Reduced forms of linear differential systems and the intrinsic Galois-Lie algebra of Katz

Generalizing the main result of (Aparicio, Compoint, Weil 2013), we prove that a system is in reduced form in the sense of Kochin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement, which was implicit in (Aparicio, Compoint, Weil 2013) and is a crucial ingredient of (Barkatou, Cluzeau, Di Vizio, Weil 2016). We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group, which is actually another fundamental ingredient of the algorithm in (Barkatou, Cluzeau, Di Vizio, Weil 2016).


Introduction
Let us consider the field of rational functions C(x), with the derivation ∂ = d dx , and a linear differential system ∂ y = A y, where A is a square matrix of order n with coefficients in C(x). One can attach to such an object an algebraic group, called the differential Galois group, whose geometric properties encode the algebraic properties of the solutions of the linear differential system. The problem of calculating explicitly the differential Galois group of ∂ y = A y is old and still difficult. Among the several references, we cite [CS99], [Hru02], [vdH07], [Fen15] and [AMP18] that do not make any assumption on the order n of the system. Implemented (or implementable) algorithms exist only for small dimensions n. See for example [Kov86], [SU93], [vHRUW99], [Hes01], [vH02], [Per02], [NvdP10], [CS18].
Instead of calculating directly the differential Galois group of ∂ y = A y, one can try to study, or calculate, the Lie algebra of the differential Galois group, called Galois-Lie algebra in what follows, which already contains a significant part of the information. This line of thoughts has been initiated by Kolchin and Kovacic, who have proved that one can transform ∂ y = A y into an equivalent system ∂ y = B y defined over a finite extension k of C(x), such that B belongs to the set of k-rational points of the Galois-Lie algebra (see [vdPS03,Proposition 1.31]). One can even prove that the Galois-Lie algebra is then "generated" by the entries of the matrix B. These ideas are formalized in [AMCW13,§2.3]. The linear differential system ∂ y = B y is called a reduced form of ∂ y = A y. A linear differential system ∂ y = A y is said to be in reduced form if A is a k-rational point of the Galois-Lie algebra.
In the present work, we prove that a system is in reduced form if and only if any differential module in a construction admits a constant basis (see Theorem 3.2). This extends the criterion for reduced form from [AMCW13] which concerned only invariant lines. We derive an explicit version using local data (see Theorem 3.8), which was somehow hidden in [AMCW13] and is a crucial ingredient of [BCDVW16]. In the latter reference, we showed how one can compute the Galois-Lie algebra of an (absolutely) irreducible linear differential system and hence (a good part of) its differential Galois group; in [DW19], it is shown how to derive the Galois group from the Galois-Lie algebra once a reduced form has been found.
The idea of focusing on the Galois-Lie algebra rather than on the differential Galois group itself is also at the origin of [Kat82], where Katz introduces another Galois group for the linear differential system ∂ y = A y, called the generic or the intrinsic Galois group. Then he considers the Lie algebra of such a group, for whom he gives a conjectural description equivalent to a well-known conjecture of Grothendieck on the algebraicity of the solutions of a linear differential system. We will call it the Katz algebra. The algorithm in [BCDVW16] deduces the Galois-Lie algebra from the Katz algebra that it calculates using the reduced forms of Kolchin and Kovacic, notably their constructive characterization that we prove here.
In the last part of the paper, we gather material from [Kat82,Ber92,And04,vdPS03] and show how our criteria for reduced forms, combined with standard Tannakian tools, clarify the structure of the Katz algebra. Namely, for a reductive group, Theorem 4.4 shows that the Katz algebra is a k-form of the Galois-Lie algebra.
Organization of the paper. In Section 2, we recall some notions on differential modules, tensor constructions and differential Galois theory. In Section 3, we prove our two theorems on the criteria for a linear differential system to be in reduced form. In Section 4, we apply the previous results to the study of the Katz algebra.

Notation and definitions
We consider a characteristic zero differential field (k, ∂), that is a characteristic zero field k with a derivation ∂ : k → k, such that ∂(a + b) = ∂(a) + ∂(b) and ∂(ab) = ∂(a)b + a∂(b), for any a, b ∈ k. We suppose that the subfield of constants C := k ∂ = {f ∈ k : ∂f = 0} is algebraically closed.

