P(M)$ is a maximal projective direct summand of M. In particular, we have

(2.8) $$ \begin{align} \mathrm{H}^/_{\bullet}(U_i) = \begin{cases} U_i , & i=0,\dots, p-2, \\ 0 & i=p-1. \end{cases} \end{align} $$

The slash homology group $\mathrm {H}^/_{\bullet }(M)$, viewed as an $H_0$-module, does not contain any direct summand isomorphic to a free $H_0$-module.

The assignment $M\mapsto \mathrm {H}^{/}_{\bullet }(M)$ is functorial in M and can be viewed as a functor $H_0{\mathrm {\mbox {-}mod}} {\longrightarrow } \mathcal {C}(\Bbbk ,\partial _0)$ or as a functor $\mathcal {C}(\Bbbk ,\partial _0) {\longrightarrow } \mathcal {C}(\Bbbk ,\partial _0)$. The latter functor is then isomorphic to the identity functor.

As in the usual homological algebra case, we say a morphism $f:M{\longrightarrow } N$ of p-complexes of A-modules is a quasi-isomorphism if, upon taking slash homology, f induces an isomorphism $f^/: \mathrm {H}^/_{\bullet }(M) \cong \mathrm {H}^/_{\bullet }(N)$. The class of quasi-isomorphisms constitutes a localizing class in $\mathcal {C}(A,\partial _0)$ ([Reference KhovanovKho16, Proposition 4]).

Definition 2.1. The p-derived category $\mathcal {D}(A,\partial _0)$ is the localization of $\mathcal {C}(A,\partial _0)$ at the class of quasi-isomorphisms.

Alternatively, $\mathcal {D}(A,\partial _0)$ is the Verdier quotient of $\mathcal {C}(A,\partial _0)$ by the class of acyclic p-complexes, that is, those p-complexes of A-modules annihilated by the slash-homology functor.

p-Extension

We now define the p-extension functor

(2.9) $$ \begin{align} \mathcal{P}: \mathcal{C}(A,d_0){\longrightarrow} \mathcal{C}(A ,\partial_0) \end{align} $$

as follows. Given a chain complex of A-modules, we repeat every term sitting in odd homological degrees $(p-1)$ times. More explicitly, for a given complex

the p-extended complex looks like

Likewise, for a chain map

the obtained morphism of p-complexes of A-modules is given by

This is clearly a functor from the abelian category of chain complexes over A into the category of p-DG modules over $(A,\partial _0)$, (p-complexes of A-modules). Denote this functor by $\widehat {\mathcal {P}}$.

Lemma 2.2. The functor $\widehat {\mathcal {P}}$ preserves the ideal of null-homotopic morphisms.

Proof. It suffices to show that $\widehat {\mathcal {P}}$ sends null-homotopic morphisms in $\mathcal {C}(A,d_0)$ to null-homotopic morphisms in $\mathcal {C}(A,\partial _0)$. Suppose $f=dh+hd$ is a null-homotopic morphism in $\mathcal {C}(A,d_0)$. We first extend $h: M_{\bullet }{\longrightarrow } N_{\bullet +1}$ to a map

$$\begin{align*}\hat{\mathcal{P}}(h): \hat{\mathcal{P}}(M)_{\bullet}{\longrightarrow} \hat{\mathcal{P}}(N)_{\bullet+p-1}.\end{align*}$$

On unrepeated terms, $\hat {\mathcal {P}}(h)$ sends $M_{2k}$ to the copy of $N_{2k+1}$ sitting as the leftmost term in the repeated $N_{2k+1}$’s, while on the repeated terms, it only sends the rightmost $M_{2k+1}$ to the unrepeated $N_{2k+2}$ and acts by zero on the other repeated $M_{2k+1}$’s. Schematically, this has the effect as in the diagram below:

Now, it is an easy exercise to check that

(2.10) $$ \begin{align} \widehat{\mathcal{P}}(f)=\sum_{i=0}^{p-1} \partial^{p-1-i}_M \circ \widehat{\mathcal{P}}(h) \circ \partial^i_N, \end{align} $$

where $\partial _{M}$ denotes the extended p-differential on $\widehat {\mathcal {P}}(M)$, and similarly for $\partial _N$. For instance, between the rightmost repeated $M_{2k+1}$ and $N_{2k+1}$, the left-hand side of equation (2.10) equals $f_{2k+1}$, while there are only two nonzero terms contributing to the right-hand side of equation (2.10), which are equal to, respectively,

$$\begin{align*}\partial_N^{p-1}\circ \widehat{\mathcal{P}}(h)=d_{2k+2}\circ h_{2k+1}, \quad \quad \partial_N^{p-2}\circ \widehat{\mathcal{P}}(h)\circ \partial_M=h_{2k}\circ d_{2k+1}. \end{align*}$$

The sum of these two nonzero terms is precisely $f_{2k+1}$ by the null-homotopy assumption on h. One similarly checks for the other repeated and unrepeated terms, and the lemma follows.

This lemma implies that $\widehat {\mathcal {P}}$ descends to a functor

(2.11) $$ \begin{align} \mathcal{P}: \mathcal{C}(A,d_0){\longrightarrow} \mathcal{C}(A,\partial_0), \end{align} $$

which we call the p-extension functor.

Proposition 2.3. The p-extension functor $\mathcal {P}$ is exact.

Proof. It suffices to show that $\mathcal {P}$ commutes with homological shifts in both categories and sends distinguished triangles to distinguished triangles.

On $\mathcal {C}(A,d_0)$, the homological shift $[1]_d$ moves every term of a complex one step to the left, while the homological shift $[1]_{\partial }$ on $\mathcal {C}(A,\partial _0)$ is given by tensoring a p-complex of A-modules with the $(p-1)$-dimensional complex

where the underlined $\Bbbk $ sits in degree $p-1$. Note that the collection of repeated terms can be identified with

where the underlined term sits in degree $kp-1$. Using the fact that

$$\begin{align*}U_{p-2}\{p-1\}\otimes U_{p-2} \{kp-1\}\cong U_0\{kp\} \oplus F, \end{align*}$$

where F is a direct sum of graded free $H_0$-modules, we see that

$$\begin{align*}U_{p-2}\{p-1\}\otimes (M_{2k-1}\otimes U_{p-2}\{kp-1\}) \cong M_{2k-1}\{kp\} \end{align*}$$

in the homotopy category $\mathcal {C}(A,\partial _0)$. From this, it follows that $U_{p-2}\{p-1\}\otimes \mathcal {P}(M)$ is homotopy equivalent to the p-complex $\mathcal {P}(M[1]_d)$. Thus, $\mathcal {P}$ commutes with homological shifts.

To show that $\mathcal {P}$ sends distinguished triangles in $\mathcal {C}(A,d_0)$ to those in $\mathcal {C}(A,\partial _0)$, we use the characterization of distinguished triangles in $\mathcal {C}(A,d_0)$. Recall that a distinguished triangle $P \rightarrow Q \rightarrow R$ in $\mathcal {C}(A,d_0)$ is a short exact sequence $0 \rightarrow P \rightarrow Q \rightarrow R \rightarrow 0$ of complexes of A-modules which split when ignoring the differentials. After applying the p-extension functor $\mathcal {P}$ to $0 \rightarrow P \rightarrow Q \rightarrow R \rightarrow 0$, one gets a short exact sequences of p-complexes which splits when ignoring the p-differentials. This is precisely the condition that $\mathcal {P}(P) \rightarrow \mathcal {P}(Q) \rightarrow \mathcal {P}(R)$ is a distinguished triangle in $\mathcal {C}(A,\partial _0)$ (see [Reference QiQi14, Lemma 4.3]).

Totalization

Another useful functor is the totalization functor $\mathcal {T}$, which we introduce next. To do so, we will need the following result.

Lemma 2.4. Let $(K_{\bullet },\partial _K)$ be a p-complex of modules over A. Then $K_{\bullet }$ is null-homotopic if and only if there exists an A-module map $\sigma : K_{\bullet }{\longrightarrow } K_{\bullet +1}$ such that $\partial _K\sigma -\sigma \partial _K=\mathrm {Id}_K$.

Proof. By definition (see equation (2.2)), a p-complex is null-homotopic if and only if there is an A-linear map $h:K_{\bullet }{\longrightarrow } K_{\bullet +p-1}$ such that

$$\begin{align*}\mathrm{Id}_K=\sum_{i=0}^{p-1} \partial_K^{p-1-i} \circ h \circ \partial_K^i.\end{align*}$$

For any linear map $\phi $ on $K_{\bullet }$, let $\mathrm {ad}_{\partial }(\phi ):=[\partial _K,\phi ]$. Now, if $\sigma $ is as given satisfies $[\partial _K,\sigma ]=\mathrm {Id}_K$, then the map $h:=-\sigma ^{p-1}$ satisfies $[\partial _K,h]=-(p-1)\sigma ^{p-2}$, and, inductively,

$$\begin{align*}\mathrm{ad}_{\partial}^r(h)=-(p-1)\dots (p-r)\sigma^{p-r-1}.\end{align*}$$

In particular, we have

$$\begin{align*}\sum_{i=0}^{p-1} (-1)^i \binom{p-1}{i} \partial_K^{p-1-i} \circ h \circ \partial_K^i=\mathrm{ad}_{\partial}^{p-1}(h)=-(p-1)!\mathrm{Id}_K=\mathrm{Id}_K. \end{align*}$$

Since $\binom {p-1}{i}=(-1)^i$ in characteristic p, the null-homotopy formula follows.

Conversely, if h is an A-linear null-homotopy map, then $ \sigma :=\mathrm {ad}_{\partial }^{p-2}(h) $ satisfies $[\partial _K, \sigma ]=\mathrm {Id}_K$. As the iterated commutator of A-linear maps, $\sigma $ is also A-linear. The lemma follows.

Let $(A,\partial _A)$ be a p-DG algebra, where $\partial _A$ has degree two.Footnote ¹ We regard A as an algebra object in the graded module category of the graded Hopf algebra $H_q=\Bbbk [\partial _q]/(\partial _q^p)$ in which $\mathrm {deg}(\partial _q)=2$. The smash product algebra $A\# H_q$ is then the graded algebra $A\otimes H_q$ containing the subalgebras $A\otimes 1$ and $1\otimes H_q$ and subject to the commutation relations

$$\begin{align*}(1\otimes \partial_q)(a\otimes 1)=a\otimes \partial_q + \partial_A(a)\otimes 1 \end{align*}$$

for any $a\in A$. Graded modules over $A\# H_q$ are also called p-DG modules over A, the collection of which will be denoted $(A,\partial _A){\mathrm {\mbox {-}mod}}$.

In analogy with the Hopf algebra $H_0$, we introduce the indecomposable balanced $H_q$-complexes

(2.12)

for $i=0, \dots , p-1$.

As a matter of notation, we will denote the q-grading shifted copy of $V_i$ by $q^a V_i$, where the lowest degree term sits in degree $a-i$. Furthermore, if M is any p-DG module over A, we will denote by $q^a M$ the p-DG module whose underlying module is the same as M but grading shifted up by .

The p-DG homotopy category $\mathcal {C}(A,\partial _A)$ can be defined, similarly as for $\mathcal {C}(A,\partial _0)$ before, by taking the quotient of the abelian category of p-DG modules by the ideal of null-homotopic morphisms, which consists of homogeneous $A\# H_q$-module maps $f:(M,\partial _M){\longrightarrow } (N,\partial _N)$ of the form

(2.13) $$ \begin{align} f = \sum_{i=0}^{p-1} \partial_N^i \circ h \circ \partial_M^{p-1-i}, \end{align} $$

where h is an A-linear homomorphism from $(M,\partial _M)$ to $(N,\partial _N)$ of degree $2-2p$.

A p-DG homomorphism $f:(M,\partial _M){\longrightarrow } (N,\partial _N)$ is called a quasi-isomorphism if, again, f induces an isomorphism of slash homology with respect to the p-differentials on M and N. Inverting quasi-isomorphisms in $\mathcal {C}(A,\partial _A)$ results in the p-DG derived category $\mathcal {D}(A,\partial _A)$.

Equip $A\# H_q$ with the zero p-differential $\partial _0$, and $\partial _0$ carries an additional -grading that is independent of the original grading on A and $H_q$. A p-complex of graded $A\# H_q$-modules thus has a -grading, where $\partial _q$ has degree $(2,0)$ and $\partial _0$ has degree $(0,-1)$. Consider the functor

(2.14) $$ \begin{align} \widehat{\mathcal{T}}: (A\# H_q,\partial_0){\mathrm{\mbox{-}mod}} {\longrightarrow} (A,\partial_A){\mathrm{\mbox{-}mod}} \end{align} $$

defined as follows. To a p-complex M of $A\# H_q$-modules

where each term $M_i$ is a graded A-module together with an internal p-differential $\partial _i$, compatible with $\partial _A$, we assign to it the singly graded A-module whose new p-DG structure is given by

$$\begin{align*}\partial_T(m):=\partial_M(m)+\partial_i(m)\in q^{-2i+2}M_{i-1}\oplus q^{-2i}M_i \end{align*}$$

if $m\in q^{-2i}M_i$.

Lemma 2.5. The functor $\widehat {\mathcal {T}}$ descends to a triangulated functor on the p-DG homotopy categories:

$$ \begin{align*} \mathcal{T}: \mathcal{C}(A\# H_q, \partial_0) {\longrightarrow} \mathcal{C}(A,\partial_A) . \end{align*} $$

Proof. If $Q_{\bullet }$ is a null-homotopic p-complex of $A\# H_q$-modules, then, by Lemma 2.4, there exist an $A\# H_q$-linear $\sigma : Q_{\bullet } {\longrightarrow } Q_{\bullet + 1}$ such that

$$\begin{align*}[\partial_Q, \sigma]= \mathrm{Id}_{Q}. \end{align*}$$

Since h commutes with the $H_q$-actions, we have

$$\begin{align*}[\partial_Q+\partial_q, \sigma]=\mathrm{Id}_{Q}. \end{align*}$$

It follows that $\mathcal {T}(Q_{\bullet })$ is null-homotopic, and the functor $\mathcal {T}$ is well defined on the p-homotopy categories. It is then an easy exercise to verify that $\mathcal {T}$ preserves the triangulated structures on both sides.

2.2 Grothendieck rings

We will be considering the Grothendieck rings of the homotopy categories $\mathcal {C}(\Bbbk ,\partial _0)$ and $\mathcal {C}(\Bbbk ,\partial _q)$.

