# Dieudonné modules

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. The 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.)

[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.