Tannakian formalism over fields with operators

We study fields with operators (briefly described below, and more thoroughly in §2), and linear groups over such fields. Given such a group G (as defined in §3), our goal is to describe the category Rep G of finite dimensional representations of G, in a manner similar to the classical Tannakian formalism. In addition to generalising the usual Tannakian formalism, this paper forms a natural generalisation and reformulation of the theory of differential Tannakian categories (Ovchinnikov [17]), and especially of the definition of differential tensor categories in Kamensky [11, § 4].

We mention that results this kind are expected to have applications to Galois theory of linear equations with various operators. The classical Galois theories of ordinary differential and difference linear equations (as explained in Put and Singer [21] and Put and Singer [20], respectively) may be approached via the classical Tannakian formalism (also in Deligne [7, § 9]). More recently, there are the Galois theory of (linear) partial differential equations (initiated by Cassidy and Singer [5]) to which the differential Tannakian theory mentioned above was applied in Ovchinnikov [18] (see also Gillet et al. [9]), as well as linear equations involving both derivatives and automorphisms (Hardouin and Singer [10]), and other variants. It is hoped that the present paper will provide the tools to approach all these Galois theories in a uniform manner, from the Tannakian point of view (of course, the classical Tannakian theory has many more applications in different areas, and we hope that similar applications will be found for the generalised theory in this paper). We sketch the definition of the Galois group (in the case of commuting automorphism and Hasse-Schmidt derivations) in §0.1 below.

The main result of the paper, describing the analogue of tensor categories, as well as the statement that shows the notion to be adequate, i.e., that it does axiomatise categories of representations, is in §3 (specifically, Definition 3.2.4 and Theorem 3.2.9). Both the definition and the statement are rather immediate once the fundamental ideas are developed, so we now turn to a brief overview of the ideas that appear in the first two sections.

Our notion of a “field with operators” comes from (a variant of) the formalism developed by Moosa and Scanlon [16,15]. This formalism includes at least the cases of differential fields (fields endowed with a derivation, or a vector field), difference fields (fields with an endomorphism), Hasse-Schmidt derivations, and their combinations. To explain the idea, consider a field k with an endomorphism. One could alternatively describe the situation by saying that we are given an action of the monoid N of natural numbers on k. More generally, one could consider the action of a monoid M on k. When M is infinite, it cannot be viewed as a scheme. However, as a set, it is the (filtered) union of finite sets, each of which can be viewed as a scheme. Furthermore, the monoid operation maps the product of two finite sets in the system into another such finite set. In other words, M is a monoid in the category of ind-finite schemes, and we are given an action of M on spec(k).

Since any set is the filtered union of its finite subsets, the description above accounts for all discrete monoid actions. However, some finite schemes do not come from finite sets. Recall that the data of a derivation on the field k over the subfield k 0 is equivalent to that of a k 0 -algebra map k -→ k[ǫ] = k ⊗ k 0 k 0 [ǫ] whose composition with the unique map k[ǫ] -→ k is the identity. Geometrically, we are given an “action” M 0 × spec(k) -→ spec(k) of the scheme M 0 = spec(k 0 [ǫ]), in such a way that the k 0 -point of M 0 acts as the identity. The finite scheme M 0 is not a monoid, but it is part of a system defining an ind-scheme, the additive formal group M, and in characteristic 0, each “action” of M 0 extends uniquely to an action of M. As with the discrete case, the further components of the system M correspond to iterative application of the operator, i.e., to higher derivations (this example, which is classical, is discussed in detail in §2. We note that one could think of the additive formal group as a “limit” of the additive group of the integers, as the generator 1 “tends to 0”).

It is therefore reasonable to define a “field with operators” simply as a field with an arbitrary ind-finite scheme monoid action. This is (essentially) the approach taken in Moosa and Scanlon [16,15], and which we adopt here. We mentioned that similar ideas appear before: for example, Buium [3, § 2.4] discusses the encoding of a certain class of operators by suitable algebra maps (In fact, the approach there is somewhat more general, see §4.6). The case of k[ǫ] goes back (at least) to Weil (unpublished), and is, in any case, classical. The case of Hasse-Schmidt derivations is discussed in Matsumura [14, § 27]. It appears that the geometric description w

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