In this post I’ll motivate and outline the construction of an important invariant of (bicommutative) Hopf algebras in characteristic $p$: its *Dieudonné module*.
This post is a combination of expositions by others; I follow [HL13, §1] for the most part.
Another helpful source is the book by Demazure [Dem].

Throughout this post, I’ll assume that all Hopf algebras are bicommutative.

## Motivation: understanding Hopf algebras

Let $k$ be a field. Our goal is to understand Hopf algebras over $k$. This is good for many things: for example, if $X$ is a homotopy-commutative H-space, then $H_*(X;k)$ is a bicommutative Hopf algebra over $k$. If $X$ is an Eilenberg–MacLane space, then the Morava K-theory $K(n)_*(X)$ will be a Hopf algebra over $\mathbf{F}_p$. This second example is particularly important as a motivation for this post, since it is an example in characteristic $p$.

The following invariants play an important role in understanding Hopf algebras.

**Definition 1.** Let $H$ be a Hopf algebra over $k$.

- The set of
**primitive elements**of $H$ is \[ \operatorname{Prim}(H) := \{\ x \in H \mid \Delta x = 1\otimes x + x \otimes 1 \ \}. \] - The set of
**grouplike elements**of $H$ is \[ \operatorname{GLike}(H) := \{\ x \in H \mid \Delta x = x\otimes x,\ \varepsilon x = 1 \ \}. \]

We call $H$ **multiplicative** if $\operatorname{Prim}(H) = \{0\}$, and **connected** if $\operatorname{GLike}(H\otimes_k \overline{k}) = \{ 1\}$.

The set $\operatorname{Prim}(H)$ inherits the structure of a $k$-vector space from $H$, and the set $\operatorname{GLike}(H)$ becomes an abelian group under the multiplication of $H$.

*Remark.* Let $A = k[a]$ denote the polynomial algebra, and give it the structure of a Hopf algebra by letting $a$ be primitive.
(As the comultiplication is a $k$-algebra map, this determines the comultiplication, and hence the Hopf algebra structure.)
Then $A$ corepresents the functor $\operatorname{Prim}$.
We will see generalisations of this later.

It turns out that to understand all Hopf algebras, it is enough to understand just the multiplicative and the connected ones (at least, if $k$ is perfect).

**Theorem 2.** *Let $k$ be a perfect field.*
*Then the functor*
\[
\mathrm{Hopf}_k^\mathrm{m} \times \mathrm{Hopf}_k^\mathrm{c} \longrightarrow \mathrm{Hopf}_k, \quad (H,H’)\longmapsto H\otimes_k H'
\]
*is an equivalence.*

*Proof*. See, e.g., [Dem], page 39. $\hspace{1em}\blacksquare$

In particular, every Hopf algebra admits an essentially unique factorisation into a multiplicative and connected Hopf algebra.
For this reason, *we henceforth assume that $k$ is perfect*.

Understanding multiplicative Hopf algebras is not the hard part.

**Proposition 3.**
*The functor $\operatorname{GLike}$ lifts to a functor*
\[
\mathrm{Hopf}_k^\mathrm{m} \longrightarrow \{ \ \text{abelian groups with a continuous action of }\operatorname{Gal}(\overline{k}/k)\ \}
\]
*that is an equivalence, with an inverse equivalence given by*
\[
M \longmapsto \left(\overline{k}[M]\right)^{\operatorname{Gal}(\overline{k}/k)}.
\]

The connected Hopf algebras are harder to understand. Still, in characteristic zero (which is a much more specific case than that of perfect fields), it is also not hard to understand.

**Proposition 4.**
*Suppose $k$ is of characteristic zero.*
*Then the functor*
\[
\operatorname{Prim} \colon \mathrm{Hopf}_k^\mathrm{c} \longrightarrow \mathrm{Vect}_k
\]
*is an equivalence, with an inverse equivalence given by*
\[
V \longmapsto \operatorname{Sym}(V).
\]

In particular, over characteristic zero, every connected Hopf algebra is *primitively generated*: the natural map $\operatorname{Sym}(\operatorname{Prim} H) \to H$ is an isomorphism.

