Tensor Constructions on z-Divisible Local Anderson Modules

Abstract

In this article we develop the multilinear theory of Drinfeld displays and use it to construct tensor products, symmetric and exterior powers of z-divisible local Anderson modules, which are the function fields analogs of p-divisible groups.

Introduction

In this paper we prove the existence of “tensor constructions” of z-divisible local Anderson modules, which are the (correct) equal characteristic analogs of p-divisible groups. They arise naturally as z-torsions of Drinfeld modules or more generally Drinfeld-Anderson A-motives. Multilinear morphisms of z-divisible local Anderson modules are defined as compatible systems of multilinear morphisms of their finite $z^n$-torsion group schemes, seen as fppf sheaves (see Definition 2.1, Definition 2.8). Tensor products, symmetric and exterior powers of z-divisible local Anderson modules are defined by the categorical universal property of tensor products, symmetric and alternating powers (see Definition 2.9).

The results of this paper are the natural generalizations and respectively analogs of our results in [9] and respectively [10]. In [9], we showed that over fields, exterior powers of π-divisible modules of dimension at most one exist, where π is a uniformizer of a non-Archimedean local field. Then, in [10] we generalized the results in [9] in one direction, where we proved that over any base scheme, the exterior powers of p-divisible groups of dimension one exist (or more generally for π-divisible modules, where π is a uniformizer of a p-adic local field, i.e., in the mixed characteristic case; however, in this case we need to assume that the base scheme is locally Noetherian). Now, in the present paper, we show, in the equal characteristic case, not only the existence of exterior powers of z-divisible local Anderson modules, but also the existence of tensor products and symmetric powers. We also do not impose any restriction on the dimension. To be more precise, in this paper we prove:

Theorem (Main Theorem)

Let $\mathcal{O}$ be the ring of integer of a non-Archimedean local field of positive characteristic, with uniformizer z, and S an $\mathcal{O}$-scheme, on which the image of z is Zariski locally nilpotent. Let $G,G_1,\dots ,G_r$ be z-divisible local Anderson modules over S. Fix $d\ge 1$. Then

• The tensor product $G_1\otimes \dots \otimes G_r$, the symmetric power ${\mathrm{Sym}}^{d}G$ and the exterior power ${\bigwedge }^{d}G$ exist.

• We have the following formulae for the heights:

  $$ht(G_1\otimes \dots \otimes G_r)=ht(G_1)\dots ht(G_r)ht({\mathrm{Sym}}^{d}G)=\left(\begin{array}{c}ht(G)+d-1\\ d\end{array}\right)ht({\bigwedge }^{d}G)=\left(\begin{array}{c}ht(G)\\ d\end{array}\right)$$

• We have the following formulae for the dimensions:

  $$\begin{gathered} \dim (G_1\otimes \dots \otimes G_r)=\sum _{i=1}^r(ht(G_1)\cdots ht(G_{i-1})\cdot \dim G_i\cdot ht(G_{i+1})\cdots ht(G_r))\dim ({\mathrm{Sym}}^{d}G)=\left(\begin{array}{c}ht(G)+d-1\\ d-1\end{array}\right)\cdot \dim G \\ \dim ({\bigwedge }^{d}G)=\left(\begin{array}{c}ht(G)-1\\ d-1\end{array}\right)\cdot \dim G \end{gathered}$$

• If G is a z-divisible module (i.e., has scalar $\mathcal{O}$-action) and has dimension one, then ${\bigwedge }^{d}G$ is a z-divisible module as well (its height and dimension are given by the above formulae).

• For any S-scheme T, the natural base change morphisms

  $$\begin{gathered} G_{1,T}\otimes \dots \otimes G_{r,T}\to {(G_1\otimes \dots \otimes G_r)}_T \\ {\mathrm{Sym}}^d(G_T)\to {({\mathrm{Sym}}^{d}G)}_T{\bigwedge }^d(G_T)\to {({\bigwedge }^{d}G)}_T \end{gathered}$$

  are isomorphisms.