Lemma 2.6. The Grothendieck rings of the tensor triangulated categories $\mathcal {C}(\Bbbk ,\partial _0)$ and $\mathcal {C}(\Bbbk ,\partial _q)$ are respectively isomorphic to

We will often abbreviate the Grothendieck rings by

(2.15a)

(2.15b)

In this paper, we will be working with (finite-dimensional) a and q bigraded complexes over $\Bbbk $ equipped with commuting differentials $\partial _0$ and $\partial _q$. On this category, one may consider the composition of slash-homology functors, first in the a-direction and then in the q-direction:

(2.16) $$ \begin{align} H_q\otimes H_0{\mathrm{\mbox{-}mod}} \xrightarrow{\mathrm{H}^{/}_{\bullet}~\textrm{in} \ a\textrm{-direction}} \mathcal{C}(H_q,\partial_0) \xrightarrow{\mathrm{H}^{/}_{\bullet}~\textrm{in} q\textrm{-direction}} \mathcal{C}(\mathcal{C}(\Bbbk,\partial_q),\partial_0). \end{align} $$

Here, the last category stands for the homotopy category with object in $\mathcal {C}(\Bbbk ,\partial _q)$. As usual with taking homology of the usual bicomplexes, these functors do not commute, and their order matters in the construction.

Corollary 2.8. The categories $\mathcal {C}(H_q,\partial _0)$ and $\mathcal {C}(\mathcal {C}(\Bbbk ,\partial _q),\partial _0)$ have Grothendieck rings isomorphic to

Proof. The bigraded abelian category $H_q\otimes H_0{\mathrm {\mbox {-}mod}}$ has its Grothendieck ring isomorphic to . An object lying in the first slash homology functor has Euler characteristic in the ideal

Similarly, a module lying inside the kernel of the composition functor has Euler characteristic in the ideal

The result follows.

2.3 p-Hochschild homology and cohomology

Now, we come to the construction of the p-DG simplicial bar complex of Mayer [Reference MayerMay42a, Reference MayerMay42b] (see also [Reference Kassel and WambstKW98]). The usual simplicial bar complex of a unital, associative algebra A is the complex:

(2.17a) $$ \begin{align} \cdots \xrightarrow{d_{n+1}} A^{\otimes(n+2)} \stackrel{d_n}{{\longrightarrow}} A^{\otimes (n+1)}\xrightarrow{d_{n-1}} \cdots \stackrel{d_2}{{\longrightarrow}} A^{\otimes 3}\stackrel{d_1}{{\longrightarrow}} A^{\otimes 2} {\longrightarrow} 0, \end{align} $$

where

(2.17b) $$ \begin{align} d_i(a_0\otimes a_1\otimes\cdots \otimes a_{i+1})=\sum_{k=0}^{i+1} (-1)^i a_0\otimes \cdots a_{k-1}\otimes a_ka_{k+1}\otimes a_{k+2} \otimes \cdots \otimes a_{i+1}. \end{align} $$

The bar complex is a free bimodule resolution of A, as the augmented complex

(2.18) $$ \begin{align} \cdots \xrightarrow{d_{n+1}} A^{\otimes(n+2)} \stackrel{d_n}{{\longrightarrow}} A^{\otimes (n+1)}\xrightarrow{d_{n-1}} \cdots \stackrel{d_2}{{\longrightarrow}} A^{\otimes 3}\stackrel{d_1}{{\longrightarrow}} A^{\otimes 2} {\longrightarrow} A {\longrightarrow} 0, \end{align} $$

is acyclic. This can be seen by constructing a left A-module map

(2.19) $$ \begin{align} \sigma: A^{\otimes n}{\longrightarrow} A^{\otimes (n+1)},\quad x\mapsto x \otimes 1 \end{align} $$

as the null-homotopy.

Let A be a $\Bbbk $-algebra. In analogy with the usual simplicial bar complex, Mayer introduced on the usual augmented bar complex (2.17a) the linear map

$$\begin{align*}\partial_H(a_0\otimes a_1\otimes\cdots \otimes a_{i+1}):=\sum_{k=0}^{i+1} a_0\otimes \cdots a_{k-1}\otimes a_ka_{k+1}\otimes a_{k+2} \otimes \cdots \otimes a_{i+1}. \end{align*}$$

Then it is an easy exercise to show that $\partial ^p\equiv 0$. Furthermore, the null-homotopy map $\sigma $ in equation (2.19) clearly satisfies

$$\begin{align*}\partial_H \sigma -\sigma \partial_H = \mathrm{Id}. \end{align*}$$

It follows that the augmented p-complex

(2.20) $$ \begin{align} (\mathbf{p}_{\bullet}^{\prime} (A),\partial_H):=\left(\cdots \xrightarrow{\partial_{H}} A^{\otimes(n+2)} \stackrel{\partial_H}{{\longrightarrow}} A^{\otimes (n+1)}\xrightarrow{\partial_{H}} \cdots \stackrel{\partial_H}{{\longrightarrow}} A^{\otimes 3}\stackrel{\partial_H}{{\longrightarrow}} A^{\otimes 2} {\longrightarrow} A {\longrightarrow} 0\right) \end{align} $$

is acyclic.

Assume next that $(A,\partial _A)$ is a p-DG algebra. Extend the p-differential on A to any $A^{\otimes (n+1)}$ by the Leibniz rule so that

$$\begin{align*}\partial_A(a_0\otimes a_1\otimes\cdots \otimes a_{n}):=\sum_{i=0}^{n} a_0\otimes \cdots \otimes \partial_A(a_i) \otimes \cdots \otimes a_{n}. \end{align*}$$

As the multiplication map $m:A\otimes \mathcal {\longrightarrow } A$ commutes with $\partial _A$, it follows that the boundary maps in $(\mathbf {p}_{\bullet }^{\prime } (A), \partial _M)$ commute with the internal differentials $\partial _A$ on each $A^{\otimes n}$. We may thus consider the total complex $(\mathbf {p}_{\bullet }^{\prime } (A),\partial _H+\partial _A)$. This construction is equivalent to the totalization $\mathcal {T}(\mathbf {p}_{\bullet }^{\prime } (A),\partial _H)$. To make a distinction, we will denote the total differential on $\mathbf {p}_{\bullet }^{\prime } (A)$ by $\partial _T:=\partial _H+\partial _A$ in order to avoid potential confusion with the other differentials $\partial _H$ and $\partial _A$.

There is a natural inclusion map

$$\begin{align*}\iota_A: A{\longrightarrow} \mathbf{p}_{\bullet}^{\prime} (A) \end{align*}$$

of p-DG bimodules over A, whose cokernel is the p-DG bimodule

$$\begin{align*}(\widetilde{\mathbf{p}}_{\bullet} (A), \partial_T):= \left(\cdots \xrightarrow{\partial_{H}} A^{\otimes(n+2)} \stackrel{\partial_H}{{\longrightarrow}} A^{\otimes (n+1)}\xrightarrow{\partial_{H}} \cdots \stackrel{\partial_H}{{\longrightarrow}} A^{\otimes 3}\stackrel{\partial_H}{{\longrightarrow}} A^{\otimes 2} {\longrightarrow} 0\right). \end{align*}$$

Recall here that each $A^{\otimes n}$ also carries its internal differential $\partial _A$.

Proof. In order to show that $\mathbf {p}^{\prime }_{\bullet }(A)$ is acyclic, it suffices to check, by Lemma 2.4, that

$$\begin{align*}(\partial_H+\partial_A)\sigma-\sigma(\partial_H+\partial_A)=\mathrm{Id}_{\mathbf{p}^{\prime}}, \end{align*}$$

where $\mathrm {Id}_{\mathbf {p}^{\prime }}$ is the identity map of $\mathbf {p}^{\prime }_{\bullet }(A)$. This is clear since we have the easily verified commutator relations

$$\begin{align*}[\partial_H,\sigma]=\mathrm{Id}_{\mathbf{p}^{\prime}},\quad [\partial_A,\sigma]=0, \quad [\partial_H,\partial_A]=0. \end{align*}$$

The last statement is similar, as one just needs to replace the last copy of A in $A^{\otimes n}$ by Q.

Definition 2.10. Suppose $(M,\partial _M)$ is a left p-DG module over A. Set $M[-1]$ to be the tensor product of M with the $(p-1)$-complex $q^{p}V_{p-2}$ (see equation (2.12)). The simplicial bar resolution for M is the p-DG module

$$\begin{align*}\mathbf{p}_{\bullet}(M):=\widetilde{\mathbf{p}}_{\bullet}(A)\otimes_A M[-1]. \end{align*}$$

It inherits the p-differential from that of $\partial _T$ and $\partial _M$ via the Leibniz rule.

Likewise, one defines the simplicial bar resolution for right p-DG modules.

Proof. To see the cofibrance of $\mathbf {p}_{\bullet }(M)$, first note that

$$\begin{align*}\mathbf{p}_{\bullet}(M)=\left(\cdots {\longrightarrow} A^{\otimes 3}\otimes M[-1] {\longrightarrow} A^{\otimes 2}\otimes M[-1] {\longrightarrow} A\otimes M[-1] {\longrightarrow} 0\right), \end{align*}$$

which carries p-differentials on the arrows and internal differentials within each term. It has a natural filtration by p-DG submodules, whose subquotients have the form

$$\begin{align*}A^{\otimes n}\otimes M[-1], \quad (n \geq 1) . \end{align*}$$

As left p-DG modules over A, such modules are clearly direct sums of free p-DG A-modules. Therefore $\mathbf {p}_{\bullet }(M)$ satisfies the ‘Property P’ criterion of [Reference QiQi14, Definition 6.3] and is thus cofibrant.

By construction, there is a short exact sequence of p-DG modules over A

$$\begin{align*}0 {\longrightarrow} M[-1] {\longrightarrow} \mathbf{p}^{\prime}_{\bullet} (A)\otimes_A M[-1] {\longrightarrow} \mathbf{p}_{\bullet} (M){\longrightarrow} 0 . \end{align*}$$

Since $\mathbf {p}_{\bullet }(M)$ is projective as a left A-module, the sequence splits when forgetting about p-differentials. By [Reference QiQi14, Lemma 4.3], the short exact sequence above gives rise to a distinguished triangle in the homotopy category

$$\begin{align*}M[-1] {\longrightarrow} \mathbf{p}^{\prime}_{\bullet} (A)\otimes_A M[-1] {\longrightarrow} \mathbf{p}_{\bullet} (M)\stackrel{[1]}{{\longrightarrow}} M . \end{align*}$$

Therefore, there is a morphism $f: \mathbf {p}_{\bullet }(M){\longrightarrow } M$ representing the $[1]$ map on the last arrow. By Proposition 2.9, f is a quasi-isomorphism.

Using the bar resolution, we recall the derived tensor product functor construction in the p-DG setting.

Definition 2.12. Let M be a left p-DG module and N be a right p-DG module over A. The p-DG derived tensor product of N and M is the object in $H_q{\mathrm {-\underline {mod}}}$

$$\begin{align*}N\otimes^{\mathbf{L}}_A M:= N\otimes_A \mathbf{p}_{\bullet}(M). \end{align*}$$

As in [Reference QiQi14, Corollary 8.9], the functor is well defined. Furthermore, it is readily seen that it is independent of cofibrant replacements one chooses for M (or N).

We define the analogue of Hochschild homology in the p-DG setting. As a shorthand notation, for an algebra A, we will denote by $A^{\mathrm {en}}:=A\otimes A^{\mathrm {op}}$ the enveloping algebra of A. Thus, an $(A,A)$-bimodule is synonymous with a left module over $A^{\mathrm {en}}$.

Definition 2.14. Let A be a p-DG algebra, and M be a bimodule over A. Then the p-DG Hochschild (co)homology is the p-complex

$$\begin{align*}p\mathrm{HH}_{\bullet}(M):=\mathrm{H}^{/}_{\bullet}(\mathbf{p}_{\bullet}(A)\otimes_{A^{\mathrm{en}}} M) \quad \quad (\textrm{resp.}~ p\mathrm{HH}^{\bullet}(M):=\mathrm{H}^{/}_{\bullet} (\mathrm{HOM}_{A^{\mathrm{en}}}(\mathbf{p}_{\bullet}(A),M)) . \end{align*}$$

The p-DG Hochschild homology is a functorial ‘categorical trace’ on the category of p-DG bimodules over a p-DG algebra, similar to the usual Hochschild homology functor. Here, the functoriality means that, a morphism of p-DG bimodules $f: M{\longrightarrow } N$ over A induces a morphism of p-Hochschild homology groups, which is defined by

(2.21) $$ \begin{align} p\mathrm{HH}_{\bullet}(f):= \mathrm{H}^/_{\bullet} ( \mathrm{Id}_{\mathbf{p}_{\bullet}(A)}\otimes f) : \mathrm{H}^/_{\bullet}(\mathbf{p}_{\bullet}(A)\otimes_{A^{\mathrm{en}}} M) {\longrightarrow} \mathrm{H}^/_{\bullet}(\mathbf{p}_{\bullet}(A)\otimes_{A^{\mathrm{en}}} N). \end{align} $$

Theorem 2.15. Given two p-DG bimodules M and N over A, there is an isomorphism of p-complexes

$$\begin{align*}p\mathrm{HH}_{\bullet}(M\otimes^{\mathbf{L}}_A N)\cong p\mathrm{HH}_{\bullet}(N\otimes^{\mathbf{L}}_A M). \end{align*}$$

Proof. This is more or less parallel to the classical Hochschild homology case. For this, we notice that by Definition 2.12,

$$\begin{align*}M\otimes^{\mathbf{L}}_A N = M\otimes_A \mathbf{p}_{\bullet}(A) \otimes_A N . \end{align*}$$

Then by Definition 2.14, one takes another tensor product over $A^{\mathrm {en}}$ with $\mathbf {p}_{\bullet }(A)$ with respect to the left A-action on M and right A-action on N. This is best visualized as putting everything on a circle:

Here, the connecting lines joining the p-DG modules in the picture stand for the usual tensor product over A.

Rotating the picture by $180^{\circ }$, one obtains the p-complex computing the p-Hochschild homology for the bimodule $N\otimes ^{\mathbf {L}}_A M$. The result follows.

2.4 Relative homotopy categories

For any ungraded algebra B over $\Bbbk $, denote by $d_0$ the zero ordinary differential and by $\partial _0$ the zero p-differential on B, while letting B sit in homological degree zero. When B is graded, the homological grading is independent of the internal grading of B.

Suppose $(A,\partial _A)$ is a p-DG algebra. There is an exact forgetful functor between the usual homotopy categories of chain complexes of graded $A\# H$-modules

$$\begin{align*}\mathcal{F}_d: \mathcal{C}(A\# H_q,d_0){\longrightarrow} \mathcal{C}(A,d_0). \end{align*}$$

An object $K_{\bullet }$ in $\mathcal {C}(A\# H_q,d_0)$ lies inside the kernel of the functor if and only if, when forgetting the $H_q$-module structure on each term of $K_{\bullet }$, the complex of graded A modules $\mathcal {F}_d(K_{\bullet })$ is null-homotopic. The null-homotopy map on $\mathcal {F}_d(K_{\bullet })$, though, is not required to intertwine $H_q$-actions.

