Last week we showed how to associate a linear algebraic object, the Dieudonne module, to a -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 is a category equipped with a bi-functor which is symmetric and associative (i.e. there is a functorial isomorphism . These isomorphisms are also required to satisfy a few compatibility conditions.

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

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

**Definition: **A unit object of tensor category is an object , an isomorphism so that the functor is an equivalence of categories from to itself.

**Definition: **Given , we have a functor given by . If this functor is representable, we denote it . If the functor is representable for all then we say that has internals homs. Explicitely, this gives us the formula .

Let be the category of -modules. Then the unit object is and internal homs are given by .

The above definitions let us produce many familiar constructions. For instance, the identity map gives a map . For -modules this is exactly the evaluation map.

If has internal homs and a unit object, we can define for each and a unit object, we define a dual and the evaluation map is .

Further, by the definition of internal homs we have . Thus the evaluation map gives us a unique map . We call reflexive if this map gives an isomorphism .

Finally given we can use the universal properties to build a morphism

These constructions motivate the following definition:

**Definition: **A rigid tensor category 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 is a functor which takes the unit object to a unit object and an isomorphism of functors which satisfies a number of associativity and commutativity axioms. Given two tensor functors one can define the notion of a morphism of tensor functors and denote the class of such objects by .

The case we are most interested in is where is an Abelian category and is -bilinear for some . Note that the isomorphism induces a morphism which makes -linear.

We now come to a pair of key definitions.

**Definition: **Let be a field , a -algebra and a rigid abelian -linear tensor category. Then we define a fiber functor valued in -modules to be a functor .

**Definition: **A Tannakian category is a rigid -linear Abelian tensor category equipped with a fiber functor . If , then we say that the category is neutral Tannakian.

**Key Example**: If is an affine group scheme over and is the category of finite dimensional representations of , then with the natural fiber functor is a neutral Tannakian category.

A key object of consideration when studying neutral Tannakian categories will be . Definitionally, this is a collection indexed over of elements of . However, we will prefer to think to think of this as a group-valued functor on -algebras so that consists of collections of elements of indexed by and the group operation is composition.

Note that for we have a natural functor (since each object of is definitionally a -rep). The key theorem is that we can reconstruct from the fiber functor.

**Theorem:** Let be the category of finite dimensional -representations of . Then the morphism is an isomorphism.

Now we have the fundamental theorem of Tannakian categories.

**Theorem: **Let be a neutral Tannakian category over . Then is representable by some affine group scheme over and there is a unique tensor equivalence of categories .

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 and a field , we can produce the constant group scheme over asociated with and we have its Cartier dual . For instance, and .

The category of -graded -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 . So for example, the category of -graded -vector spaces is equivalent to the category . As a corollary, we see that if the fiber functor of factors through the category of -graded vector spaces, then we get a map of group schemes.

## Isocrystals with Additional Structure

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 is a perfect field of characteristic with Witt vectors , the lift of Frobenius and , then an isocrystal is a finite dimensional -vector space equipped with an automorphism which is -linear. This is a -linear Tannakian category which we denote .

In practice it’s good to remember that this formalism arose because we wanted to understand -divisible groups. These -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 in the Shimura data. Analogously we want to consider categories of isocrystals with “-data”.

Suppose is a reductive group over and that embeds naturally into for some finite -vector space . Then given an element , we can take the space and equip it with the automorphism . This is an isocrystal and we have encoded -structure via . One can check that if there exists so that , then give isomorphic isocrystals. Thus, the collection of isocrystals so that is a -twist of an element of is parametrised by the set where is the aforementioned -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 , an element of gives an isocrystal structure on the vector space underlying that representation. This motivates the following definition.

**Definition: **A -isocrystal is a functor such that the natural fiber functors commute: .

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

We now think of as a category whose objects are elements and whose morphisms are so that .

**Proposition: **The natural functor 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 -structure only in certain cases. These are:

- is algebraically closed and is connected.
- is semi-simple simply connected and 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 is Tannakian and so is equivalent to the category of representations of some group scheme . Now suppose is an isocrystal. Then by the Tannakian formalism we get a homomorphism . has a natural -grading given by the slope decomposition. Thus we get a slope map . The composition of these two maps allows us to define a functorial Newton map .

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 -structure. For instance, in the case of -isocrystals and algebraically closed, each isocrystal produces a map . This will factor through some maximal torus and so becomes a map which is the same data as an -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.

Dearest Bertie,

Some comments:

1) Typo in your first paragraph. You say that the Dieudonné module is determined by its slopes when, of course, you mean that the Dieudonné isocrystal is determined by its slopes.

2) Typo ’50 treatise’ should perhaps be ‘Alex Youcis level tour de force of mathematical exposition’ or, maybe, ’50 page treatise’. 🙂

3) Just a pedantic comment, but “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.” should perhaps be that you’ve axiomatized tensor products *and* duals.

4) Typo in your definition of fiber functor, unless you’re using some much more general notion than I am. Namely, fiber functors are supposed to exact and faithful.

5) Some comments about the confusiong of whether things belong over or , but we discussed this in your lecture.

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