There is a Dieudonné-like classification of z-divisible local Anderson modules in terms of linear-algebraic objects; indeed it is shown in [11] that the category of z-divisible local Anderson modules is equivalent to the category of Drinfeld displays and in [7] that it is anti-equivalent to the category of local shtukas. But this alone is not enough to show that multilinear morphisms are also in bijection under this equivalence, and it requires hard work to prove this statement. In fact the core of this paper is showing that the equivalence between z-divisible local Anderson modules and Drinfeld displays preserves multilinear morphisms.¹ A similar phenomenon appears in the case of the category of truncated Barsotti-Tate groups or p-divisible groups, where they are equivalent to truncated Displays or Dieudonné modules respectively, but showing that their multilinear morphisms are preserved under these equivalences requires much work (see [10]). As we will see (in section 5), the fact that multilinear categories of z-divisible local Anderson modules and Drinfeld displays are equivalent implies that tensor products are also preserved under this equivalence (after we establish the existence of tensor products), and so, one can say that the equivalence between these categories is a “monoidal” equivalence.

In a few words, the reason why in the equal characteristic case we have a much more general result (the existence of all tensor constructions and no restriction on the dimension) is that in this case, there is no “slope” constraints: recall that an isocrystal is the isocrystal associated with a p-divisible group if and only if its Dieudonné-Manin slopes are between 0 and 1 and this in turn is imposed by the fact that p-divisible groups are equipped with two morphisms F and V satisfying $FV=p=VF$. However, in the equal characteristic case, there is no such condition on the slopes of the isocrystals of z-divisible local Anderson modules (cf. [6]). Now, the category of isocrystals (in either characteristic) admits all tensor constructions, but the slopes leave the interval $[0,1]$ under these constructions (cf. [13]).

Drinfeld modules and Drinfeld-Anderson A-motives, which are global objects and z-divisible local Anderson modules, which are their local counterparts, play a crucial role in the global and local Langlands program over function fields of positive characteristic. In fact, Drinfeld modular varieties (analogs of Shimura varieties), which are the moduli spaces of these objects carry actions of the related Galois group and ${\mathrm{GL}}_n$, and their cohomology spaces realize the Langlands correspondence [15], [14]. The existence of tensor constructions for Drinfeld-Anderson A-motives was conjectured by Drinfeld and proved by Anderson and Thakur [1], [2] in order to give a “geometric” construction of (or explanation for) the tensor constructions of the Galois representations attached to these geometric objects. Similarly, our result provides natural geometric objects that explain why tensor constructions of the Galois representations of z-divisible local Anderson modules are the Galois representations of z-divisible local Anderson modules. The properties of the tensor constructions, listed in the above theorem, will also enable one to define natural morphisms between moduli spaces of z-divisible local Anderson modules, which can then be used to study the relation between their cohomology spaces from the perspective of the (local) Langlands program. In fact, this is the content of an upcoming paper of the author. Such a study in the mixed characteristic case, using the results of [10] is being done in [12].

Let us now briefly sketch the proof of the Main Theorem. We will do it for the tensor products, as the case of symmetric and exterior powers are similar. In what follows, all schemes are defined over the finite field ${\mathbb{F}}_q$ of characteristic p. The main steps of the proof are the following:

• A key ingredient of the proof is Drinfeld displays, the theory of which was developed in [11]. Here we study their multilinear theory and in particular construct their tensor products.

• We also develop a multilinear theory for Ver-sheaves, which are the dual objects to finite ${\mathbb{F}}_q$-shtukas. They were defined and studied in [11] (for details about finite ${\mathbb{F}}_q$-shtukas, see [7, §2, §5] and [5]).

• Let $G_0,G_1,\dots ,G_r$ be z-divisible local Anderson modules over the base scheme S. Let ${\mathcal{P}}_i$ be the nilpotent Drinfeld display attached to $G_i$. We construct a homomorphism

  $$\beta :\mathrm{Mult}({\mathcal{P}}_1\times \dots \times {\mathcal{P}}_r,{\mathcal{P}}_0)\to \mathrm{Mult}(G_1\times \dots \times G_r,G_0)$$

  by constructing homomorphisms

  $${\beta }_n:\mathrm{Mult}({\mathcal{P}}_{1,n}\times \dots \times {\mathcal{P}}_{r,n},{\mathcal{P}}_{0,n})\to \mathrm{Mult}(G_{1,n}\times \dots \times G_{r,n},G_{0,n})$$

  where $G_{i,n}$ is the $z^n$-torsion subgroup of $G_i$ and ${\mathcal{P}}_{i,n}$ is its associated Ver-sheaf, and then taking inverse limit.