Likewise, there is an exact forgetful functor

$$\begin{align*}\mathcal{F}_{\partial}: \mathcal{C}(A\# H_q,\partial_0){\longrightarrow} \mathcal{C}(A,\partial_0). \end{align*}$$

Similarly, an object $K_{\bullet }$ in $\mathcal {C}(A\# H_q,\partial _0)$ lies inside the kernel of the functor if and only if, when forgetting the $H_q$-module structure on each term of $K_{\bullet }$, the p-complex of A modules $\mathcal {F}_{\partial }(K_{\bullet })$ is null-homotopic. The null-homotopy map on $\mathcal {F}_{\partial }(K_{\bullet })$, though, is not required to intertwine $H_q$-actions.

Definition 2.16. Given a p-DG algebra $(A,\partial _A)$, the relative homotopy category is the Verdier quotient

$$\begin{align*}\mathcal{C}^{\partial_q}(A,d_0):=\dfrac{\mathcal{C}(A\# H_q,d_0)}{\mathrm{Ker}(\mathcal{F}_d)}.\end{align*}$$

Likewise, the relative p-homotopy category is the Verdier quotient

$$\begin{align*}\mathcal{C}^{\partial_q}(A,\partial_0):=\dfrac{\mathcal{C}(A\# H_q,\partial_0)}{\mathrm{Ker}(\mathcal{F}_{\partial})}.\end{align*}$$

The subscripts in the definitions are to remind the reader of the $H_q$-module structures on the objects.

The categories $\mathcal {C}^{\partial _q}(A,d_0)$ and $\mathcal {C}^{\partial _q}(A,\partial _0)$ are triangulated. By construction, there is a factorization of the forgetful functor

Let us briefly remark on the triangulated structures of the relative homotopy categories $\mathcal {C}^{\partial _q}(A,d_0)$ and $\mathcal {C}^{\partial _q}(A,\partial _0)$. By construction the shift functors $[\pm 1]$ are inherited from those of $\mathcal {C}(A\# H_q,d_0)$ and $\mathcal {C}(A\# H_q,\partial _0)$. In the first case, the functor $[\pm 1]_d$ just shifts complexes one step to the left or right. In the second case, the functor $[1]_{\partial }$ is given by tensoring with the p-complex $U_{p-2}\{p-1\}$ and $[-1]_{\partial }$ by tensoring with $U_{p-2}\{-1\}$ (see equations (2.3) and (2.4)).

For the usual homotopy category $\mathcal {C}(A,d_0)$ of an algebra A, standard distinguished triangles arise from short exact sequences

$$\begin{align*}0 {\longrightarrow} M_{\bullet} \stackrel{f}{{\longrightarrow}} N_{\bullet} \stackrel{g}{{\longrightarrow}} L_{\bullet} {\longrightarrow} 0 \end{align*}$$

of complexes of A-modules that are termwise split exact. The class of distinguished triangles in $\mathcal {C}(A,d_0)$ are declared to be those that are isomorphic to standard ones. Similarly, termwise split short exact sequences of p-complexes of A-modules lead to standard distinguished triangles in $\mathcal {C}(A,\partial _0)$ ([Reference QiQi14, Lemma 4.3]). For distinguished triangles in the relative homotopy categories, we have the following construction.

Proof. We will only show the first statement. The proof of the second one is entirely similar.

By construction, distinguished triangles are those in $\mathcal {C}^{\partial _q}(A,d_0)$ that are isomorphic to standard distinguished triangles arising from short exact sequences of $A\# H_q$-modules that are termwise $A\#H_q$-split exact. Forgetting about the $H_q$-actions, such sequences are also termwise A-split exact.

Now, let $f: M_{\bullet } {\longrightarrow } N_{\bullet } $ be the injection as in the statement. The cone of f in $\mathcal {C}(A\# H_q, d_0)$ is given by

$$\begin{align*}C_{\bullet} (f) \cong \left( M_{\bullet} [1]_d \oplus N_{\bullet} , d_{C}:= \begin{pmatrix} d_{M[1]_d} & f \\ 0 & d_N \end{pmatrix} \right). \end{align*}$$

The cone fits into a short exact sequence of $A\#H_q$-modules that are termwise $A\# H_q $-split:

$$\begin{align*}0{\longrightarrow} N_{\bullet} {\longrightarrow} C_{\bullet} (f){\longrightarrow} M_{\bullet} [1]_d{\longrightarrow} 0. \end{align*}$$

Associated with this sequence is the (rotated) standard distinguished triangle

$$\begin{align*}N_{\bullet} {\longrightarrow} C_{\bullet} (f) {\longrightarrow} M_{\bullet} [1]_d {\longrightarrow} N_{\bullet} [1]_d \end{align*}$$

in $\mathcal {C}(A\# H_q, d_0)$, which descends to a standard distinguished triangle in $\mathcal {C}^{\partial _q}(A,d_0)$.

To prove the statement, it then suffices to show that, in the relative homotopy category, we have an isomorphism $C_{\bullet } (f) \cong L_{\bullet } $. Consider the map

$$\begin{align*}g^{\prime} : C_{\bullet} (f) = (M_{\bullet} [1]_d \oplus N_{\bullet} , d_{C}) {\longrightarrow} L_{\bullet} , \quad (m,n)\mapsto g(n). \end{align*}$$

It is easily checked that $g^{\prime }$ is a surjective map of chain complexes, and the kernel is isomorphic to $C_{\bullet } (\mathrm {Id}_M)$. Thus, we have a short exact sequence of chain complexes of $A\# H_q$-modules

$$\begin{align*}0{\longrightarrow} C_{\bullet} (\mathrm{Id}_M) {\longrightarrow} C_{\bullet} (f) \stackrel{g^{\prime}}{{\longrightarrow}} L_{\bullet} {\longrightarrow} 0. \end{align*}$$

Now, under $\mathcal {F}_d$, the sequence termwise splits over A:

$$\begin{align*}\mathcal{F}_d (C_{\bullet} (f)) \cong \left( \mathcal{F}_d(M_{\bullet} [1]_d\xrightarrow{\mathrm{Id}_M} M_{\bullet} ) \oplus \mathcal{F}_d(L_{\bullet} ) \right) \cong \left(\mathcal{F}_d(C_{\bullet} (\mathrm{Id}_M))\oplus \mathcal{F}_d(L_{\bullet} )\right). \end{align*}$$

It follows that we have a distinguished triangle in $\mathcal {C}(A,d_0)$

$$\begin{align*}0\cong \mathcal{F}_d (C_{\bullet} (\mathrm{Id}_M)) {\longrightarrow} \mathcal{F}_d (C_{\bullet} (f)) \xrightarrow{\mathcal{F}_d(g^{\prime})} \mathcal{F}_d (L_{\bullet} ) {\longrightarrow} \mathcal{F}_d (C_{\bullet} (\mathrm{Id}_M)[1]_d)\cong 0, \end{align*}$$

implying that $g^{\prime }$ is an isomorphism under $\mathcal {F}_d$. The result follows.

We also record the following useful fact.

Proposition 2.18. The p-extension functor $\mathcal {P}: \mathcal {C}(A\# H_q, d_0){\longrightarrow } \mathcal {C}(A\# H_q, \partial _0)$ descends to a functor, still denoted by $\mathcal {P}$, between the relative homotopy categories:

$$\begin{align*}\mathcal{P}: \mathcal{C}^{\partial_q} (A, d_0){\longrightarrow} \mathcal{C}^{\partial_q}(A, \partial_0) .\end{align*}$$

2.5 Relative p-Hochschild homology

In this paper, instead of the absolute version of p-Hochschild homology, we will need a relative version of p-Hochshild homology for a p-DG algebra, which we define now. An important reason for introducing the relative homotopy category is that the relative p-Hochschild homology functor descends to this category.

Let $(A,\partial _A)$ be a p-DG algebra. Equip the zero differential $d_0$ and p-differential $\partial _0$ on A, and denote the resulting trivial (p)-DG algebras by $(A_0,d_0)$ and $(A_0,\partial _0)$. Likewise, for a (p-)DG bimodule M over A, we temporarily denote by $M_0$ the A-bimodule equipped with zero (p-) differentials.

The usual Hochschild homology of $M_0$ over $(A_0,d_0)$ in this case carries a natural $H_q$-action since the $H_q$-action commutes with all differentials in the simplicial bar complex (2.17) for $A_0$.

Definition 2.19. The relative Hochschild homology of a p-DG bimodule $(M,\partial _M)$ over $(A,\partial _A)$ is the usual Hochschild homology of $M_0$ over $(A_0,d_0)$ equipped with the induced $H_q$-action from $\partial _M$ and $\partial _A$, and denoted

$$\begin{align*}\mathrm{HH}^{\partial_q}_{\bullet}(M):=\mathrm{HH}_{\bullet}(A_0,M_0). \end{align*}$$

Replacing the usual simplicial bar complex by Mayer’s p-simplicial bar complex $\mathbf {p}_{\bullet }(A_0)$ for $(A_0,\partial _0)$, we make the following definition.

Definition 2.20. The relative p-Hochschild homology of M is the p-complex of

$$\begin{align*}p\mathrm{HH}^{\partial_q}_{\bullet} (M):=\mathrm{H}^/_{\bullet}(A_0 \otimes_{A_0^{\mathrm{en}}}^{\mathbf{L}} M_0)=\mathrm{H}^/_{\bullet}(\mathbf{p}(A_0) \otimes_{A_0^{\mathrm{en}}} M_0) . \end{align*}$$

Similar to p-Hochschild homology, the relative case is also covariant functor: If $f: M {\longrightarrow } N$ is a morphism of p-DG bimodules over A, it induces

$$\begin{align*}p\mathrm{HH}_{\bullet}^{\partial_q}( f ):=\mathrm{H}^/_{\bullet}(\mathrm{Id}_{A_0}\otimes f): \mathrm{H}^/_{\bullet}(A_0 \otimes_{A_0^{\mathrm{en}}}^{\mathbf{L}} M_0) {\longrightarrow} \mathrm{H}^/_{\bullet}(A_0 \otimes_{A_0^{\mathrm{en}}}^{\mathbf{L}} N_0) . \end{align*}$$

We also have the trace-like property for relative p-Hochschild homology.

Proposition 2.22. Given two p-DG bimodules M and N over A, there is an isomorphism of p-complexes of $H_q$-modules

$$\begin{align*}p\mathrm{HH}^{\partial_q}_{\bullet}(M\otimes^{\mathbf{L}}_A N)\cong p\mathrm{HH}^{\partial_q}_{\bullet}(N\otimes^{\mathbf{L}}_A M). \end{align*}$$

Our next goal is to show that we may relax the requirement that we utilize the simplicial bar resolution when computing the relative Hochschild homology. For the next theorem, we use the fact that in the simplicial bar complex $\mathbf {p}_{\bullet }(A_0)$, all the p-complex maps are $H_q$-equivariant since they are just sums of multiplications maps of A tensored with identities maps on A.

Theorem 2.23. Let M be a p-DG bimodule over A. Suppose $f:Q_{\bullet } {\longrightarrow } M$ is a p-complex resolution of M over $(A_0,\partial _0)$ which is $H_q$-equivariant, and each term of $Q_{\bullet }$ is projective as an $A_0^{\mathrm {en}}$-module. Then f induces an isomorphism of $H_q$-modules

$$\begin{align*}\mathrm{H}^{/}_{\bullet}(A_0\otimes_{A_0^{\mathrm{en}}}Q_{\bullet})\cong p\mathrm{HH}^{\partial_q}_{\bullet}(M). \end{align*}$$

Proof. By definition, there is a short exact sequence over $(A_0,\partial _0)$:

$$\begin{align*}0{\longrightarrow} M {\longrightarrow} C_{\bullet}(f){\longrightarrow} Q_{\bullet}[1]{\longrightarrow} 0. \end{align*}$$

The cone $C_{\bullet }(f)$, by construction, is equal to

$$\begin{align*}C_{\bullet}(f) \cong \dfrac{Q_{\bullet} \otimes \Bbbk[\partial_0]/(\partial_0^p)\oplus M}{ \left\{ (x\otimes \partial_0^{p-1}, f(x))|x\in Q_{\bullet} \right\}} , \end{align*}$$

which is then an acyclic p-complex of bimodules over $(A,\partial _0)$. Since the $H_q$-actions on the modules $Q_{\bullet }$ and M commute with the $\partial _0$ action, the above short exact sequence is an exact sequence of $H_q$-modules.

Taking the tensor product of $\mathbf {p}_{\bullet }(A_0)$ with the sequence, we get a short exact sequence since $\mathbf {p}_{\bullet }(A_0)$ is projective as a module over $A_0^{\mathrm {en}}$

$$\begin{align*}0{\longrightarrow} \mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} M {\longrightarrow} \mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} C_{\bullet}(f){\longrightarrow} \mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} Q_{\bullet}[1]{\longrightarrow} 0 . \end{align*}$$

The middle term $\mathbf {p}_{\bullet }(A_0)\otimes _{A_0^{\mathrm {en}}} C_{\bullet }(f)$ is a p-complex of $A_0$-bimodules equipped with a filtration, whose subquotients are isomorophic to grading shifts of $C_{\bullet }(f)$. Therefore, it is acyclic. It follows that we have an isomorphism in $\mathcal {D}((A_0^{\mathrm {en}}\# H_q,\partial _0)$

$$\begin{align*}\mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} Q_{\bullet}\cong \mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} M . \end{align*}$$

Taking p-Hochschild homology, we obtain an isomorphism of $H_q$-modules

$$\begin{align*}\mathrm{H}^/_{\bullet}(\mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} Q_{\bullet}) \cong \mathrm{H}^/_{\bullet}(\mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} M). \end{align*}$$

Similarly, since $Q_{\bullet }$ is projective as a bimodule over $A_0$, tensoring Mayer’s short exact sequence of bimodules with $Q_{\bullet }$ over $A_0^{\mathrm {en}}$ remains exact:

$$\begin{align*}0{\longrightarrow} A_0 \otimes_{A_0^{\mathrm{en}}} Q_{\bullet} {\longrightarrow} \mathbf{p}^{\prime}_{\bullet} (A_0)\otimes_{A_0^{\mathrm{en}}} Q_{\bullet}{\longrightarrow} \mathbf{p}_{\bullet}(A_0)[1]\otimes_{A_0^{\mathrm{en}}} Q_{\bullet} {\longrightarrow} 0. \end{align*}$$

The middle term is, as above, acyclic since $\mathbf {p}_{\bullet }^{\prime }(A_0)$ is. It follows that

$$\begin{align*}A_0 \otimes_{A_0^{\mathrm{en}}} Q_{\bullet} \cong \mathbf{p}_{\bullet}(A_0)\otimes_{A_0^{\mathrm{en}}} Q_{\bullet} \end{align*}$$

as objects in $\mathcal {D}(H_q,\partial _0)$. Taking slash homology on both sides gives the desired result.