The assumption of characteristic zero is a sudden drop in generality compared to the previous results.
Over characteristic $p$, this result breaks down: the natural map $\operatorname{Sym}(\operatorname{Prim} H) \to H$ need not be injective or surjective for a general connected Hopf algebra.
We would like to ‘fix’ this result by constructing a ‘better’ version of the primitives for characteristic $p$.
This turns out to be possible, and this improved invariant is the *Dieudonné module* of the Hopf algebra.
It will again be a subset of $H$ endowed with additional structure, and it will contain the primitives as a subset.
The main players in the structure of Dieudonné modules will be the *Frobenius* and *Verschiebung* maps.

## Frobenius and Verschiebung

Henceforth, $k$ is a perfect field of characteristic $p$. This means that it has a Frobenius automorphism \[ \mathrm{Frob}_k \colon k \longrightarrow k,\quad x \longmapsto x^p. \]

If $H$ is a Hopf algebra over $k$, we define $H^{(p)}$ to be the base-change $H\otimes_k k$ over $\mathrm{Frob}_k$. It is customary to denote the element $x\otimes 1$ in $H^{(p)}$ by $x^{(p)}$. The scalar multiplication on $H^{(p)}$ is determined by the relation $\lambda^p \cdot x^{(p)} = (\lambda x)^{(p)}$ for $\lambda \in k$ and $x \in H$.

**Definition 5.**
The **Frobenius** of $H$ is the map
\[
F \colon H^{(p)} \longrightarrow H, \quad x\otimes \lambda \longmapsto x^p \cdot \lambda.
\]

The Verschiebung is essentially the dual construction of the Frobenius map. To spell it out, we need a little notation (taken from [Dem]). We write $TS^p H$ for the subset of $H^{\otimes p}$ on the symmetric tensors. Write $s\colon H^{\otimes p}\to TS^p H$ for the symmetrisation map, and write \[ \alpha_H \colon H^{(p)} \longrightarrow TS^p H, \quad x\otimes \lambda \longmapsto \lambda (x\otimes \dotsb \otimes x). \] Then the composite of $\alpha_H$ with the quotient $TS^p H \to TS^p H / \operatorname{im}s$ is bijective. The inverse then gives rise to a map $\lambda_H \colon TS^p H \to H^{(p)}$ with the property that $\lambda_H \circ s = 0$ and $\lambda_H \circ \alpha_H = \operatorname{id}$.

**Definition 6.**
The **Verschiebung** of $H$ is the composite
\[
V \colon H \longrightarrow H^{(p)}, \quad x \longmapsto \lambda_H (\Delta x).
\]

**Proposition 7.**
The composite $FV \colon H \to H$ is $[p]_H$, and the composite $VF \colon H^{(p)} \to H^{(p)}$ is $[p]_{H^{(p)}}$.

## Witt vectors and the Witt Hopf algebra

I will be assuming a basic familiarity with Witt vectors. A brief summary is the following (see [Ser, Ch. II, §5]): there is a functor \[ W \colon \{\ \text{perfect fields of characteristic }p \ \} \longrightarrow \{ \ \text{complete DVR’s with }v(p)=1 \ \} \] such that

- $W(k)$ has residue field $k$;
- for every map $f \colon k \to \ell$, there is a unique lift $F \colon W(k)\to W(\ell)$ compatible with the quotients $W(k)/p \cong k$ and $W(\ell)/p \cong \ell$.

We call $W(k)$ the ($p$-typical) **Witt vectors** of $k$.
Note that in particular we obtain a lift of Frobenius, which we denote by $\varphi\colon W(k)\to W(k)$.