• The most important and non-trivial result here is that β is an isomorphism. This result is based on several technical lemmas and propositions. Essentially, we show that each ${\beta }_n$ is an isomorphism, by explicitly constructing an inverse ${\eta }_n$.

• Now let ${\mathcal{P}}_1\otimes \dots \otimes {\mathcal{P}}_r$ be the tensor product of ${\mathcal{P}}_i$, and let ⨂G be the associated z-divisible local Anderson module. Since β is an isomorphism, it follows, rather formally, that indeed ⨂G is the tensor product of $G_i$. This settles the question of the existence of tensor products for z-divisible local Anderson modules.

• Étale z-divisible local Anderson modules are the same as continuous representations of the étale fundamental group of S with coefficients in $\mathcal{O}[\frac{1}{z}]/\mathcal{O}$. Since the category of representations of a group has all the tensor constructions, the tensor products of étale z-divisible local Anderson modules exist as well.

• In order to show the formulae for the heights and dimensions, we use the fact that these invariants are preserved under the equivalence between Drinfeld displays and z-divisible local Anderson modules. Then, using the above construction of the tensor product $⨂G_i$, we need to compute the rank and dimension of $⨂{\mathcal{P}}_i$.

• We compute the rank and dimension of $⨂{\mathcal{P}}_i$.

Notations 0.1

• p is a prime number different from 2, $q=p^f$ is a power of p and ${\mathbb{F}}_q$ is the finite field with q elements.

• For natural numbers m and n, the binomial coefficient $\left(\begin{array}{c}n\\ m\end{array}\right)$ is defined to be zero when $m>n$.

• $\mathcal{O}={\mathbb{F}}_q〚z〛$ is the ring of formal power series in variable z and coefficients in ${\mathbb{F}}_q$.

• S is a fixed $\mathcal{O}$-scheme.

• Denote by ζ the image of z in ${\mathcal{O}}_S$ that we assume is locally nilpotent on S.

• If R is a ring, r is an element of R and M is an R-module, we denote by $M/r$ the quotient $M/rM$.

• For a ring R, we denote by ${\mathfrak{Mod}}_R$ the category of R-modules.

• For a scheme T, we denote by ${\mathfrak{Sch}}_T$ the category of schemes over T.

• Let X be a scheme over a base scheme S. We identify X with the sheaf ${\mathrm{Mor}}_S(\mathrm{\_},X)$ on the big fppf site of S.

• If a base scheme S is fixed, unless otherwise specified, by a presheaf on S we mean an fppf presheaf on the big fppf site of S, which is nothing but a contravariant functor on ${\mathfrak{Sch}}_S$.

• Let X be a scheme over a base scheme S and $f:T\to S$ a morphism. We denote by $X_T$ the fiber product $X{\times }_{S}T$. If $\mathcal{F}$ is a sheaf on a Grothendieck site over S, we denote by $f^{⁎}\mathcal{F}$ the pullback of $\mathcal{F}$ along f. So $f^{⁎}X$ and $X_T$ are identified as sheaves.

• Let $\mathcal{F}$ and $\mathcal{G}$ be sheaves on a Grothendieck site. We denote by $\underset{\_}{\mathrm{Mor}}(\mathcal{F},\mathcal{G})$ the presheaf of morphisms from $\mathcal{F}$ to $\mathcal{G}$. If $\mathcal{F}$ and $\mathcal{G}$ are sheaves of abelian groups, we denote by $\underset{\_}{\mathrm{Hom}}(\mathcal{F},\mathcal{G})$ the presheaf of homomorphisms. Note that since $\mathcal{G}$ is a sheaf, these presheaves are in fact sheaves (cf. [16, Lemma 03EM]).

• In this paper, all group schemes are assumed to be commutative.

• In this paper, unless otherwise stated, all schemes are defined over ${\mathbb{F}}_q$.

• For a scheme X (defined over ${\mathbb{F}}_q$) we denote by $\sigma :X\to X$ the $f^{\text{th}}$-th iterate of the absolute Frobenius, and continue to call it the absolute Frobenius of X. This is the ${\mathbb{F}}_q$-morphism which is the identity on the underlying topological space of X and is the ${\mathbb{F}}_q$-linear morphism $x\mapsto x^q$ on ${\mathcal{O}}_X$.

• For a group scheme G defined over a base scheme S, we denote by $F_G:G\to G^{(q)}$ the relative Frobenius morphism of G over S.