3.1 p-DG bimodules over the polynomial algebra

Let $\Bbbk $ be a field of characteristicFootnote ³ $p>2$. The graded polynomial algebra $R_n=\Bbbk [x_1,\ldots ,x_n]$ has a natural module-algebra structure over the graded Hopf algebra $H_q=\Bbbk [\partial _q]/(\partial _q^p)$, where $\partial _q \in H_q$ acts on the generators by $\partial _q(x_i)=x_i^2$ for $i=1,\ldots ,n$ and is extended to the full algebra by the Leibniz rule $\partial _q(fg)=f\partial _q(g)+f\partial _q(g)$ for any $f,g\in R_n$. In particular $\partial _q(1)=0$. Here, the degree of each $x_i$ and $\partial _q$ are both two, and will be referred to as the q-degree.

The differential $\partial _q$ is invariant under the permutation action of the symmetric group $S_n$ on the indices of the variables. Therefore, let the subalgebra of polynomials symmetric in variables $x_i$ and $x_{i+1}$ with its inherited $H_q$-module structure be denoted by

$$\begin{align*}R^i_n=\Bbbk[x_1,\ldots,x_{i-1},x_i+x_{i+1},x_i x_{i+1},x_{i+2},\ldots,x_n]. \end{align*}$$

More generally, given any subgroup $G\subset S_n$, the invariant subalgebra $R_n^G$ inherits an $H_q$-algebra structure from $R_n$ (and is thus a p-DG algebra). In particular, we will also use the $H_q$-subalgebra $ R^{i,i+1}_n:= R^{S_3}_n $, where $S_3$ is the subgroup generated by permuting the indices i, $i+1$ and $i+2$.

When the number of variables n is clear from the context or is irrelevant of the statements, we will abbreviate $R_n$ by just R in what follows.

The $(R,R)$-bimodule $B_i=R \otimes _{R^i} R$ has the structure of an $H_q$-module (and is thus a p-DG bimodule) where the differential acts via the Leibniz rule: for any $h\otimes g\in R\otimes _{R^i} R$,

$$\begin{align*}\partial_q(h \otimes g)=\partial_q(h) \otimes g+ h \otimes \partial_q(g). \end{align*}$$

The tensor category of $(R,R)$-bimodules generated by the $B_i$, has an $H_q$-module structure where the action comes from the comultiplication in $H_q$. We denote this category by $(R,R) \# H_q {\mathrm {\mbox {-}mod}}$.

Let be a linear function. We twist the $H_q$-action on the bimodule $B_i$ to obtain a bimodule $B_i^f$ defined as follows. As an $(R_n,R_n)$-bimodule, it is the same as $B_i$, but the action of $H_q$ is twisted by defining

(3.1a) $$ \begin{align} \partial_q(1 \otimes 1)=(1 \otimes 1)f. \end{align} $$

Similarly we define ${}^f B_i$ where now

(3.1b) $$ \begin{align} \partial_q(1\otimes1)=f(1 \otimes 1). \end{align} $$

For $R_n$ as a bimodule over itself, it is clear that $^fR_n \cong R_n^f$ as p-DG bimodules. It follows that there are $p^n$ ways to put an $H_q$-module structure on a rank-one free module over $R_n$. Each such $H_q$-module is quasi-isomorphic to a finite-dimensional p-complex. Choose numbers $b_i\in \{1,\dots ,p \}$ such that $b_i\equiv a_i~(\mathrm {mod}~p)$, $i=1,\dots , n$, and define the $H_q$-ideal of R

(3.2) $$ \begin{align} I=(x_1^{p+1-b_1},\cdots, x_{n}^{p+1-b_n}). \end{align} $$

Then the natural quotient map

(3.3) $$ \begin{align} \pi:R_n^f \twoheadrightarrow R_n^f/(I\cdot R_n^f) \end{align} $$

is readily seen to be a quasi-isomorphism.

Lemma 3.1. For each $f=\sum _{i=1}^n a_ix_i$, the rank-one p-DG module $R_n^f$ has finite-dimensional slash homology:

$$\begin{align*}\mathrm{H}^/_{\bullet}(R_n^f)\cong \bigotimes_{i=1}^n V_{p-b_i} \{p-b_i\}. \end{align*}$$

In particular, if any $a_i$ of $f=\sum _i a_ix_i$ is equal to one, then $\mathrm {H}^/_{\bullet }(R_n^f)=0$.

By Definition 2.16, a morphism $f \colon A \longrightarrow B$ in the homotopy category $\mathcal {C}((R,R) \# H_q,d_0)$ is a relative isomorphism if $\mathcal {F}_d(f)$ is an isomorphism in the homotopy category $\mathcal {C}((R,R),d_0)$. Localizing $\mathcal {C}((R,R) \# H_q,d_0)$ at all relative isomorphisms produces the relative homotopy category of $(R,R)$-bimodules $\mathcal {C}^{\partial _q}(R,R,d_0)$. Similarly we have the p-version of the relative homotopy category $\mathcal {C}^{\partial _q}(R,R,\partial _0)$.

3.2 Elementary braiding complexes

Proof. The fact that these maps are $(R,R)$-bimodule homomorphisms is well known. See, for instance, [Reference Elias and KhovanovEK10] for more details.

In order to check that the homomorphism is compatible with the $H_q$-module structure, we must check that $rb_i(\partial _q(1))=\partial _q(rb_i(1))$. Clearly, $rb_i(\partial _q(1))=0$. On the other hand

$$ \begin{align*} \partial_q(rb_i(1))&=\partial_q(x_{i+1} \otimes 1 - 1 \otimes x_{i}) \\ &=x_{i+1}^2 \otimes 1 - 1 \otimes x_{i}^2 +(x_{i+1} \otimes 1 - 1 \otimes x_{i}) (-x_i - x_{i+1}) \\ &=0, \end{align*} $$

where the third term in the second equation above comes from the twist on $B_i$.

The second homomorphism clearly respects the $H_q$-structure since $\partial _q(1 \otimes 1)=0=\partial _q(1)$.

We now have complexes of $(R,R) \# H_q$-modules

(3.4) $$ \begin{align} T_i := \left(t B_i \xrightarrow{br_i} R\right) , \quad \quad \quad T_i' := \left(R \xrightarrow{rb_i} q^{-2} t^{-1} B_i^{-(x_i+x_{i+1})}\right). \end{align} $$

In the coming sections we will, for presentation reasons, often omit the various shifts built into the definitions of $T_i$ and $T_i'$.

We associate respectively to the left and right crossings $\sigma _i$ and $\sigma _i^{\prime }$ between the ith and $(i+1)$st strands in equation (3.5) the chain complexes of $(R,R)\#H_q$-bimodules $T_i$ and $T_i'$:

(3.5)

More generally, if $\beta \in \mathrm {Br}_n$ is a braid group element written as a product $\sigma _{i_i}^{\epsilon _1}\cdots \sigma _{i_k}^{\epsilon _k}$ in the elementary generators, where $\epsilon _i\in \{\emptyset , \prime \}$, we assign the chain complex of $(R,R)\# H_q$-bimodules

(3.6) $$ \begin{align} T_{\beta}:=T_{i_1}^{\epsilon_1}\otimes_R\cdots \otimes_R T_{i_k}^{\epsilon_k}. \end{align} $$

The proof of the theorem, which is like the analogous proof in [Reference Khovanov and RozanskyKR16], will take up the rest of this section. We will repeatedly apply Proposition 2.17 to simplify complexes in the relative homotopy category.

3.3 Reidemeister II

The following lemma is crucial to proving the Reidemeister II braid relation and should be compared with [Reference Khovanov and RozanskyKR16, Lemma 4.3].

Lemma 3.5. There exists an isomorphism of $(R,R) \#H_q$-modules

$$\begin{align*}B_i \otimes_R B_i \cong B_i \oplus q^2 B_i^{x_i+x_{i+1}} \end{align*}$$

defined by

$$\begin{align*}\phi=\begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} \colon B_i \otimes_R B_i \longrightarrow B_i \oplus q^2 B_i^{x_i+x_{i+1}}, \quad \phi_1(1 \otimes 1 \otimes 1)= \begin{pmatrix} 1 \otimes 1 \\ 0 \end{pmatrix}, \quad \phi_2( 1 \otimes (x_{i+1}-x_{i}) \otimes 1 ) = \begin{pmatrix} 0 \\ 1 \otimes 1 \end{pmatrix}, \end{align*}$$

$$\begin{align*}\psi=(\psi_1, \psi_2) \colon B_i \oplus q^2 B_i^{x_i+x_{i+1}} \longrightarrow B_i \otimes_R B_i , \quad \psi_1(1 \otimes 1)=1 \otimes 1 \otimes 1, \quad \psi_2(1 \otimes 1)=1 \otimes (x_{i+1}-x_i) \otimes 1. \end{align*}$$

Proposition 3.7. There are isomorphisms in the homotopy category $\mathcal {C}((R,R) \# H_q, d_0)$

$$\begin{align*}T_i \otimes_R T_i' \cong \mathrm{Id} \cong T_i' \otimes_R T_i. \end{align*}$$

Proof. In order to prove the isomorphism $T_i' \otimes _R T_i\cong \mathrm {Id}$, we tensor the complexes for $T_i'$ and $T_i$ to obtain

(3.7)

By Lemma 3.5, the complex (3.7) is isomorphic to

(3.8)

where

$$ \begin{align*} \gamma_1(1 \otimes 1)= \frac{1}{2}(x_i-x_{i+1}) \otimes 1, \quad \gamma_2(1 \otimes 1)= -\frac{1}{2}(1 \otimes 1), \quad \delta_1(1 \otimes 1)=1 \otimes 1, \quad \delta_2(1 \otimes 1)=1 \otimes (x_{i+1}-x_i). \end{align*} $$

Contracting out the terms $B_i$, one gets that equation (3.8) is homotopic to

(3.9)

Finally, contracting out the terms $B^{-(x_i+x_{i+1})}$, one gets $T^{\prime }_i \otimes _R T_i \cong R$. Note that each homotopy $\gamma $ used in the contractions has the property $\partial (\gamma )=0$.

3.4 Reidemeister III

Let $B_{i,i+1,i}=R \otimes _{R^{i,i+1}}R$, where

$$ \begin{align*} R^{i,i+1}=\Bbbk[x_1,\ldots,x_{i-1},e_1(x_i,x_{i+1},x_{i+2}), e_2(x_i,x_{i+1},x_{i+2}),e_3(x_i,x_{i+1},x_{i+2}),x_{i+3},\ldots,x_n]. \end{align*} $$

The following lemma is a rephrasing of [Reference Khovanov and RozanskyKR16, Lemma 4.7]. We will use the map $\phi _2$ from the statement of Lemma 3.5.

Lemma 3.8. There exists a short exact sequence of $(R,R) \# H_q$-modules which splits upon restricting to the category of $(R,R)$-bimodules

where

$$ \begin{align*} f_1(1 \otimes 1)=1 \otimes 1 \otimes 1, \quad \quad \quad f_2=\phi_2 \circ (\mathrm{Id} \otimes br_{i+1} \otimes \mathrm{Id}). \end{align*} $$

Proof. We will prove the first isomorphism. The second follows from the first part and Proposition 3.7.

By definition

We will use the notation $f_1$ and $f_2$ defined in Lemma 3.8 and $\phi _1, \phi _2, \psi _1,$ and $\psi _2$ defined in Lemma 3.5. There is a short exact sequence of $(R,R) \# H_q$-modules which splits when forgetting the $H_q$-actions involved:

$$ \begin{align*} 0 \longrightarrow E_1 \longrightarrow T_i T_{i+1} T_i \longrightarrow E_2 \longrightarrow 0 \end{align*} $$

which we write vertically in diagram (3.10) below. Most of the maps in the complexes below transform a $B_i$ or $B_{i+1}$ into an R (via $br_i$ or $br_{i+1}$) and act as the identity on the other tensor factors. We use the following shorthand notation. A ‘$+$’ indicates that the coefficient of such a map is $1$, and a ‘$-$’ indicates that the coefficient of such a map is $-1$. For example, the negative sign below stands for

$$ \begin{align*} - := - br_i \otimes \mathrm{Id} \otimes \mathrm{Id} \colon B_i \otimes R \otimes B_i \longrightarrow R \otimes R \otimes B_i. \end{align*} $$

(3.10)

All of the chain maps are annihilated by $\partial _q$ and clearly $E_2$ is homotopically equivalent to $0$. Thus, $T_i T_{i+1} T_i$ is isomorphic to $E_1$ in the relative homotopy category.

There is an isomorphism of complexes $E_1 \cong E_3$, via a basis change respecting the $H_q$-structures, given by the diagram

There is a short exact sequence of complexes of $(R,R)$-bimodules compatible with the $H_q$-structure which splits when forgetting the $H_q$-structure

Since $E_4$ is contractible, $T_i T_{i+1} T_i \cong E_5$. In the same exact way, one deduces that $T_{i+1} T_i T_{i+1} \cong E_5$, which proves the proposition.

4.1 Triply graded theory

In this section, we categorify the HOMFLYPT polynomial of any link using analogous arguments from [Reference CautisCau17], [Reference Robert and WagnerRW20] and [Reference RouquierRou17] adapted to the p-DG setting.

For the next definition, we will allow complexes of Soergel bimodules to sit in half-integer degrees in the Hochschild (a) and topological (t) degrees when considering the usual complexes of vector spaces. We then modify the elementary braiding complexes of equation (3.4) to be

(4.1) $$ \begin{align} T_i := (at)^{-\frac{1}{2}}q^{-2} \left(t B_i \xrightarrow{br_i} R\right) , \quad \quad \quad T_i' := (at)^{\frac{1}{2}} q^{2} \left(R \xrightarrow{rb_i} q^{-2} t^{-1} B_i^{-(x_i+x_{i+1})}\right). \end{align} $$

Let $\beta \in \mathrm {Br}_n$ be a braid group element on n strands. By Theorem 3.4, there is a chain complex of $(R_n,R_n)\# H_q$-bimodules $T_{\beta }$, well defined up to homotopy, associated with $\beta $. Then we write

(4.2) $$ \begin{align} T_{\beta} = \left(\dots\stackrel{d_0}{{\longrightarrow}} T_{\beta}^{i+1} \stackrel{d_0}{{\longrightarrow}} T_{\beta}^{i} \stackrel{d_0}{{\longrightarrow}} T_{\beta}^{i-1}\stackrel{d_0}{{\longrightarrow}}\dots\right). \end{align} $$

Definition 4.1. The untwisted $H_q$-HOMFLYPT homology of $\beta $ is the object

$$\begin{align*}\widehat{\mathrm{HHH}}^{\partial_q}(\beta):=a^{-\frac{n}{2}}t^{\frac{n}{2}}\mathrm{H}_{\bullet} \left(\dots {\longrightarrow} \mathrm{HH}_{\bullet}^{\partial_q}(T_{\beta}^{i+1}) \xrightarrow{d_t} \mathrm{HH}_{\bullet}^{\partial_q}(T_{\beta}^{i}) \xrightarrow{d_t} \mathrm{HH}_{\bullet}^{\partial_q}(T_{\beta}^{i-1}){\longrightarrow} \dots\right) \end{align*}$$

in the category of triply graded $H_q$-modules, where $d_t:=\mathrm {HH}_{\bullet }^{\partial _q}(d_0)$ is the induced map of $d_0$ on Hochschild homology.