**Example 8.**
If $k = \mathbf{F}_p$, then $W(k)\cong \mathbf{Z}_p$, the $p$-adic integers.
The lift of Frobenius is the identity on $\mathbf{Z}_p$, because the Frobenius on $\mathbf{F}_p$ is the identity.

**Example 9.**
If $k=\mathbf{F}_q$ with $q=p^n$, then $W(k)$ is $\mathbf{Z}_p$ with a $(q-1)$-st root of unity adjoined.

For those who want to know more, I also found the blog post by [Kim] to be a very good introduction to Witt vectors.

What we need is a sort of dual notion.

**Definition 10.** Let $\mathrm{Wt}_n^k$ denote the $k$-algebra $k[a_0,\dotsc,a_{n-1}]$, and let $\mathrm{Wt}^k$ denote $k[a_0,a_1,\dotsc]$.

It turns out that $\mathrm{Wt}_n^k$ has a unique Hopf algebra structure such that for every $m\leq n-1$, the polynomial
\[
\Phi_m := a_0^{p^m} + p a_1^{p^{m-1}} + \dotsb + p^m a_{m}
\]
is primitive.
For example, $\mathrm{Wt}_1^k$ is the Hopf algebra $k[a_0]$ with $a_0$ being primitive (i.e., the Hopf algebra corepresenting the functor $\mathrm{Prim}$).
Similarly, $\mathrm{Wt}^k$ has a unique Hopf algebra structure such that for every $m$ the polynomial $\Phi_m$ is primitive.
We call $\mathrm{Wt}_n^k$ the **$n$-truncated Witt Hopf algebra**, and call $\mathrm{Wt}^k$ the **Witt Hopf algebra**.

*Remark.*
The reason I call this a dual notion to Witt vectors is because we have an isomorphism of abelian groups
\[
\operatorname{Hom}_{\mathrm{CAlg}_k}(\mathrm{Wt}^k,\ k) \cong W(k).
\]
Here the comultiplication of $\mathrm{Wt}^k$ gives the Hom-set its abelian group structure.
For the $n$-truncated Witt vectors, we have the analogous isomorphism:
\[
\operatorname{Hom}_{\mathrm{CAlg}_k}(\mathrm{Wt}^k_n,\ k) \cong W_n(k).
\]

### Verschiebung on the Witt Hopf algebra

Notice that $\mathrm{Wt}_n^k$ is canonically isomorphic to $\mathrm{Wt}_n^{\mathbf{F}_p} \otimes_{\mathbf{F}_p} k$. The Frobenius automorphism of $\mathbf{F}_p$ is the identity, so over $\mathbf{F}_p$, every Hopf algebra $H$ is canonically isomorphic to $H^{(p)}$. Combining these two facts, we get an isomorphism \[ (\mathrm{Wt}_n^k)^{(p)} \cong \mathrm{Wt}_n^k. \] In particular, the Frobenius and Verschiebung maps are now endomorphisms of $\mathrm{Wt}_n^k$. The Verschiebung becomes particularly easy under this isomorphism: it is given by \[ V(a_m) = \begin{cases} \ a_{m-1} &\text{if }m>0, \\ \ 0 &\text{if }m=0. \end{cases} \] In particular, we may even view $V$ as a map $\mathrm{Wt}_n^k \to \mathrm{Wt}_{n-1}^k$.

## Dieudonné modules

From now on, all Hopf algebras are assumed to be connected. We still let $k$ denote a perfect field of characteristic $p$.

**Definition 12.**
Let $H$ be a connected Hopf algebra over $k$.
For $n\geq 1$, let
\[
\operatorname{DM}_n(H) := \{\ x \in H \mid \text{there exists a Hopf algebra map }f\colon \mathrm{Wt}_n^k \to H \text{ with }f(a_{n-1}) = x \ \}.
\]
Let $\operatorname{DM}(H)$ denote the union of $\operatorname{DM}_n(H)$ over all $n$.