Differential modules
for any f ∈ k and any m ∈ M . For a detailed exposition on differential modules, see [vdPS03,§2.2]. We denote by M ∇ or ker ∇ the set of horizontal elements of M, that is elements m ∈ M satisfying ∇(m) = 0. This is a C-vector space of dimension at most n.
Main properties of differential modules. Given a basis 1 e := (e 1 , · · · , e n ) of M over k, the action of ∇ with respect to the basis e is described by a square matrix A ∈ M n (k) as follows: For any y ∈ k n such that e y represents an element of M , we have ∇(e y) = e(∂ y − A y). Thus horizontal elements of M correspond to solutions over k of the differential system We say that [A] : ∂ y = A y is the linear differential system associated to M with respect to the basis e.
If f = eP , with P ∈ GL n (k), is another basis of M , then the horizontal elements of M are of the form f z, with z ∈ k n , where z verifies the linear differential system: We say that two matrices A, B ∈ M n (k) are equivalent over k if there exists a gauge transformation P ∈ GL n (k) Notice that one can extend the scalars of M to a field extension k ′ of k, equipped with an extension of ∂. The Leibnitz rule allows to extend ∇ to M ⊗ k k ′ , so that it makes sense to consider gauge transformations in GL n (k ′ ), as counterpart of basis changes of M ⊗ k k ′ = (M ⊗ k k ′ , ∇).
Algebraic constructions. A construction of linear algebra is a finite iteration of the usual constructions (direct sums ⊕, tensor products ⊗, duals * , spaces of homomorphisms Hom, symmetric powers Sym r , for some r ≥ 1, and exterior powers ∧ r ). Given a construction of linear algebra, we denote by Constr(M ) the finitedimensional k-vector space obtained by applying the construction to M . If e is a basis of M , we will denote by Constr(e) the induced basis of Constr(M ). If we consider a basis change in the vector space M : f = eP , for some P ∈ GL n (k), then the change-of-basis matrix for the vector space Constr(M ) will be denoted by Constr(P ), i.e. Constr(f ) = Constr(e)Constr(P ). The constructions of linear algebra apply functorially to differential modules. Indeed, given a differential module M = (M, ∇) over k, the operator ∇ induces a C-linear map from Constr(M ) → Constr(M ), that we will also denote by ∇, defining a differential module structure over Constr(M ). We will call the latter a construction of M (or tensor construction) and denote it Constr(M) = (Constr(M ), ∇).
Remark 2.1. For our purposes, we will not need to consider the subquotients of the constructions above.
Let M = (M, ∇) be a differential module and let −A be the matrix of ∇ with respect to a basis e as defined above. For any construction Constr(M), the matrix of ∇ with respect to Constr(e) will be denoted by −constr(A). Example 2.2. Let M = (M, ∇) be a differential module over k and let −A be the matrix of ∇ with respect to a basis e. We consider the differential module End(M) = (End k (M ), ∇). If ϕ ∈ End k (M ) then ∇(ϕ) is the endomorphism of M defined by If F is the matrix of ϕ with respect to e, one can check that ∇(ϕ)(e) = e(∂F − AF + F A). If we denote by square matrices F the elements of End(M ) with respect to the basis induced by e, then the linear differential system associated to End(M) with respect to the basis induced by e is ∂F = AF − F A.
The horizontal elements of End(M) are the elements ϕ ∈ End k (M ) such that ∇(ϕ(m)) = ϕ(∇(m)), ∀m ∈ M. Thus the horizontal elements of the differential module End(M) are exactly the k-endomorphisms of M which commute with ∇. They are called differential module endomorphisms of M. They form a C-algebra denoted by E(M) which is called the eigenring of M.
Among all the possible constructions, the differential module M ⊗ k M * will play a special role in the exposition below. If we identify it canonically to End(M), then the linear differential system associated to M ⊗ k M * in the basis induced by e is exactly ∂F = AF − F A. The set E([A]) of matrices F ∈ M n (k) satisfying the above matrix differential equation is called the eigenring of the system [A] : ∂ y = A y. It is isomorphic (as a C-algebra) to E(M).

