# P-Divisible Groups Mini Course: Talk 3

Last week we showed how to associate a linear algebraic object, the Dieudonne module, to a $p$-divisible group. Furthermore, we saw how in the algebraically closed case the Dieudonne module is determined by its slopes. Today’s talk generalizes the previous setting in two ways. Firstly we will broaden our perspective to that of isocrystals with additional structure. This is analogous to how when we study Abelian varieties we often keep track of extra data such as the endomorphism ring and polarizations. Secondly, we move to the more general language of Tannakian categories. This will make the key results more straightforward to prove and understand and is generally very useful to know in many situations, such as when one is studying Shimura varieties or related constructions.

## Crash Course in Tannakian Categories

In what follows I’m purposely going to be vague in my definitions. The purpose is not to write a 50 treatise on Tannakian formalism: those already exist and by infinitely stronger mathematicians than myself. Rather I want to present some of the key results and discuss when one might expect Tannakian categories to be useful.

Definition: A tensor category $\mathcal{C}$ is a category  equipped with a bi-functor $\otimes: \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ which is symmetric and associative (i.e. there is a functorial isomorphism $X \otimes Y \simeq Y \otimes X$. These isomorphisms are also required to satisfy a few compatibility conditions.

Examples: Key examples of the above include categories of vector spaces, $R$-modules, representations of  some group.

Given a tensor category, we can define a number of other structures.

Definition: A unit object $( \mathbf{1}, e)$ of tensor  category $\mathcal{C}$ is an object $\mathbf{1} \in \mathcal{C}$, an isomorphism $e: \mathbf{1} \to \mathbf{1} \otimes \mathbf{1}$ so that the functor $X \mapsto \mathbf{1} \otimes X$  is an equivalence of categories from $\mathcal{C}$ to itself.

Definition: Given $X, Y \in \mathcal{C}$, we have a functor given by $T \mapsto \mathrm{Hom}(T \otimes X, Y)$. If this functor is representable, we denote it $\underline{ \mathrm{ Hom}}(X, Y)$. If the functor is representable for all $X, Y \in \mathcal{C}$ then we say that $\mathcal{C}$ has internals homs. Explicitely, this gives us the formula $\mathrm{Hom}(T, \underline{\mathrm{Hom}}(X, Y)) \simeq Hom(T \otimes X, Y)$.

Let $\mathcal{C}$ be the category of $R$-modules. Then the unit object is $R$ and internal homs are given by $\underline{\mathrm{Hom}}(X,Y)=\mathrm{Hom}_R(X,Y)$.

The above definitions let us produce many familiar constructions. For instance, the identity map $id: \underline{ \mathrm{ Hom}}(X, Y) \to \underline{ \mathrm{ Hom}}(X, Y)$ gives a map $ev: \underline{ \mathrm{Hom}}(X, Y) \otimes X \to Y$. For $R$-modules this is exactly the evaluation map.

If $\mathcal{C}$ has internal homs and a unit object, we can define for each $X \in \mathcal{C}$ and a unit object, we define a dual $\hat{X} = \underline{ \mathrm{ Hom}}(X, \mathbf{1})$ and the evaluation map is $ev: \hat{X} \otimes X \to \mathbf{1}$.

Further, by the definition of internal homs we have $Hom(X, \hat{\hat{X}}) \simeq Hom(X \otimes \hat{X}, \mathbf{1})$. Thus the evaluation map gives us a unique map $X \to \hat{\hat{X}}$. We call $X \in \mathcal{C}$ reflexive if this map gives an isomorphism $X \simeq \hat{\hat{X}}$.

Finally given  $X_1, X_2, Y_1, Y_2$ we can use the universal properties to build a morphism $\underline{ \mathrm{ Hom}}(X_1, Y_1) \otimes \underline{ \mathrm{ Hom}}(X_2, Y_2) \to \underline{ \mathrm{ Hom}}(X_1 \otimes X_2, Y_1 \otimes Y_2)$

These constructions motivate the following definition:

Definition: A rigid tensor category $\mathcal{C}$ is a tensor category equipped with  internal homs and a unit object such that all objects are reflexive and the morphism constructed in the previous paragraph is an isomorphism.

The above is nothing crazy. All we’ve done is axiomatized the notion of a tensor product and shown that the constructions we enjoy for vector spaces extend to this setting.

A tensor functor of rigid tensor categories $\mathcal{C}, \mathcal{C}'$ is a functor  $F: \mathcal{C} \to \mathcal{C'}$ which takes the unit object to a unit object and an isomorphism of functors $c: \otimes \circ (F, F) \to F \circ \otimes$ which satisfies a number of associativity and commutativity axioms. Given two tensor functors $(F,c) , (G,d)$ one can define the notion of a morphism of tensor functors and denote the class of such objects by $\mathrm{Hom}^{\otimes}(F, G)$.