By construction, the space $\widehat {\mathrm {HHH}}^{\partial _q}(\beta )$ is triply graded by the topological (t) degree, the Hochschild (a) degree as well as the quantum (q) degree. When necessary to emphasize each graded homogeneous piece of the space, we will write $\widehat {\mathrm {HHH}}^{\partial _q}_{i,j,k}(\beta )$ to denote the homogeneous component concentrated in t-degree i, a-degree j and q-degree k.

The following theorem is a particular case of the main result of [Reference Khovanov and RozanskyKR16], where we have only kept track of the degree two p-nilpotent differential in finite characteristic p. The detailed verification given below, however, uses the main ideas of [Reference RouquierRou17] and differs from that of [Reference Khovanov and RozanskyKR16]. This proof serves as the model for the other link homology theories in this paper.

As a convention for the framing number of braid closure, if a strand for a component of link is altered as in the left of equation (4.3), then we say that the framing of the component is increased by $1$ (with respect to the blackboard framing). If a strand for a component of link is altered as in the right of equation (4.3), then we say that the framing of the component is decreased by $1$.

(4.3)

Denote by $\mathtt {f}_i(L)$ the framing number of the ith strand of a link L. Then, under the two Reidemeister moves of equation (4.3), $\mathtt {f}_i(L)$ adds or subtracts $1$ respectively when changing from the corresponding left local picture to the right local picture.

Our main goal in this section is to establish this result. Due to Theorem 3.4, the proof reduces to showing the invariance under the two Markov moves.

4.2 Doubly graded theory

We next seek to define the analogue of $\widehat {\mathrm {HHH}}$ in the category of p-complexes. This construction will serve as a precursor to the finite-dimensional $\mathfrak {sl}_2$-homology theory defined in the next section.

We first would like to define $pT_{\beta }$ to be the p-complex of Soergel bimodules associated with $\beta $ by

(4.4) $$ \begin{align} pT_{\beta} :=\mathcal{P}(T_{\beta}). \end{align} $$

In other words, $pT_{\beta }$ should be a p-complex of the form

$$\begin{align*}pT_{\beta} = \left( \cdots {\longrightarrow} T_{\beta}^{2k} \xrightarrow{d_0} T_{\beta}^{2k-1} =\cdots = T_{\beta}^{2k-1} \xrightarrow{d_0} T_{\beta}^{2k-2} {\longrightarrow} \cdots \right), \end{align*}$$

where every term in odd topological (the same as homological) degree is repeated $p-1$ times. We will denote the boundary maps in the p-extended complex $pT_{\beta }$ by $\partial _0$, in contrast to the usual topological differential $d_0$.

Remark 4.3 (Half grading shifts for p-complexes).

Here, we point out that, unlike in the ordinary homotopy category of complexes, we do not need to formally introduce half grading shifts in $\mathcal {C}(\Bbbk ,\partial )$ in odd prime characteristic. For instance, one may set

(4.5) $$ \begin{align} a^{[\frac{1}{2}]}:=a^{\frac{p+1}{2}}[-1]_{\partial}^a , \quad \quad t^{[\frac{1}{2}]}:=t^{\frac{p+1}{2}}[-1]_{\partial}^t. \end{align} $$

Using $[-1]^a_{\partial }\circ [-1]^a_{\partial }=[-2]^{a}_{\partial }=a^{-p}$, one sees that, as functors, $a^{[\frac {1}{2}]}\circ a^{[\frac {1}{2}]} = a$. Likewise, $t^{[\frac {1}{2}]}\circ t^{[\frac {1}{2}]}=t$.

The same half grading shift functors can also be interpreted as

(4.6) $$ \begin{align} a^{[\frac{1}{2}]}:=a^{\frac{1-p}{2}}[1]_{\partial}^a , \quad \quad t^{[\frac{1}{2}]}:=t^{\frac{1-p}{2}}[1]_{\partial}^t. \end{align} $$

The two seemingly different definitions actually agree, as both are given by taking tensor product with the p-complex $U_{p-2}\{\frac {p-1}{2}\}$ in the a or t direction.

However, the p-extension functor $\mathcal {P}$ does not intertwine between the ordinary half graded complexes and p-complexes, since, if we were to set $\mathcal {P}(a^{\frac {1}{2}})=a^{[\frac {1}{2}]}$, then we would have

$$\begin{align*}\mathcal{P}([1]_d^a)=\mathcal{P}(a)=\mathcal{P}(a^{\frac{1}{2}} \circ a^{\frac{1}{2}})=a^{[\frac{1}{2}]} \circ a^{[\frac{1}{2}]}=a \neq [1]^a_{\partial}. \end{align*}$$

Thus, we do not naively p-extend the braiding complexes (4.1) via $\mathcal {P}$.

We emphasize that the proof of invariance under both Markov II moves of our theory forces us to collapse the a and t gradings in this construction. The resulting homology theory will be doubly graded. More specifically, let us first collapse the a and t gradings in $\mathrm {HHH}$ into a single grading satisfying $a=q^2t$, then we will categorically specialize $t=[1]^t_d$ into $[1]^t_{\partial }$ by p-extension. We then lose the a-grading below, which is determined by the t-grading and q-grading. The t-degree remains independent of the q-degree on Soergel bimodules.

We then define

(4.7) $$ \begin{align} pT_i := q^{-3} \left(B_i \xrightarrow{br_i} R[-1]^t_{\partial}\right) , \quad \quad \quad pT_i' := q^{3} \left(R[1]^t_{\partial} \xrightarrow{rb_i} q^{-2} B_i^{-(x_i+x_{i+1})}\right). \end{align} $$

Since $pT_i$ and $pT_i^{\prime }$ are, up to t-grading shifts, obtained by applying $\mathcal {P}$ to $T_i$ and $T_i^{\prime }$, it follows that they satisfy the same braid relations and can be used to define $pT_{\beta }$ similarly as done in Theorem 3.4.

Definition 4.4. Let $\beta \in \mathrm {Br}_n$ be a braid group element written as a product $\sigma _{i_i}^{\epsilon _1}\cdots \sigma _{i_k}^{\epsilon _k}$ in the elementary generators, where $\epsilon _i\in \{\emptyset , \prime \}$. We assign to $\beta $ the p-chain complex of $(R_n,R_n)\# H_q$-bimodules

(4.8) $$ \begin{align} pT_{\beta}:=pT_{i_1}^{\epsilon_1}\otimes_R\cdots \otimes_R pT_{i_k}^{\epsilon_k}. \end{align} $$

The boundary maps in the t-direction will be denoted $\partial _0$ for any $pT_{\beta }$.

Definition 4.5. The untwisted $H_q$-HOMFLYPT p-homology of $\beta $ is the object

$$\begin{align*}p\widehat{\mathrm{HHH}}^{\partial_q}(\beta):= q^{-n} \mathrm{H}^/_{\bullet} \left(\dots {\longrightarrow} p\mathrm{HH}_{\bullet}^{\partial_q}(pT_{\beta}^{i+1}) \xrightarrow{\partial_t} p\mathrm{HH}_{\bullet}^{\partial_q}(pT_{\beta}^{i}) \xrightarrow{\partial_t} p\mathrm{HH}_{\bullet}^{\partial_q}(pT_{\beta}^{i-1}){\longrightarrow} \dots\right) \end{align*}$$

in the homotopy category of bigraded $H_q$-modules. Here, $\partial _t$ stands for the induced map of the topological differentials on p-Hochschild homology groups $\partial _t:=p\mathrm {HH}_{\bullet }^{\partial _q}(\partial _0)$.

In the definition of the $H_q$-HOMFLYPT p-homology, we have applied the p-extensions in both the topological and the Hochschild directions so that they can be collapsed into a single degree. The reason will become clearer later when categorifying the Jones polynomial at roots of unity. Therefore, in contrast to $\widehat {\mathrm {HHH}}(\beta )$, $p\widehat {\mathrm {HHH}}(\beta )$ is only doubly graded, and we will adopt the notation $p\widehat {\mathrm {HHH}}_{i,j}(\beta )$ as above to stand for its homogeneous components in topological degree i and q degree j. Further, the overall grading shift in the definition will be utilized in the invariance under the Markov II moves below.

The proof of Theorems 4.2 and 4.6 will occupy the next few subsections after we introduce the $H_q$-equivariant (p-)Koszul resolutions.

4.3 Koszul complexes

We recall here the Koszul resolution $C_n$ of the polynomial algebra $R_n$ as well as its adaption in the p-DG setting. In the one variable case, we have a short exact sequence of $\Bbbk [x]$-bimodules:

(4.9) $$ \begin{align} 0 {\longrightarrow} q^2\Bbbk[x]\otimes \Bbbk[x] \xrightarrow{x\otimes 1-1\otimes x}\Bbbk[x]\otimes \Bbbk[x] \stackrel{m}{{\longrightarrow}} \Bbbk[x]{\longrightarrow} 0. \end{align} $$

In order to make the maps $H_q$-equivariant, we twist the $H_q$-action on the leftmost bimodule:

(4.10) $$ \begin{align} 0 {\longrightarrow} q^2\Bbbk[x]^x\otimes \Bbbk[x]^x \xrightarrow{x\otimes 1-1\otimes x}\Bbbk[x]\otimes \Bbbk[x] \stackrel{m}{{\longrightarrow}} \Bbbk[x]{\longrightarrow} 0. \end{align} $$

In the usual homotopy category of $(\Bbbk [x], \Bbbk [x])\# H_q$-modules, we have then an $H_q$-equivariant relative replacement of $\Bbbk [x]$

(4.11) $$ \begin{align} 0{\longrightarrow} q^2a\Bbbk[x]^x\otimes \Bbbk[x]^x \xrightarrow{x\otimes 1-1\otimes x}\Bbbk[x]\otimes \Bbbk[x]{\longrightarrow} 0, \end{align} $$

where we have inserted a in the leftmost nonzero term to emphasize the homological degree that it sits in. Denote by $(C_1,d)$ this $H_q$-equivariant Koszul resolution in the one-variable case.

In the p-homotopy category, a relative replacement is then obtained by applying the p-extension functor $\mathcal {P}$ to $C_1$:

(4.12) $$ \begin{align} 0{\longrightarrow} q^2a^{p-1}\Bbbk[x]^x\otimes \Bbbk[x]^x= \dots = q^2a\Bbbk[x]^x\otimes \Bbbk[x]^x \xrightarrow{x\otimes 1-1\otimes x}\Bbbk[x]\otimes \Bbbk[x]{\longrightarrow} 0. \end{align} $$

We abbreviate this p-resolution by $pC_1$.

For the ease of notation, it will be useful to denote both replacements by

$$\begin{align*}0{\longrightarrow} q^2\Bbbk[x]^x\otimes \Bbbk[x]^x [1]^a_{*}\xrightarrow{x\otimes 1-1\otimes x}\Bbbk[x]\otimes \Bbbk[x]{\longrightarrow} 0, \end{align*}$$

where $*\in \{d, \partial \}$ will label $[1]$ as either the usual homological or p-homological shift in a-degree.

For the polynomial ring $R_n=\Bbbk [x_1,\dots , x_n]$, one takes the n-fold tensor product over $\Bbbk $ of the bimodule resolution $C_1$ of $\Bbbk [x_1]$ to get $(C_n=C_1^{\otimes n},d)$. We also define

(4.13) $$ \begin{align} (pC_n,\partial):=pC_1\otimes_{\Bbbk} \cdots \otimes_{\Bbbk} pC_1 \end{align} $$

as the n-fold tensor product of the one-variable resolution. Note that $pC_n$ is homotopic to, but bigger than, the p-complex of bimodules $\mathcal {P}(C_n)$.

Example 4.7. In characteristic $3$, the $3$-complex $3C_2$ is the total complex of the cube

Under the total p-differential, the copy of $\Bbbk [x,y]^{x+y}\otimes \Bbbk [x,y]^{x+y}$ sitting in the southwest corner generates an acyclic $3$-subcomplex, modulo which one obtains the total $3$-complex of $\mathcal {P}(C_2)$:

Using this Koszul resolution, one immediately obtains, via the following result, that the relative p-Hochschild homology is completely determined by the classical relative Hochschild homology. Therefore, it may appear that one does not gain any more information by introducing $p\mathrm {HHH}_{\bullet }^{\partial }$. However, this construction is essential for the collapsed homology theory to be defined in the next section.

Proposition 4.8. Let M be an $(R_n,R_n) \#H_q$-bimodule. Then $p\mathrm {HH}_{\bullet }^{\partial _q}(M)$ is determined by $\mathrm {HH}_{\bullet }^{\partial _q}(M)$ by

(4.14) $$ \begin{align} p\mathrm{HH}^{\partial_q}_i (M) = \begin{cases} \mathrm{HH}_{2k-1}^{\partial_q}(M) & (k-1)p+1 \leq i \leq kp-1\\ \mathrm{HH}_{2k}^{\partial_q}(M) & i=kp. \end{cases} \end{align} $$

Proof. Using the Koszul resolution, we haveFootnote ⁴

$$\begin{align*}p\mathrm{HH}^{\partial}_{\bullet}(M) = \mathrm{H}^/_{\bullet}(M\otimes_{R_n^{\mathrm{en}}} \mathcal{P}(C_n))= \mathrm{H}^/_{\bullet}(\mathcal{P}(M\otimes_{R_n^{\mathrm{en}}} C_n))=\mathcal{P}(\mathrm{H}_{\bullet} (M\otimes_{R_n^{\mathrm{en}}}C_n)). \end{align*}$$

The result follows.