Picard-Vessiot extensions
We introduce very briefly some notions of differential Galois theory, with the main purpose of fixing the notation. There exists several detailed introduction to the topic. We refer to [vdPS03] for a general introduction and to [AMCW13] for more specific notions which are needed in this paper.
Let us consider the linear differential system [A] : ∂ y = A y. To any such system we can attach a k-algebra R, with an extension of ∂, having the following properties: 3. R has no proper non-trivial ideals stable by ∂, i.e., it is a simple differential ring.
We say that R is a Picard-Vessiot ring of k for [A]. It is an integral domain and its ring of constants R ∂ coincides with C. Its quotient field K = Frac(R) is generated (as a field) by the entries of U and its subfield of constant is again C. We call K a Picard-Vessiot extension of k for [A]. . Hence one can define a Picard-Vessiot extension of k for a given differential module M = (M, ∇) as a Picard-Vessiot extension of k for the differential system associated to M with respect to a basis of M .
2. Let K be a Picard-Vessiot extension of k for differential module M = (M, ∇). The Leibnitz rule allows to endow M ⊗ k K with a natural structure of differential module over K, which will be denoted by M ⊗ k K. The definition of K implies that M ⊗ k K is trivial, i.e., M ⊗ k K admits a basis over K of horizontal elements. One can show that if a module is trivial, then all its algebraic constructions and their subquotients are trivial. See [vdPS03, Exercice 2.12,5].
3. Let V := (M ⊗ k K) ∇ be the C-vector space of the horizontal elements of M ⊗ k K. As already pointed out, it has dimension n.
We give now a definition that we will use in the main theorem.
i.e., an element m ⊗ g with m ∈ M and g ∈ K such that ∇(m ⊗ g) = 0. If m is a horizontal element in some construction Constr(M ), then it is called an invariant of M.
For the convenience of the reader, we reprove the following classical lemma that we will use in this work.
Lemma 2.5. Let M = (M, ∇) be a differential module and K a Picard-Vessiot extension of k for M. Fix a basis e of M and let [A] : ∂ y = A y be the associated linear differential system. The following statements are equivalent: Proof. Let us assume that there exists m ∈ M such that ∇(m) = f m, for some f ∈ k. Then the line L generated by m over k is a differential module L, and L ⊗ k K ⊂ M ⊗ k K is a trivial differential module (see Remark 2.3).
Thus there exists g ∈ K, g = 0, such that m ⊗ g is a horizontal element of L ⊗ k K. It follows that we have: Let us now suppose that we are in the situation described in the second assertion. If we define m := e v, then Remark 2.6. In particular, Lemma 2.5 above implies that, if m ⊗ g is a semi-invariant, then necessarily ∇(m) = (−∂(g)/g)m, or equivalently m generates a ∇-stable line L contained in a construction Constr(M ) of M . If m ′ = hm, for some h ∈ k, then ∇(m ′ ) = (−∂(g)/g + ∂(h)/h)m ′ and ∇(m ′ ⊗ (g/h)) = 0. Moreover, if m ⊗ g is a semi-invariant of M, such that m generates a line L in a construction of M , and c ∈ C is a nonzero constant, then c(m ⊗ g) is another semi-invariant, corresponding to the same line L. One can prove that all the semi-invariants can be obtained in this way.
Roughly speaking, semi-invariants of M correspond to exponential solutions of ∂ y = constr(A) y and invariants correspond to rational solutions. For more details on these definitions, see [AMCW13,§3.4].
For further reference we recall the following lemma: : ∂ y = A y be the linear differential system associated to M with respect to a fixed basis e. We suppose that there exist an algebraic extension k ′ /k and a matrix P ∈ GL n (k ′ ) such that, for any invariant of M given by a horizontal element in some construction Constr(M) of coordinates v with respect to the basis Constr(e), the vector Constr(P ) −1 v has constant coordinates. Then the same property holds for any invariant of M ⊗ k k, where k is the algebraic closure of k.