The case we are most interested in is where $\mathcal{C}$ is an Abelian category and $\otimes$ is $R$-bilinear for some $R$. Note that the isomorphism $X \to X \otimes \mathbf{1}$ induces a morphism $\mathrm{End}(\mathbf{1}) \to \mathrm{End}(X)$ which makes $\mathcal{C}$ $R$-linear.

We now come to a pair of key definitions.

Definition: Let $k$ be a field , $R$ a $k$-algebra and $\mathcal{C}$ a rigid abelian $k$-linear tensor category. Then we define a fiber functor valued in $R$-modules to be a functor $\omega: \mathcal{C} \to \mathrm{Mod}_R$.

Definition: A Tannakian category is a rigid $k$-linear Abelian tensor category $\mathcal{C}$ equipped with a fiber functor $\omega: \mathcal{C} \to \mathrm{Mod}_R$. If $R=k$, then we say that the category is neutral Tannakian.

Key Example: If $G$ is an affine group scheme over $k$ and $\mathrm{Rep}_k(G)$ is the category of finite dimensional representations of $G$, then $\mathrm{Rep}_k(G)$ with the natural fiber functor $\omega^G: \mathrm{Rep}_k(G) \to \mathrm{Vec}_k$ is a neutral Tannakian category.

A key object of consideration when studying neutral Tannakian categories will be $\mathrm{Aut}^{\otimes}(\omega)$. Definitionally, this is a collection indexed over $X \in \mathcal{C}$ of elements of $\mathrm{Aut}(\omega(X))$. However, we will prefer to think to think of this as a group-valued functor $\underline{\mathrm{Aut}}(\omega(X))$ on $k$-algebras so that $\underline{\mathrm{Aut}}( \omega (X))(R)$  consists of collections of elements of $\mathrm{Aut}(\omega (X) \otimes R)$ indexed by $X \in \mathcal{C}$ and the group operation is composition.

Note that for $\mathcal{C}= \mathrm{Rep}_k(G)$ we have a natural functor $G \mapsto \underline{\mathrm{Aut}}^{\otimes}( \omega^G)$ (since each object of $\mathcal{C}$ is definitionally a $G$-rep). The key theorem is that we can reconstruct $G$ from the fiber functor.

Theorem: Let $\mathrm{Rep}_k(G)$ be the category of finite dimensional $k$-representations of $G$. Then the morphism $G \to \underline{\mathrm{Aut}}^{\otimes}( \omega^G)$ is an isomorphism.

Now we have the fundamental theorem of Tannakian categories.

Theorem: Let $(\mathcal{C}, \omega )$ be a neutral Tannakian category over $k$. Then $\underline{\mathrm{Aut}}^{\otimes}( \omega)$ is representable by some affine group scheme $G$ over $k$ and there is a unique tensor equivalence of categories $(\mathcal{C}, \omega) \to (\mathrm{Rep}_k(G), \omega^G)$.

These results together are pretty crazy. The first says that since we can recover an affine group scheme from its category of finite dimensional representations, the two categories contain equivalent data. The second says that any neutral Tannakian category can be thought of as representations of a group scheme. This is cool because often times a Tannakian category won’t obviously come from representations. For instance categories of Hodge structures are often Tannakian. As we shall see, the same is true for isocrystals.

Before discussing isocrystals, we briefly look at how gradings factor into the Tannakian picture. Given an Abelian group $M$ and a field $k$, we can produce the constant group scheme over $k$ asociated with $M$ and we have its Cartier dual $D(M)$. For instance, $D(\mathbb{Z})= G_m, D(\mathbb{Z}/ n \mathbb{Z})= \mu_n$ and $\mathbb{D} := D(\mathbb{Q})=D( \varinjlim \frac{1}{n} \mathbb{Z})=\varprojlim G_m$.

The category of $M$-graded $k$-vector spaces is  neutral Tannakian and has a natural fiber functor. Thus it must be equivalent to the category of representations of some group scheme. It turns out that the correct group scheme is $D(M)$. So for example, the category of $\mathbb{Z}$-graded $k$-vector spaces is equivalent to the category $\mathrm{Rep}_k(G_m)$. As a corollary, we see that if the fiber functor $\omega^G$ of $\mathrm{Rep}_k(G)$ factors through the category of $M$-graded vector spaces, then we get a map $D(M) \to G$ of group schemes.

I’ve spent a lot of time talking about Tannakian categories. Lets see how this shows up with regards to isocrystals. Recall that we defined an isocrystal last time as follows. If $k$ is a perfect field of characteristic $p$ with Witt vectors $W(k)$, $\sigma$ the lift of Frobenius and $K := W(k)[\frac{1}{p}]$, then an isocrystal is a finite dimensional $K$-vector space $V$ equipped with an automorphism $F: V \to V$ which is $\sigma$-linear. This is a $\mathbb{Q}_p$-linear Tannakian category which we denote $\mathrm{Isoc}(K)$.