**Example 13.**
The set $\operatorname{DM}_1(H)$ is equal to the set of primitive elements.
Indeed, $\mathrm{Wt}_1^k$ is the Hopf algebra $k[a_0]$ with $a_0$ primitive, and maps of Hopf algebras map primitive elements to primitive elements.
(And since the coalgebra structure is completely determined by $a_0$ being primitive, any primitive element of $H$ indeed gives rise to a map $\mathrm{Wt}_1^k \to H$.)

**Lemma 14.**
*Evaluation at $a_{n-1}$ induces a bijection*
\[
\operatorname{Hom}_{\mathrm{Hopf}_k}(\mathrm{Wt}_n^k, H) \cong \operatorname{DM}_n(H).
\]

*Proof.* Surjectivity is obvious.
For injectivity, note that $\mathrm{Wt}_n^k$ is generated (as a $k$-algebra) by the elements
\[
a_{n-1}, \quad a_{n-2} = V a_{n-1}, \quad \dotsc, \quad a_0 = V^{n-1}a_{n-1}.
\]
The Verschiebung is natural in maps of Hopf algebras, so the map $f$ is determined by where it sends $a_{n-1}$. $\hspace{1em}\blacksquare$

The Frobenius and Verschiebung of $H$ give rise to endomorphisms $F$ and $V$ of $\operatorname{DM}(H)$ as follows. If $f \colon \mathrm{Wt}_n^k \to H$ is a map, then it induces a map $f^{(p)} \colon (\mathrm{Wt}_n^k)^{(p)} \to H^{(p)}$. Earlier we identified $(\mathrm{Wt}_n^k)^{(p)}$ with $\mathrm{Wt}_n^k$, so we can think of $f^{(p)}$ as an element in $\operatorname{DM}_n(H^{(p)})$. This turns out to give a bijection \[ \operatorname{DM}_n(H) \to \operatorname{DM}_n(H^{(p)}). \] Using this, $F$ and $V$ descend to automorphisms of $\operatorname{DM}_n(H)$, and these are compatible as $n$ varies.

### The $\mathbf{Z}_p$-module structure

The set $\operatorname{DM}(H)$ has more structure still: it is naturally a $W(k)$-module. First we discuss how it gets the structure of an abelian group. The above lemma stating that $\operatorname{Hom}_{\mathrm{Hopf}_k}(\mathrm{Wt}_n^k, H) \cong \operatorname{DM}_n(H)$ gives $\operatorname{DM}_n(H)$ an abelian group structure. These are compatible in the following way. First, observe that $\operatorname{DM}_n(H)\subseteq \operatorname{DM}_{n+1}(H)$ for all $n$. Indeed, if $f \colon \mathrm{Wt}_n^k \to H$ sends $a_{n-1}$ to $x$, then $fV \colon \mathrm{Wt}_{n+1}^k \to X$ sends $a_n$ to $x$. Thus the inclusion $\operatorname{DM}_n(H)\subseteq \operatorname{DM}_{n+1}(H)$ corresponds to the map \[ V^* \colon \operatorname{Hom}_{\mathrm{Hopf}_k}(\mathrm{Wt}_n^k, H) \longrightarrow \operatorname{Hom}_{\mathrm{Hopf}_k}(\mathrm{Wt}_{n+1}^k, H). \] Because $V$ is a map of Hopf algebras, the induced map on Hom-groups is a group homomorphism. Thus $\operatorname{DM}(H)$ becomes an abelian group.

*Warning.*
This abelian group structure is not obtained by restricting the addition on $H$ to $\operatorname{DM}_n(H)$.
However, the group structure on $\operatorname{DM}_1(H)$ does agree with that on $\operatorname{Prim}(H)$.