The differential Galois group
The differential Galois group of In fact, ϕ(U ) must be a solution matrix of [A], therefore U −1 ϕ(U ) must be an invertible matrix with constant coefficients. The choice of another fundamental matrix of solutions leads to a conjugated representation. We will sometimes simply call G the differential Galois group Gal ∂ ([A]), identifying it with its image via the morphism above and without mentioning the matrix U , unless the context makes it necessary. We are not explaining here any result on the Galois correspondence and we refer the interested reader to the literature. For the purpose of this paper, we mostly need to know that, if k • is the relative algebraic closure of k in K, then K/k • is a Picard-Vessiot extension for [A] over k • . Moreover the field of constants is still C and we have Aut ∂ (K/k • ) = G • , where G • is the connected component of G containing 1. This means that the differential Galois group of [A] over k • coincides with the automorphisms of G that fix k • and can be identified with G • .
The Galois-Lie algebra gal([A]). Since the differential Galois group G = Gal ∂ ([A]) is an algebraic group over C, one can naturally consider its Lie algebra g := gal([A]) called Galois-Lie algebra, that is the tangent space to G • at 1. If we look at its C-rational points, we have: As it is an algebraic Lie algebra over C, gal([A]) is generated as a C-vector space by a finite subset of M n (C). It will be useful to notice that the same subset of matrices of M n (C) generates, as a k-vector space, the algebra gal(

Characterization of reduced forms
We keep the notation of the previous section.
Definition 3.1. We say that a linear differential system [A] : ∂ y = A y is in reduced form when A belongs to gal([A])(k). Let M = (M, ∇) be a differential module and e be a k-basis of M . We say that e is a reduced basis if the system associated to M with respect to the basis e is in reduced form.
As pointed out in the introduction, Kolchin and Kovacic has proved that a reduced form always exists on a finite extension of k. See [vdPS03, Proposition 1.31]. Criteria for reduced forms have been studied by Aparicio, Compoint and Weil in [AMCW13]. Their main result is generalized in Theorems 3.2 and 3.8 below.
Theorem 3.2. Let [A] : ∂ y = A y be the linear differential system associated to a differential module M = (M, ∇) over k, with respect to a fixed basis e. The following assertions are equivalent: If moreover M is completely reducible (i.e. it is direct sum of irreducibles), then the assertions above are equivalent to: 4. Any invariant of M has constant coordinates.
Proof. Lemma 2.5 shows that "1 ⇔ 2" is a reformulation of [AMCW13, Theorem 1]. Since "3 ⇒ 2" is tautological, it is enough to prove that "2 ⇒ 3". Let N = (N, ∇) be a construction of M and W be a ∇-stable sub-k-vector space of N of dimension d. It follows that ∧ d W is ∇-stable sub-k-vector space of dimension 1 of ∧ d N . By assumption, there exists a non-zero element of ∧ d W , whose coordinates w with respect to the basis induced by e on ∧ d N are in C. Hence, in the basis induced by e, the map is represented by a matrix with coefficients in C. It follows that ker Ψ has a basis of vectors with coordinates in C, with respect to the basis induced by e. Since ker Ψ = W , we have proved "2 ⇒ 3". By definition, a horizontal element of a construction Constr(M) corresponds to a solution of the associated differential system ∂ y = constr(A) y with respect to the basis Constr(e Definition 3.3. We say that k ′ is a reduction field when M ⊗ k k ′ admits a reduced basis.
When k ′ is a reduction field, then the differential Galois group of M ⊗ k k ′ is connected, see [AMCW13, Lemma 32], and the Galois correspondence implies that k ′ ∩ K = k • .
Remark 3.4. Let m ∈ Constr(M ) generate a line which is invariant by ∇ and let e be a reduced basis. By Theorem 3.2 above, we can choose m such that m = Constr(e) v, with v ∈ C n . Moreover since ∇(m) = f m for some f ∈ k, there exists g ∈ K verifying ∂(g) = f g. This means that g v is a solution vector of the system associated to Constr(M) with respect to Constr(e). We may say, a little bit informally, that invariants and semi-invariants have constant coordinates with respect to this reduced basis.