In practice it’s good to remember that this formalism arose because we wanted to understand $p$-divisible groups. These $p$-divisible groups will show up in our studies of Abelian varieties. However, we typically study Abelian varieties keeping track of some extra structure such as endomorphisms or polarizations or level structure. For Shimura varieties this is encoded in the choice of a reductive group $G$ in the Shimura data. Analogously we want to consider categories of isocrystals with “$G$-data”.

Suppose $G$ is a reductive group over $\mathbb{Q}_p$ and that $G$ embeds naturally into $GL(V)$ for some finite $\mathbb{Q}_p$-vector space $V$. Then given an element $b \in G(K)$, we can take the space $V \otimes_{\mathbb{Q}_p} K$ and equip it with the automorphism $b( id_V \times \sigma)$. This is an isocrystal and we have encoded $G$-structure via $b$. One can check that if there exists $g, b, b' \in G(K)$ so that $b'=gb\sigma(g)^{-1}$, then $b, b'$ give isomorphic isocrystals. Thus, the collection of isocrystals $(F, \phi)$ so that $\phi$ is a $\sigma$-twist of an element of $G(K)$ is parametrised by the set $B(G) := G(K)/ \sim$ where $\sim$ is the aforementioned $\sigma$-conjugacy relation.

The above is somewhat awkward since it relies on a natural faithful representation of our reductive group. For many classical groups this is fine but for some reductive groups the definition is unclear. We can improve things by realizing that for each representation of $G$, an element of $B(G)$ gives an isocrystal structure on the vector space underlying that representation. This motivates the following definition.

Definition: A $G$-isocrystal is a functor $N: \mathrm{Rep}_{\mathbb{Q}_p}(G) \to Isoc(K)$ such that the natural fiber functors commute: $\omega \otimes_{\mathbb{Q}_p} K = N \circ \omega^G$.

Note that a $GL_n$-isocrystal is the same thing as an isocrystal.

We now think of $B(G)$ as a category whose objects are elements $b \in G(K)$ and whose morphisms $b \to b'$ are $g \in G(K)$ so that $b'=gb \sigma(g)^{-1}$.

Proposition: The natural functor $B(G) \to Isoc(K)$ we constructed above is an isomorphism of categories.

This proof is an exercise in applying the first key theorem of Tannakian categories given above.

Warning: The above construction is the correct notion for an isocrystal with $G$-structure only in certain cases. These are:

1. $k$ is algebraically closed and $G$ is connected.
2. $G=GL_n$
3. $G$ is semi-simple simply connected and $L$ is finite.

In general, one drops the compatibility condition on fiber functors in the definition but this makes the theory more difficult as (for instance) the above proposition is no longer true.

The category $\mathrm{Isoc}(K)$ is Tannakian and so is equivalent to the category of representations of some group scheme $D_K$. Now suppose $\mathrm{Rep}_{\mathbb{Q}_p}(G) \to \mathrm{Isoc}(K)$ is an isocrystal. Then by the Tannakian formalism we get a homomorphism $D_K \to G$. $\mathrm{Isoc}(K)$ has a natural $\mathbb{Q}$-grading given by the slope decomposition. Thus we get a slope map $\nu: \mathbb{D} \to D_K$. The composition of these two maps allows us to define a functorial Newton map $N: \mathrm{Isoc}(K) \to Hom(\mathbb{D}, G)$.

The Newton map is crucial in further constructions. Intuitively it should be thought of as a generalization of the slope decomposition of an isocrystal to the case of $G$-structure.  For instance, in the case of $GL(V)$-isocrystals and $K$ algebraically closed, each isocrystal produces a map $\mathbb{D} \to GL(V)$. This will factor through some maximal torus and so becomes a map $\mathbb{D} \to \prod\limits_{\mathrm{dim}(V)} G_m$ which is the same data as an $n$-tuple of rationals. These are precisely the slopes of the isocrystal.

## Resources

The well-read reader will notice that I have heavily borrowed the above from Dat, Orlik, and Rapoport’s Period Domains over Finite and p-adic Fields. I haven’t read this whole text but I intend to soon.

The original references for this material are Kottwitz’s Isocrystals with Additional Structure I, II. Isocrystals I Isocrystals II

The following paper by Rappoport-Richartz discusses the newton map in some detail. Rapoport – Richartz

Milne has written an extensive introduction to Tannakian categories with Deligne.

## One thought on “P-Divisible Groups Mini Course: Talk 3”

1. Dearest Bertie,

5) Some comments about the confusiong of whether things belong over $K$ or $\mathbb{Q}_p$, but we discussed this in your lecture.