Remark 4.9. The result is true in more generality. If A is an (ungraded) algebra equipped with the zero p-differential, then its p-Hochschild homology is entirely determined by its usual Hochschild homology. This result is essentially due to Spanier [Reference SpanierSpa49] but is also proved in more generality by [Reference Kassel and WambstKW98].

Indeed, if $P_{\bullet } {\longrightarrow } A$ is any projective resolution of A over $A\otimes A^{\mathrm {op}}$, then $\mathcal {P}(P_{\bullet })$ provides a p-resolution of A, and

(4.15) $$ \begin{align} p\mathrm{HH}_{\bullet}(A)=\mathrm{H}^/_{\bullet} (\mathcal{P}(P_{\bullet})\otimes_{A\otimes A^{\mathrm{op}}}A)=\mathrm{H}^/_{\bullet}(\mathcal{P}(P_{\bullet}\otimes_{A\otimes A^{\mathrm{op}}}A)). \end{align} $$

When computing the last slash homology, one may safely forget about the module structures involved in $\mathcal {P}(P_{\bullet }\otimes _{A\otimes A^{\mathrm {op}}}A)$ and think of it as a direct sum, possibly infinite copies, of chain-complexes of the form

(4.16) $$ \begin{align} 0{\longrightarrow} \underline{\Bbbk} {\longrightarrow} 0,\quad \quad \quad 0{\longrightarrow} \underline{\Bbbk}{\longrightarrow} \Bbbk {\longrightarrow} 0, \end{align} $$

where the underlined term sits in some homological degree i. Under the p-extension functor, the last complex extends to a contractible p-complex, while the first complex becomes

$$\begin{align*}0{\longrightarrow} \Bbbk {\longrightarrow} 0, \end{align*}$$

if i is even, or the $(p-1)$-dimensional

$$\begin{align*}0{\longrightarrow} \Bbbk=\cdots = \Bbbk{\longrightarrow} 0 \end{align*}$$

if i is odd. The slash homology computation then follows.

4.4 Markov I

The usual HOMFLYPT homologies of two braid compositions $\beta _1\beta _2$ and $\beta _2\beta _1$ are isomorphic due to the trace-like property of the usual Hochschild homology functor. The relative Hochschild homology also remembers the $H_q$-action.

The same property also holds for the HOMFLYPT $pH_q$-homology groups.

Proof. This follows from Proposition 2.22, since we have the functorial isomorphism

$$\begin{align*}p\mathrm{HH}^{\partial_q}_{\bullet}(pT_{\beta_1}^i\otimes_{R_n} pT_{\beta_2}^i)\cong p\mathrm{HH}^{\partial_q}_{\bullet}(pT_{\beta_2}^i\otimes_{R_n} pT_{\beta_1}^i) \end{align*}$$

for all . Alternatively, this follows from combining the previous result with Proposition 4.8.

4.5 Markov II

In order to prove the second Markov move, one needs to show that for a (complex of) Soergel bimodules M over the polynomial p-DG algebra $R_n$, that HOMFLYPT (p)$H_q$-homologies of the bimodules (4.17) are isomorphic (up to shifts and twists).

(4.17)

Let $\Lambda \langle x_{n+1} \rangle $ be the exterior algebra in the variable $x_{n+1}$. Recall that $R_n=\Bbbk [x_1,\ldots ,x_n]$, and let $M \in (R_n,R_n) \#H_q {\mathrm {\mbox {-}mod}}$. Set $C_1^{\prime }=\Bbbk [x_{n+1}] \otimes \Lambda \langle x_{n+1} \rangle \otimes \Bbbk [x_{n+1}] \cong C_1$. Letting $C_n$ denote the Koszul resolution of $R_n$, we have inductively that $C_{n+1}=C_n \otimes C_1^{\prime }$.

As in the proof of Theorem 2.15, the Hochschild homology of M is depicted by the closure diagram

where the single strands connecting the boxes indicate tensor products over the one-variable polynomial rings labelling those strands.

The proof of second Markov move essentially reduces to a computation of the partial Hochschild homology with respect to the last variable $x_{n+1}$. This operation is diagramatically represented in equation (4.18).

(4.18)

This leads to an analysis of $C_1^{\prime } \otimes _{\Bbbk [x_{n+1}]^{\mathrm {en}}} T_n$ in Proposition 4.12.

The following technical result will be the heart of establishing the invariance under the Markov II moves. We start with the usual HOMFLYPT homology case under the Hopf algebra $H_q$-action.

Proof. Both identities are proved in a similar way. For the first statement, we note that by definition

$$ \begin{align*} \mathrm{HH}^{\partial_q}_{\bullet}((M\otimes \Bbbk[x_{n+1}])\otimes_{R_{n+1}} T_n) = \mathrm{H}_{\bullet}^v(C_{n+1} \otimes_{R_{n+1}^{\mathrm{en}}} ((M \otimes \Bbbk[x_{n+1}])\otimes_{R_{n+1}} T_n)), \end{align*} $$

where the (vertical) homology $\mathrm {H}_{\bullet }^v$ above is taken with respect to the differential coming from the Koszul complex $C_{n+1}$.

Note that

(4.19) $$ \begin{align} C_{n+1} \otimes_{R_{n+1}^{\mathrm{en}}} ((M \otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) &= (C_n \otimes C_1^{\prime}) \otimes_{R_{n+1}^{\mathrm{en}}} ((M \otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) \nonumber \\ & \cong C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} (C_1^{\prime} \otimes_{\Bbbk[x_{n+1}]^{\mathrm{en}}} T_n)). \end{align} $$

These isomorphisms, in terms of diagrammatics, can be interpreted as taking closures of the following diagrammatic equalities:

Observe that $C_1^{\prime } \otimes _{\Bbbk [x_{n+1}]^{\mathrm {en}}} T_n$ is a bicomplex of $(R_{n+1},R_{n+1})$-bimodules

Here, the grading shift conventions follow from equation (3.4). For ease of notation, we will mostly ignore them within this proof below.

It follows that there is a short exact sequence of bicomplexes of $(R_{n+1},R_{n+1})$-bimodules

$$ \begin{align*} 0 \longrightarrow Y_1 \longrightarrow C_1^{\prime} \otimes_{\Bbbk[x_{n+1}]^{\mathrm{en}}} T_n \longrightarrow Y_2 \longrightarrow 0, \end{align*} $$

where the terms of the sequence are defined by

(4.20)

Here, $\widetilde {R}_{n+1}$ is equal to $R_{n+1}$ as a left $R_{n+1}$-module, but the right action of $R_{n+1}$ is twisted by the permutation $\sigma _n \in S_{n+1}$ and $\tilde {br}(a \otimes b)=br(a \sigma _n(b))$. It is a straightforward exercise to check that all maps above are equivariant with respect to the $H_q$-action. We show it for the map

$$ \begin{align*} \phi := (x_{n+1}-x_n) \otimes 1 + 1 \otimes (x_{n+1}-x_n) \colon R_{n+1}^{x_n+3x_{n+1}} \longrightarrow {}^{x_{n+1}}B_n^{x_{n+1}}. \end{align*} $$

One calculates

$$ \begin{align*} \phi(\partial_q(1)) & = \phi(x_n+x_{n+1}+2x_{n+1}) \\[6pt] & = (x_{n+1}^2-x_n^2) \otimes 1 + 1 \otimes (x_{n+1}^2-x_n^2) +2x_{n+1}(x_{n+1} -x_n) \otimes 1 +2x_{n+1} \otimes (x_{n+1}-x_n), \end{align*} $$

and

$$ \begin{align*} \partial_q(\phi(1)) = & ~ \partial((x_{n+1}-x_n)\otimes 1 + 1\otimes (x_{n+1}-x_n)) \\[6pt] = & ~(x_{n+1}^2-x_n^2) \otimes 1 + 1 \otimes (x_{n+1}^2-x_n^2) + x_{n+1}(x_{n+1}-x_n) \otimes 1 + \\[6pt] & ~(x_{n+1}-x_n) \otimes x_{n+1} + x_{n+1} \otimes (x_{n+1}-x_n) + 1 \otimes (x_{n+1}-x_n)x_{n+1}. \end{align*} $$

Comparing the terms of $\phi (\partial _q(1))$ and $\partial _q(\phi (1))$ it suffices to check the identity inside the bimodule $B_n$

$$ \begin{align*} x_{n+1}^2 \otimes 1 - x_{n+1} \otimes x_n = 1 \otimes x_{n+1}^2 - x_n \otimes x_{n+1}. \end{align*} $$

Rearranging the terms of the above equation, we must show

(4.21) $$ \begin{align} x_{n+1}^2 \otimes 1 + x_{n} \otimes x_{n+1} = 1 \otimes x_{n+1}^2 + x_{n+1} \otimes x_{n}. \end{align} $$

Adding $x_n x_{n+1} \otimes 1$ to both sides of equation (4.21) and using the fact that symmetric functions in $x_n$ and $x_{n+1}$ may be brought through a tensor product finishes the proof that $\phi (\partial _q(1))=\partial _q(\phi (1))$.

There is a splitting of the short exact sequence (4.20) regarded as a short exact sequence of $(R_n,R_n)$-bimodules, given by

(4.22)

where

$$ \begin{align*} \theta(f(x_1,\ldots,x_{n-1})x_n^i x_{n+1}^j) =f(x_1,\ldots,x_{n-1}) x_n^i \otimes x_{n}^j. \end{align*} $$

We briefly explain why $\theta $ is a well-defined bimodule homomorphism. By definition $\theta (x_n^i x_{n+1}^j)= x_n^i \otimes x_{n}^j$. Note that $\theta (x_n^i x_{n+1}^j)=\theta (x_n^i \cdot x_{n+1}^j)=x_n^i \theta (x_{n+1}^j)=x_n^i \otimes x_{n}^j $, where we viewed $x_n^i$ as acting on the left of $x_{n+1}^j$. Similarly, $\theta (x_n^i x_{n+1}^j)=\theta (x_n^i \cdot x_{n}^j)=\theta (x_n^i)x_n^j= x_n^i \otimes x_{n}^j $, where we viewed $x_n^j$ as acting on the right of $x_{n}^i$.

The short exact sequence (4.20) plugged back into equation (4.19) gives us a short exact sequence

(4.23) $$ \begin{align} 0\rightarrow C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1) \rightarrow C_n \otimes_{R_n^{\mathrm{en}}}(M \otimes \Bbbk[x_{n+1}])\otimes_{R_{n+1}} T_n \rightarrow C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_2) \rightarrow 0, \end{align} $$

which is split as a sequence of $(R_n,R_n)$-bimodules. Taking homology with respect to the vertical differentials gives rise to a long exact sequence

(4.24)

$$ \begin{align*} \cdots \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1)) \rightarrow \mathrm{HH}_i^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_2)) \rightarrow \cdots. \end{align*} $$

Due to the splitting exactness of equation (4.23), the long exact sequence breaks up into short exact sequences of the form

(4.25) $$ \begin{align} 0 \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1)) \rightarrow \mathrm{HH}_i^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_2)) \rightarrow 0, \end{align} $$

one for each .

So far we have ignored the topological (horizontal) differential in diagram (4.20). Each term in the short exact sequence (4.25) carries the topological differential $d_t$. Taking homology with respect to $d_t$ gives us another long exact sequence

(4.26)

We claim that $\mathrm {H}^h_j(\mathrm {H}_i^v(C_n \otimes _{R_n^{\mathrm {en}}} (M \otimes _{R_n} Y_2)))=0$ for all j, (which we will show shortly), which implies that

$$ \begin{align*} \mathrm{H}^h_j\mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1)) \cong \widehat{\mathrm{HHH}}_{j,i}^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n). \end{align*} $$

The definition (4.20) of $Y_1$ shows that $ \mathrm {H}^h_j\mathrm {H}_i^v(C_n \otimes _{R_n^{\mathrm {en}}} (M \otimes _{R_n} Y_1)) $ is the homology of the two-term complex

Taking grading shifts back into account, this complex has homology concentrated in the second nonzero term, which is isomorphic to

$$ \begin{align*} \mathrm{H}^h_j\mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} M \otimes_{R_n} R_{n}^{2x_n}) \cong \widehat{\mathrm{HHH}}_{j,i}^{\partial_q}(M)^{2x_n}, \end{align*} $$

where the latter space is twisted as an $H_q$-module by $2x_n$. This is part $(i)$ of the proposition.

We now prove the claim that $\mathrm {H}^h_j\mathrm {H}_i^v(C_n \otimes _{R_n^{\mathrm {en}}} (M \otimes _{R_n} Y_2))=0$. Note that $Y_2$ fits into a short exact sequence of $(R_{n+1},R_{n+1})$-bimodules

$$\begin{align*}0{\longrightarrow} Y_2^{\prime \prime} {\longrightarrow} Y_2\stackrel{\psi}{{\longrightarrow}} Y_2^{\prime} {\longrightarrow} 0, \end{align*}$$

where the surjective map $\psi $ is given by

(4.27)

Since the kernel of the multiplication map $br \colon B_n \rightarrow R_{n+1}$ is generated as an $(R_{n+1},R_{n+1})$-bimodule by $v=x_{n+1} \otimes 1 -1 \otimes x_{n+1}$, it is easy to check that $x_n v =v x_{n+1}$ and v generates a copy of bimodule isomorphic to $\widetilde {R}_{n+1}^{x_n+x_{n+1}}$. Thus, the kernel of $\psi $ is given by

(4.28)

Clearly, $\mathrm {H}^v_{\bullet }(Y_2^{\prime \prime })=0$, and it follows that $\mathrm {H}^v_{\bullet }(Y_2)\cong \mathrm {H}^v_{\bullet } (Y_2^{\prime })$. Taking homology with respect to the topological (horizontal) differential $d_t$ then yields $\mathrm {H}^h_j\mathrm {H}_i^v(Y_2')=0$ and $\mathrm {H}^h_j\mathrm {H}^v_i(Y_2)=0$.