Gauge transformation to a reduced form with local conditions
Assumption 3.6. In this section, we suppose that the field k is a subfield of the field of meromorphic functions over a region D of C in the variable x such that x ∈ k and ∂ = d dx . Remark 3.7. The field of rational functions k = C(x) satisfies the assumption above, as well as most differential fields occurring in the concrete examples. Indeed let k be any differential field. Then the entries of the matrix A of the linear differential system [A] : ∂ y = A y, generate a differential field k, which is a finitely generated differential extension of Q. Seidenberg's embedding theorem (see [Sei56,Sei52]) ensures that k can be embedded isomorphically in a differential field of meromorphic functions on an open region D of C.
We consider a linear differential system [A] : ∂ y = A y, with coefficients in k. For all points x 0 ∈ D such that x 0 is not a pole of A, the system [A] has a fundamental matrix U (x) with the following properties: Theorem 3.8. Let us consider a linear differential system ∂ y = A y, defined over a field k, associated to a completely reducible differential module M with respect to a fixed basis e. We choose a point x 0 ∈ D such that A does not have a pole at x 0 . Then, there exists a finite extension k ′ of k and P ∈ GL n (k ′ ), such that: . As a consequence, the latter set of equations generates a proper ideal in the ring of polynomials in n 2 variables, which is finitely generated because of the noetherianity. The Nullstellensatz then ensures that there exists a solution P ∈ GL n (k ′ ), where k ′ /k is a finite extension of k. Hence the first property is verified. The second property then follows from Lemma 2.7 and Theorem 3.2.
Remark 3.10. Assume that M admits a reduced basis e. Consider an invariant m in some construction Constr(M) having (constant) coordinates v ∈ C N with respect to the basis Constr(e). Then we have v = Constr( U (x)) c and the constant vector c is an invariant (in the usual sense of representation theory) of the Galois group Gal ∂ ([A]) in its representation induced by U (x). Now, by construction, U (x 0 ) = Id (the identity matrix); as Constr acts as a group morphism, we see that In a reduced basis, this observation allows to identify the invariants of M and those of Gal ∂ ([A]).

The intrinsic Galois-Lie algebra of Katz
Let M = (M, ∇) be a differential module over k. Remark 4.2. For ϕ ∈ End(M ), the functor constr acts on ϕ as a Lie algebra morphism, see the points before example 2.2. In particular, for any invariant m ∈ Constr(M ) and ϕ ∈ g Katz , we have constr(ϕ)(m) = 0.
By noetherianity, g Katz is a stabilizer of a finite family of differential modules contained in some constructions of M. This shows that g Katz is an algebraic Lie algebra. It is the central object of the famous Grothendieck-Katz conjecture on p-curvatures. The following properties of g Katz are known, see [Kat82,And04,vdPS03], but are reproved for self-containedness. 2. If m generates a ∇-stable line in some construction of M , which defines g Katz as a stabilizer, and ϕ ∈ g Katz , then constr(ϕ)(m) = αm and ∇(m) = βm, for some α, β ∈ k. We conclude that: Therefore ∇(constr(ϕ)) belongs to g Katz .
3. The Tannakian correspondence will show that g Katz ∼ = (g ⊗ k K) G . Indeed, we have seen that g Katz is the stabilizer of a line L. Then L C := (L ⊗ C K) ∇ is a C-line defined in the corresponding construction on the Cvector space of solutions V : In fact, the same argument shows that (g ⊗ C K) G stabilizes any differential module in a construction and thus (g ⊗ C K) G = g Katz . This ends the proof.
Theorem 4.4. Let M = (M, ∇) be a differential module over k, G the differential Galois group of M, g the Galois-Lie algebra and g Katz the Katz algebra. Let k ′ be a reduction field so that M ⊗ k k ′ admits a reduced basis e. If M is completely reducible, then g ⊗ C k ′ = g Katz ⊗ k k ′ .