To obtain the $W(k)$-module structure, we first put a $W_n(k)$-module structure on $\operatorname{DM}_n(H)$. Recall that \[ W_n(k) \cong \operatorname{Hom}_{\mathrm{CAlg}_k}(\mathrm{Wt}_n^k,\ k). \] Using this, we construct an action of $W_n(k)$ on the Hopf algebra $\mathrm{Wt}_n^k$ by Hopf algebra maps, i.e., we construct a map \[ W_n(k) \to \operatorname{Hom}_{\mathrm{Hopf}_k}(\mathrm{Wt}_n^k, \mathrm{Wt}_n^k) \] as follows. For $x \in W_n(k)$, let $[x]$ denote the corresponding map $\mathrm{Wt}_n^k \to k$. The action of $x$ we define to be the composite \[ \mathrm{Wt}_n^k \to \mathrm{Wt}_n^k \otimes_k \mathrm{Wt}_n^k \to \mathrm{Wt}_n^k \otimes_k k \cong \mathrm{Wt}_n^k \] where the first map is the comultiplication of $\mathrm{Wt}_n^k$, and the second map is given by $\mathrm{id}\otimes [x]$.

To glue these to a $W(k)$-module structure on $\operatorname{DM}_n(H)$, we need to compatibly normalise these actions. Let $\varphi \colon W(k) \to W(k)$ denote the lift of Frobenius. We let $W(k)$ act on $\mathrm{Wt}_n^k$ via the composite \[ W(k) \to W(k) \to W(k)/p^n \cong W_n(k) \] of $\varphi^{n-1}$ with the natural quotient map. In this way we have an action of $W(k)$ on $\operatorname{DM}_n(H)$ for every $n$, and the inclusions $\operatorname{DM}_n(H)\to \operatorname{DM}_{n+1}(H)$ are $W(k)$-linear. This yields a $W(k)$-module structure on $\operatorname{DM}(H)$.

*Remark.* The normalisation we did above ensures that if $\lambda \in k$, then $\tau(\lambda) \colon \operatorname{DM}_n(H)\to\operatorname{DM}_n(H)$ (where $\tau$ denotes the Teichmüller character $k\to W(k)$) is given by multiplication by $\lambda$.

### The main result

**Definition 15.**
The **Dieudonné ring** of $k$ is the non-commutative ring $D_k$ obtained by adjoining two non-commuting variables $F$ and $V$ to $W(k)$, subject to the relations
\[
F\lambda = \varphi(\lambda) F, \qquad V\varphi(\lambda) = \lambda V,\qquad FV = VF = p,
\]
where $\lambda \in k$, and where $\varphi \colon W(k) \to W(k)$ is the lift of Frobenius.

Our constructions above give $\operatorname{DM}(H)$ the structure of a left $D_k$-module. The main result is that this $D_k$-module remembers everything about $H$. A proof of the following result can be found in [Dem].

**Theorem 16.**
*The Dieudonné module assembles to a fully faithful functor*
\[
\operatorname{DM}\colon \mathrm{Hopf}_k^\mathrm{c} \longrightarrow \mathrm{LMod}_{D_k}
\]
*whose essential image consists of the* nilpotent *Dieudonné modules: those $M$ such that for every $x\in M$, there exists an $n$ such that $V^n x = 0$.*

*Remark.*
A nilpotent Diedonné module is automatically a $p$-torsion abelian group.
Indeed, if $x \in M$ is such that $V^n x = 0$, then by repeatedly using the relation $FV = p$, we find that $p^n x = 0$ as well.
This is the reason why in some sources, a Dieudonné module is defined to be a $p$-torsion abelian group with automorphisms $F$ and $V$ satisfying the identities described in Definition 15.
(And since a $p$-torsion abelian group has a unique $\mathbf{Z}_p$-module structure, this definition is equivalent to ours.)

## References

[Dem] Michel Demazure. *Lectures on $p$-divisible groups*.

[HS13] Michael Hopkins and Jacob Lurie. *Ambidexterity in $K(n)$-local stable homotopy theory.* Available online.

[Kim] Dongryul Kim. *Witt vectors*. Blog post.

[Ser] Jean-Pierre Serre. *Local fields*.