The computation of $\mathrm {HH}_{\bullet }^{\partial _q}((M\otimes \Bbbk [x_{n+1}]) \otimes _{R_{n+1}} T_n') $ is very similar. We only outline the necessary changes. Again, first note that $C_1^{\prime } \otimes _{\Bbbk [x_{n+1}]^{\mathrm {en}}} T_n^{\prime }$ is a bicomplex of $(R_{n+1},R_{n+1})$-bimodules

(4.29)

Ignore the grading shifts for now for ease of notation. There is a short exact sequence of bicomplexes of $(R_{n+1},R_{n+1})$-bimodules

$$\begin{align*}0{\longrightarrow} Z_1 {\longrightarrow} C_1^{\prime} \otimes_{\Bbbk[x_{n+1}]^{\mathrm{en}}} T_n^{\prime} {\longrightarrow} Z_2{\longrightarrow} 0, \end{align*}$$

whose terms are defined by

(4.30)

As in the previous part, there is a splitting of bicomplexes of $(R_n,R_n)$-bimodules given by

(4.31)

where $\phi $ was defined earlier. Thus, we get short exact sequences of the form

(4.32) $$ \begin{align} 0 \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Z_1)) \rightarrow \mathrm{HH}_i^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n^{\prime}) \rightarrow \mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Z_2)) \rightarrow 0 \end{align} $$

for each . Taking horizontal homology for this short exact sequence gives us a long exact sequence. However, since the homology of $\mathrm {H}_j^h\mathrm {H}_i^v(C_n \otimes _{R_n^{\mathrm {en}}} (M \otimes _{R_n} Z_1))$ is clearly always zero, we get that

$$ \begin{align*} \widehat{\mathrm{HHH}}_{j,i}((M \otimes \Bbbk[x_{n+1}])\otimes_{R_{n+1}} T_n') \cong \mathrm{H}^h_j(\mathrm{H}_i^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Z_2))) \end{align*} $$

for all . We need to analyze the latter homology space.

There is a morphism of bicomplexes

(4.33)

whose kernel is a vertical complex connected by the identity map. Thus,

$$\begin{align*}\widehat{\mathrm{HHH}}_{j,i}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n') \cong \mathrm{H}_j^h\mathrm{H}_i^v((C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Z_2^{\prime})))\cong \widehat{\mathrm{HHH}}_{j,i}(M)^{-2x_n}.\end{align*}$$

The result follows.

The above proof serves as a model for the Markov II invariance of $p\widehat {\mathrm {HHH}}$. We only present the necessary changes.

Proof. To begin with, one replaces the Koszul complex $C_n$ utilized in the proof of Proposition 4.12 with the p-extended Koszul complex $pC_n$. Also, one needs to replace the vertical homology taken there by vertical slash homology (see Remark 4.12).

For part $(i)$, one adapts equation (4.19) intoFootnote ⁵

(4.34) $$ \begin{align} pC_{n+1} \otimes_{R_{n+1}^{\mathrm{en}}} ((M \otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} pT_n) &= (pC_n \otimes pC_1^{\prime}) \otimes_{R_{n+1}^{\mathrm{en}}} ((M \otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} pT_n) \nonumber \\[8pt] & \cong pC_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} (pC_1^{\prime} \otimes_{\Bbbk[x_{n+1}]^{\mathrm{en}}} pT_n)), \end{align} $$

where $pC_1^{\prime }\cong pC_1$ is the p-extended complex of bimodules

(4.35) $$ \begin{align} 0{\longrightarrow} q^2 \Bbbk[x_{n+1}]^{x_{n+1}}\otimes \Bbbk[x_{n+1}]^{x_{n+1}}[1]^t_{\partial} \xrightarrow{x_{n+1}\otimes 1-1\otimes x_{n+1}} \Bbbk[x_{n+1}]\otimes \Bbbk[x_{n+1}]{\longrightarrow} 0. \end{align} $$

Then, diagram (4.20) becomes the following p-extended version in the Hochschild direction:

(4.36)

Here, $\phi $ is the map that sends $1$ to $(x_{n+1}-x_n) \otimes 1 + 1 \otimes (x_{n+1}-x_n)$

Now, as in the previous proof, one needs to show that $pC_n \otimes _{R_n^{\mathrm {en}}} (M \otimes _{R_n} pY_2)$ does not contribute to $p\widehat {\mathrm {HHH}}^{\partial _q}$. This is easier since now $pY_2$ is an acyclic p-complex. Furthermore, $pY_1$ is quasi-isomorphic to the p-complex $q R_n^{2x_n}$ sitting in t-degree zero. Hence, we obtain the isomorphism

$$\begin{align*}p\widehat{\mathrm{HHH}}^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} pT_n) \cong q^{-n} \mathrm{H}^/_{\bullet}(pC_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} R_n^{2x_n})) \cong p\widehat{\mathrm{HHH}}^{\partial_q}(M)^{2x_n}. \end{align*}$$

For the second isomorphism, again there is a short exact sequence of bicomplexes of $(R_{n+1},R_{n+1})$-bimodules

(4.37)

Getting rid of contractible summands, we see that $pZ_2$ is homotopy equivalent to

(4.38) $$ \begin{align} q^3 R_{n+1}[1]^t_{\partial} \xrightarrow{x_{n+1}-x_n} q R_{n+1}^{-(x_n+x_{n+1})}, \end{align} $$

which is, in turn, quasi-isomorphic to $q R_n^{-2x_n}$. The result follows after accounting for the shift built into $p\widehat {\mathrm {HHH}}^{\partial _q}$.

Remark 4.14. A closer examination of the proof of Corollary 4.13 shows the necessity of collapsing t and a gradings for the construction of $p\widehat {\mathrm {HHH}}^{\partial _q}$. Indeed, a comparison between equations (4.20) and (4.36), (4.30) and (4.37) shows that, if there were an a and t bigrading as for $\widehat {\mathrm {HHH}}$, then the grading shifts arising from the positive and negative Markov II moves would not match. This is caused by the fact that the homological shift $[1]^t_{\partial }$ and grading shift t functors are different when $p>2$, and thus

$$\begin{align*}(t^{[\frac{1}{2}]})^{\circ 2}=t\neq [1]^t_{\partial}. \end{align*}$$

Consequently, there does not seem to exist an overall compensation factor such as $a^{-\frac {n}{2}}t^{\frac {n}{2}}$ in Definition 4.1 that would make a triply graded $p\widehat {\mathrm {HHH}}$ invariant under both Markov II moves.

Theorem 4.15. Let $\beta _1$ and $\beta _2$ be two braids whose closures represent the same link L of r components up to framing. Suppose the framing numbers of the closures $\widehat {\beta }_1$ of $\beta _1$ and $\widehat {\beta }_2$ of $\beta _2$ differ by $\mathtt {f}_i(\widehat {\beta }_1)-\mathtt {f}_i(\widehat {\beta }_2)=a_i$, $i=1,\dots , r$. Then

$$ \begin{align*} \widehat{\mathrm{HHH}}^{\partial_q}(\beta_1) \cong \widehat{\mathrm{HHH}}^{\partial_q}(\beta_2)^{2\sum_{i=1}^r a_i x_i} \end{align*} $$

and

$$ \begin{align*} p \widehat{\mathrm{HHH}}^{\partial_q}(\beta_1) \cong p \widehat{\mathrm{HHH}}^{\partial_q}(\beta_2)^{2 \sum_{i=1}^r a_i x_i}, \end{align*} $$

where the generator of the polynomial action for the ith component is denoted $x_i$ and $\widehat {\mathrm {HHH}}^{\partial }(\beta _2)^{2 x_i}$ means we twist the $H_q$-module structure on the ith component by $2x_i$.

4.6 Unlinks and twistings

In this section, we compute $\mathrm {HHH}^{\partial _q}$ and $p\mathrm {HHH}^{\partial _q}$ for the identity element of the braid group $\mathrm {Br}_n$ and define an unframed link invariant in .

For the unknot, recall from the previous section the Koszul resolution $C_1$ of $\Bbbk [x]$, as a bimodule, is given by

Tensoring this complex with $\Bbbk [x]$ as a bimodule yields

Thus, the homology of the unknot (up to shift) is identified with the bigraded $H_q$-module

$$ \begin{align*} \Bbbk[x]\oplus q^2 a\Bbbk[x]^{2x}. \end{align*} $$

More generally, via the Koszul complex $C_n=C_1^{\otimes n}$, we have that the homology of the n-component unlink $L_0$ is equal to

(4.39) $$ \begin{align} \widehat{\mathrm{HHH}}^{\partial_q}(L_0)\cong a^{-\frac{n}{2}}t^{\frac{n}{2}} \mathrm{HH}_{\bullet}(R_n) \cong a^{-\frac{n}{2}}t^{\frac{n}{2}} \bigotimes_{i=1}^n \left( \Bbbk[x_i] \oplus q^2 a \Bbbk[x_i]^{2x_i} \right). \end{align} $$

Alternatively, up to the grading shift $ a^{-\frac {n}{2}}t^{\frac {n}{2}} $, we may identify $\widehat {\mathrm {HHH}}^{\partial _q}(L_0)$ with the exterior algebra over $R_n$ generated by the differential forms $dx_i$ of bidegree $aq^2$, $i=1,\dots , n$, subject to the condition that each $dx_i$ accounts for a twisting of $H_q$-module structure by $2x_i$.

It follows that, as for the ordinary HOMFLYPT homology, given a framed link L of n components arising as a braid closure $\widehat {\beta }$, its untwisted HOMFLYPT $H_q$-homology $\widehat {\mathrm {HHH}}^{\partial _q} (\beta ) $ is a module over

$$\begin{align*}\widehat{\mathrm{HHH}}_{0,0,\bullet}^{\partial_q}(L_0)\cong R_n,\end{align*}$$

and thus one may consider a twistingFootnote ⁶ of the $H_q$-module structure on $\widehat {\mathrm {HHH}}^{\partial _q}(\beta )$ via the functor $ R_n^f\otimes _{R_n}(\mbox {-}) $, where f is a linear polynomial in $x_1, \dots , x_n$, (see Section 3.1).

Definition 4.16. Let L be an n-strand framed link arising from the closure of a braid $\beta $. Label the components of L by $1$ through k, and set the (linear) framing factor of $\beta $ to be the linear polynomial

$$\begin{align*}\mathtt{f}_{\beta} = -\sum_{i=1}^k 2 \mathtt{f}_i x_i. \end{align*}$$

1. (1) The $H_q$-HOMFLYPT homology of $\beta $ is the triply graded $H_q$-module

   $$\begin{align*}\mathrm{HHH}^{\partial_q}(\beta):= \widehat{\mathrm{HHH}}^{\partial_q}(\beta)^{\mathtt{f}_{\beta}}\cong R_k^{\mathtt{f}_{\beta}}\otimes_{R_k} \widehat{\mathrm{HHH}}^{\partial_q}(\beta). \end{align*}$$

2. (2) Likewise, the $H_q$-HOMFLYPT p-homology is the doubly graded $H_q$-module

   $$\begin{align*}p\mathrm{HHH}^{\partial_q}(\beta):= p\widehat{\mathrm{HHH}}^{\partial_q}(\beta)^{\mathtt{f}_{\beta}}\cong R_k^{\mathtt{f}_{\beta}}\otimes_{R_k} p\widehat{\mathrm{HHH}}^{\partial_q}(\beta). \end{align*}$$

Proof. For the first statement, just notice that the twisting of the p-DG structure by the framing factor takes care of the Markov II move.

Next, the finite dimensionality of the homology theories follows, by construction, from the fact that ${{}^{f_{i_1}}B_{i_1}^{g_{i_1}}}\otimes _R \cdots \otimes _R {{}^{f_{i_k}}B_{i_k}^{g_{i_k}}}$ is an $H_q$-module with $2^k$-step filtration whose subquotients are isomorphic to $R^f$ as left $R\# H_q$-modules, and thus Corollary 3.2 applies.

The Poincaré polynomial of $\mathrm {HHH}^{\partial _q}(\beta )$, which is independent of the $H_q$-module structure on $\mathrm {HHH}^{\partial _q}(\beta )$, is well known to be a Laurent polynomial in a and t (i.e., in ), and Laurent series in q (i.e., in ). Specializing $t=-1$ recovers the HOMFLYPT polynomial (see, e.g., [Reference Khovanov and RozanskyKR08b, Reference Khovanov and RozanskyKR16]). On the other hand, in the construction of $p\mathrm {HHH}^{\partial _q}(\beta )$, we have categorically specialized the a and t grading shifts according to the relation $a=q^{2}t$, and then transformed t into $[1]^t_{\partial }$. The Grothendieck ring of t and q bigraded p-complexes up to homotopy is equal to (c.f. Corollary 2.8). In this ring, $[1]^t_{\partial }$ descends to $-1\in \mathcal {O}_p$. In turn, the a variable is then evaluated at $-q^2$. Hence, the Poincare polynomial of $p\mathrm {HHH}^{\partial _q}(\beta )$, taking value in

is equal to the HOMFLYPT polynomial with $a=-q^2$. This is just the Jones polynomial in a formal variable q, and the second part follows.

Finally, taking slash homology of the homology theories has the effect, on the level of Grothendieck groups, of passing from onto $\mathbb {O}_p$ (here we need part (i) showing that both $\mathrm {HHH}^{\partial _q}(\beta )$ and $p\mathrm {HHH}^{\partial _q}(\beta )$ are quasi-isomorphic to finite-dimensional $H_q$-modules). Therefore, taking slash homology of $\mathrm {HHH}^{\partial _q}(\beta )$ and $p\mathrm {HHH}^{\partial _q}(\beta )$ is equivalent to categorically specializing q at a primitive pth root of unity. This finishes the proof of the corollary.

5.1 A singly graded homology

Consider the $H_q$-Koszul complex in one variable:

(5.1) $$ \begin{align} C_1: 0 {\longrightarrow} a q^2 \Bbbk[x]^x\otimes \Bbbk[x]^{x} \stackrel{d_C}{{\longrightarrow}} \Bbbk[x]\otimes \Bbbk[x]{\longrightarrow} 0, \end{align} $$

where $d_C$ is the map $d_C(f)=(x^2\otimes 1+1\otimes x^2)f$. We regard the differential on the arrow as an endomorphism of the Koszul complex, of bidegree $(-1,2)$.

Proof. The commutator map $\phi :=[d_C,\partial _q]$ is given by

where $\phi $ sends the bimodule generator $1\otimes 1 \in \Bbbk [x]^x\otimes \Bbbk [x]^{x}$ to

$$ \begin{align*} \phi(1\otimes 1) & = d_C(\partial_q(1\otimes 1)) -\partial_q d_C(1\otimes 1) =d_C(x\otimes 1+1\otimes x)-\partial_q(x^2\otimes 1+1\otimes x^2)\\ & = (x\otimes 1+1\otimes x)(x^2\otimes 1+1\otimes x^2)-2(x^3\otimes 1+1\otimes x^3) \\ & = (x\otimes 1+1\otimes x)(x^2\otimes 1+1\otimes x^2)-2(x\otimes 1+1\otimes x)(x^2\otimes 1-x\otimes x+1\otimes x^2)\\ & = (x\otimes 1+1\otimes x)(-x^2\otimes 1+2x\otimes x-1\otimes x^2)\\ & =-(x\otimes 1+1\otimes x)(x\otimes 1-1\otimes x)^2. \end{align*} $$

We may thus choose a null-homotopy to be

where h is given by multiplication by the element $1\otimes x^2-x^2\otimes 1$ and acts on the rest of the complex by zero. The result follows.

The Koszul complex $C_n$ inherits the endomorphism $d_C$ by forming the n-fold tensor product from the one-variable case. It follows, that for a given p-DG bimodule M over $R_n$, there is an induced differential, still denoted $d_C$, given via the identification

(5.2) $$ \begin{align} \mathrm{HH}_{\bullet}^{\partial_q}(M)\cong \mathrm{H}_{\bullet} (M\otimes_{R_n^{\mathrm{en}}} C_n), \end{align} $$

where the induced differential acts on the right-hand side by $\mathrm {Id}_M \otimes d_C$. By construction, $d_C$ has Hochschild degree $-1$ and q-degree $2$.

Lemma 2.4 immediately implies the following.

Remark 5.3. The differential, first observed by Cautis [Reference CautisCau17], has the following more algebro-geometric meaning. Identifying $\mathrm {HH}^1(R_n)$ as vector fields on $\mathrm {Spec}(R_n)=\mathbb {A}^n$, $\mathrm {HH}^1(R_n)$ acts as differential operators on $\mathrm {HH}_{\bullet }(M)$ for any $R_n$-bimodule M, regarded as a coherent sheaf on $\mathbb {A}^n\times \mathbb {A}^n \cong T^* (\mathbb {A}^n)$. Under this identification, $d_C$ is given by, up to scaling by a nonzero number, contraction with the vector field

$$\begin{align*}\zeta_C:=\sum_{i=1}^n x_i^2\frac{\partial}{\partial x_i}. \end{align*}$$

On the other hand, this is also the vector field that defines the p-DG structure on $R_n$ by derivation. Therefore, via the Gerstenhaber module structure on $\mathrm {HH}_{\bullet }(M)$, the two actions naturally commute with each other on $\mathrm {HH}_{\bullet }(M)$.

In a more general context, Hochschild homology is a Gerstenhaber module over Hochschild cohomology viewed as a Gerstenhaber algebra. We may view $d_C$ and $\partial _q$ as the same element $\zeta $ in Hochschild cohomology, but the element $d_C$ acts on homology via cap product $\zeta \cap \bullet $ and the element $\partial _q$ acts via a Lie algebra action $\mathcal {L}_{\zeta }(\bullet )$. The compatibility of these actions is given by the equation

$$\begin{align*}\zeta \cap \mathcal{L}_{\zeta}(x)=[\zeta,\zeta] \cap x + \mathcal{L}_{\zeta}(\zeta \cap x). \end{align*}$$

Since $[\zeta ,\zeta ]=0$, these actions commute.

Now, we are ready to introduce a collapsed p-homology theory of a braid closure. Let $\beta \in \mathrm {Br}_n$ be an n-stranded braid. We have associated to $\beta $ a usual chain complex of $H_q$-equivariant Soergel bimodules $T_{\beta }$ as in equation (4.2), of which we take $p\mathrm {HH}^{\partial _q}_{\bullet }$ for each term:

(5.3)

Here, $\partial _C$ is a p-differential arising from $d_C$ as follows. By Proposition 4.8, the p-Hochschild homology groups in a column above are identified with the terms in

(5.4)

where each term in odd Hochschild degree is repeated $p-1$ times. Here, the horizontal differential is the p-Hochschild induced map of the topological differential, which we have denoted by $\partial _t$ to indicate its origin. On the arrows connecting even and odd Hochschild degree terms, we put the map $d_C$ while keeping the repeated terms connected by identity maps. This defines a p-complex structure, denoted $\partial _C$, in each column in diagram (5.3). The p-differential $\partial _C$ commutes with the $H_q$-action on each term by Corollary 5.2. It follows that, applying the totalization construction $\mathcal {T}$ of Lemma 2.5, we obtain a bigraded p-bicomplex of $H_q$-modules, with a horizontal (topological) p-differential $\partial _t$, a vertical p-differential $\partial _C$ and internal p-differential $\partial _q$. Denote the total p-differential $\partial _T:= \partial _t+\partial _C+\partial _q$, which collapses the double grading into a single q-grading.

Definition 5.4. Let $\beta $ be an n stranded braid. The untwisted $\mathfrak {sl}_2$ p-homology of $\beta $ is the slash homology group

$$\begin{align*}p\widehat{\mathrm{H}}(\beta):= q^{-n}\mathrm{H}^/_{\bullet}(p\mathrm{HH}^{\partial_q}_{\bullet}(pT_{\beta}),\partial_T), \end{align*}$$

viewed as an object in $\mathcal {C}(\Bbbk ,\partial _q)$.

The homology group $p\widehat {\mathrm {H}}(\beta )$ is only singly graded as an object in $\mathcal {C}(\Bbbk ,\partial _q)$. By construction, $p\widehat {\mathrm {H}}(\beta )$ is the slash homology with respect to the $\partial _T$ action on $\oplus _{i,j} p\mathrm {HH}_i^{\partial _q}(pT_{\beta }^j)$ (see diagram (5.4)). The latter space is doubly graded by the topological degree and q-degree with values in (the Hochschild a degree is already forced to be collapsed with the q degree to make the Cautis differential $\partial _C$ homogeneous). However, as in the proof of Corollary 4.13, the Markov II invariance for the homology theory requires one to collapse the t-grading onto the a-grading, thus also onto the q-grading. We will use $p\widehat {\mathrm {H}}_{i}(\beta )$ to stand for the homogeneous subspace sitting in some q-degree i.

Let us also emphasize an important point about the vertical grading collapsing as the following remark.

Remark 5.5. A special remark is needed here about the grading specialization. In order to p-extend the Koszul complex (5.1) into a p-Koszul complex with $\partial _C$ of degree two, we are forced to make the functor specialization from $[1]^a_d=a$ into $q^2[1]^q_{\partial }$ so that the p-extended complex looks like

(5.5) $$ \begin{align} pC_1:~ 0 {\longrightarrow} q^4 \Bbbk[x]^x\otimes \Bbbk[x]^{x}[1]^q_{\partial} \stackrel{d_C}{{\longrightarrow}} \Bbbk[x]\otimes \Bbbk[x]{\longrightarrow} 0. \end{align} $$

Forming iterated tensor products of $pC_1$ determines the correct vertical q-degree shifts in each column of diagram (5.3) of the p-Hochschild homology groups.

Notice that, on the level of Grothendieck groups, this has the effect of specializing the formal variable a into $-q^2$.

When $[1]^t_{\partial }=[1]^q_{\partial }$ and $a=q^2[1]_{\partial }^q$, the grading shifts in equation (4.7) translate into

(5.6) $$ \begin{align} pT_i := q^{-3} \left(B_i \xrightarrow{br_i} R[-1]^q_{\partial}\right) , \quad \quad \quad pT_i' := q^{3}\left(R[1]^q_{\partial} \xrightarrow{rb_i} q^{-2} B_i^{-(x_i+x_{i+1})}\right). \end{align} $$

This also explains the necessity of p-extension in the collapsed t and a direction in $p\mathrm {HHH}$ in the previous section: The homological shift in that direction needs to be p-extended to agree with the homological shift in the q-direction.

Furthermore, the bigrading in diagram (5.3) is now interpreted as a single grading, with both $\partial _C$ and $\partial _t$ raising q-degree by two.

This approach to a categorification of the Jones polynomial, at generic values of q, was first developed by Cautis [Reference CautisCau17]. We follow the exposition of Robert and Wagner from [Reference Robert and WagnerRW20] and the closely related approach of Queffelec, Rose and Sartori [Reference Queffelec, Rose and SartoriQRS18].

5.2 Framed topological invariance

In this subsection, we establish the topological invariance of the untwisted homology theory.

Proof. The proof of the theorem will mostly be parallel to that of Proposition 4.12 and Corollary 4.13. It amounts to showing that taking slash homology of $p\mathrm {HH}_{\bullet }^{\partial }(\beta )$ with respect to $\partial _T$ satisfies the Markov II move.

We start by discussing the normal $H_q$-equivariant Hochschild homology version. Let L be a link in obtained as a braid closure $\widehat {\beta }$, where $\beta \in \mathrm {Br}_n$ is an n-stranded braid. Recall that the homology groups $\mathrm {HH}^{\partial }_{\bullet }(L)$ are defined by tensoring a complex of Soergel bimodules M determined by $\beta $ with the Koszul complex $C_n$ and computing its termwise vertical (Hochschild) homology. The differential $d_C$ is defined on the Koszul complex $C_n$. To emphasize its dependence on n, we will write $d_C$ on $C_n$ as $d_n$ in this proof and likewise write $\partial _n$ for the p-extended differential on $pC_n$.

Since

$$ \begin{align*} C_{n+1}=C_n\otimes C_1^{\prime}=C_n \otimes \Bbbk[x_{n+1}] \otimes \Lambda \langle dx_{n+1} \rangle \otimes \Bbbk[x_{n+1}], \end{align*} $$

the vertical differential may be inductively defined as

(5.7) $$ \begin{align} d_{n+1}=d_n \otimes\mathrm{Id} + \mathrm{Id}\otimes d_1^{\prime}. \end{align} $$

Here, we have set $C_1^{\prime }=\Bbbk [x_{n+1}] \otimes \Lambda \langle dx_{n+1} \rangle \otimes \Bbbk [x_{n+1}] $ equipped with part of the Cautis differential

$$\begin{align*}{d}^{\prime}_{1}:=x_{n+1}^2 \otimes \iota_{\frac{\partial}{\partial x_{n+1}} }\otimes 1+ 1 \otimes \iota_{\frac{\partial}{\partial x_{n+1}} }\otimes x_{n+1}^2 . \end{align*}$$

The notation $\iota $ denotes the contraction of $dx_{n+1}$ with $\frac {\partial }{\partial x_{n+1}}$ .

From the proof of Proposition 4.12 (see equation (4.23)), we have a vector space decomposition

$$\begin{align*}\mathrm{HH}_{\bullet}^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) \cong \mathrm{H}_{\bullet}^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1)) \oplus \mathrm{H}_{\bullet}^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_2)). \end{align*}$$

Here, $Y_1$ and $Y_2$ are the terms of $C_1^{\prime }\otimes _{\Bbbk [x_{n+1}]^{\mathrm {en}}} T_n$ (see equation (4.20) for the definition). We claim that, instead of a direct sum decomposition, we obtain a filtration of $ \mathrm {HH}_{\bullet }^{\partial _q}((M\otimes \Bbbk [x_{n+1}]) \otimes _{R_{n+1}} T_n) $ as a module over $\Bbbk [d_C]/(d_C^2)$:

(5.8) $$ \begin{align} 0\rightarrow \mathrm{H}_{\bullet}^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_2)) \rightarrow \mathrm{HH}_{\bullet}^{\partial_q}((M\otimes \Bbbk[x_{n+1}]) \otimes_{R_{n+1}} T_n) \rightarrow \mathrm{H}_{\bullet}^v(C_n \otimes_{R_n^{\mathrm{en}}} (M \otimes_{R_n} Y_1)) \rightarrow 0. \end{align} $$

Indeed, since $d_C$ acts on the $Y_1$ and $Y_2$ tensor factors via $d_1^{\prime }$, it suffices to check that $d_1^{\prime }$ preserves the submodule arising from $Y_2$ and presents the part arising from $Y_1$ as a quotient. To do this, we reexamine the sequence (4.20) under vertical (Hochschild) homology. The part $Y_2$, under vertical homotopy equivalence, contributes to the horizontal (topological) complex (see equation (4.27))

(5.9a) $$ \begin{align} {Y}^{\prime}_2:=\left(R_{n+1}[1]^t_d \xrightarrow{d_t=\mathrm{Id}} R_{n+1}\right) \end{align} $$

sitting entirely in Hochschild degree $0$. Likewise, the part $Y_1$ contributes to the horizontal

(5.9b) $$ \begin{align} {Y}^{\prime}_1:= \left(R_{n+1}^{x_n+3x_{n+1}}[1]^t_d [1]^a_d \xrightarrow{d_t=2(x_{n+1}-x_n)}R_{n+1}^{2x_{n+1}}[1]^a_d \right) \end{align} $$

sitting entirely in Hochschild degree $1$. Since $d_1^{\prime }$ decreases Hochschild degree by one, $ {Y}^{\prime }_1 $ must be preserved under $d_1^{\prime }$, acting upon it trivially, and $ {Y}^{\prime }_2 $ is equipped with the (zero) quotient action by $d_1^{\prime }$.

Similar behavior happens under p-extension and degree collapsing with respect to the $\partial _C$ action (c.f. the proof and notation of Corollary 4.13). Write $\partial _1^{\prime }$ as the p-extended differential of $d_1^{\prime }$. Consider the degree collapsed diagram obtained from tensoring equation (5.6) with equation (5.5) while taking vertical slash homology:

(5.10)

After taking (vertical) p-Hochschild homology, the part of the term $pY_2$ that is not killed arises from

(5.11a) $$ \begin{align} p{Y}^{\prime}_2:=q^{-3}\left(R_{n+1} \xrightarrow{\mathrm{Id}} R_{n+1}[-1]^q_{\partial} \right) \end{align} $$

sitting entirely horizontally inside the lower horizontal arrow of equation (5.10). On the other hand, the part $pY_1$ contributes to the horizontal

(5.11b) $$ \begin{align} p{Y}^{\prime}_1:= q^3R_{n+1}^{x_n+3x_{n+1}}[1]^q_{\partial} \xrightarrow{2(x_{n+1}-x_n)} q R_{n+1}^{2x_{n+1}} \end{align} $$

sitting in the top horizontal line of the square (5.10). Since $\partial _1^{\prime }$ acts vertically down, upon taking the slash homology with respect to $\partial _T$, it now follows that the term arising from $pY_2^{\prime }$, on which $\partial _T$ acts just via $\partial _t+\partial _q$, contributes nothing to the total slash homology, as this term is the cone of the identity map in the homotopy category of p-complexes.

Now, translating the exact sequence (5.8) as in the final step in the proof of Corollary 4.13, we obtain that

$$\begin{align*}\mathrm{H}_{\bullet}^/( p\mathrm{HH}_{\bullet} (( M\otimes \Bbbk[x_{n+1}])\otimes_{R_{n+1}} T_n ),\partial_T) \cong \mathrm{H}_{\bullet}^/(p\mathrm{HH}_{\bullet}( M \otimes_{R_n} pY_1^{\prime}),\partial_T) \cong q\mathrm{H}_{\bullet}^/(p\mathrm{HH}_{\bullet}(M),\partial_T)^{2x_n}. \end{align*}$$

The q factor is cancelled out in the overall shift of $p\widehat {\mathrm {H}}$. This finishes the first part of Markov II move.

The other case of the Markov II move is entirely similar, which we leave the reader as an exercise.

Finally, the finite dimensionality of $p\widehat {\mathrm {H}}(\beta )$ follows from Corollary 3.2. The theorem follows.