Proof. As M is completely reducible (or equivalently the Galois group G is reductive), [AMCW13,Lemma 26] and Remark 3.10 show that the Galois-Lie algebra g is defined as the Lie algebra of matrices which annihilate a given invariant m (a Chevalley invariant ) in some construction Constr(M ) ⊗ k k ′ . As M is completely reducible, Theorem 3.2 implies that, in the reduced basis e, the invariant m has constant coefficients, i.e., we have m = Constr(e) v, with v ∈ C n . Thus (see Remark 3.10), a matrix N = (n ij ) i,j is in g if and only if constr(N ) v = 0. This relation yields a system of linear equations L((n i,j ) i,j ) = 0 for the entries (n i,j ) i,j of N , with coefficients in C. Now m is a Chevalley invariant of ∇ so that it is annihilated by any matrix N = ( n ij ) i,j ∈ g Katz ⊗ k k ′ with coefficients in k ′ . It follows that the entries ( n i,j ) i,j of N are determined by the same system L(( n i,j ) i,j ) with constant coefficients. Consequently, g ⊗ C k ′ and g Katz ⊗ k k ′ share the same constant basis so that g ⊗ C k ′ = g Katz ⊗ k k ′ .
We say that g Katz is defined over C when, in the representation associated with the basis induced by a basis e of M over k, it is generated by constant matrices (i.e., when it is a k-form of a C-algebra).
Corollary 4.5. Let k ′ be a reduction field so that M ⊗ k k ′ admits a reduced basis e. Then g Katz ⊗ k k ′ is defined over C. Moreover, if A is the matrix of the associated linear differential system with respect to e, then A ∈ g Katz ⊗ k k ′ .
Proof. Let L be a ∇-invariant line in some Constr(M ) such that g Katz is the stabilizer of L. By Theorem 3.2, we can choose a generator m of L ⊗ C k ′ whose coordinates are constant with respect to the reduced basis. This shows that g Katz ⊗ k k ′ (but not g Katz in general) is defined over C. The fact that A ∈ g Katz ⊗ k k ′ follows from the definition of a reduced basis and from Theorem 4.4.
Example 4.6. We illustrate the previous results on an example where everything can be checked by hand calculations. Let The Galois group G is a central extension of the infinite dihedral group (see [AMCW13, Example 6.1]). The connected component G • of G containing 1 is the multiplicative group G m with Lie algebra g m generated by Furthermore, using [BCDVW16], we can see that the Katz algebra is 1-dimensional and it is generated by We have ∇(N 1 ) := N ′ 1 − [A, N 1 ] = 1 2x N 1 . Over k = C(x), no nonzero multiple of N 1 is conjugated to a constant matrix. However, over the finite extension k ′ := C( √ x), √ xN 1 is is conjugated to a constant matrix. Indeed, we have N 1 = P −1 DP with We see that g Katz ⊗ k k ′ is now generated by the (constant) generator of the Galois-Lie algebra g m and, applying the gauge transformation P , we have the reduced form This shows that g Katz ⊗ k k ′ = g m ⊗ C k ′ and P [A] ∈ g Katz ⊗ k k ′ . Note that g Katz is not defined over C whereas g Katz ⊗ k k ′ is.   In [And04], André warns that spaces which are stable under g Katz may not be stable under ∇. We can see this easily using reduced forms. Consider a linear differential system [A] : ∂ y = A y having all its solutions algebraic over k. Then g Katz = 0 so that anything is stable under g Katz . In a reduced basis, the matrix of the linear differential system is the zero matrix and ∇ coincides with ∂ = d dx . A random vector in k n is not stable under ∇ even though it is stable under g Katz . However, any line defined over C is clearly stable under ∂ and hence under ∇. The next result builds on this observation to characterize which spaces, among those which are stable under g Katz , are stable under ∇.
Proposition 4.8. Let M = (M, ∇) be a completely reducible differential module over k and k ′ denote a reduction field for M. Let W be a subspace of a construction Constr(M ) ⊗ k ′ . Then W is stable under ∇ if and only if both conditions below are fulfilled: 1. W is stable under g Katz ⊗ k k ′ ; 2. W is defined over C.
Proof. The "only if" part follows from the definition of the Katz algebra (Point 1.) and from Theorem 3.2 (Point 2.). Now, assume that W is stable under g Katz ⊗ k k ′ and defined over C. Let C 1 , . . . , C s denote a constant basis of W . Let A be the matrix of the linear differential system associated with M ⊗ k k ′ in a reduced basis. We have A = j f j (x)N j , where f j (x) ∈ k ′ and the N j form a (constant) basis of g Katz (and of g). As the C i are constant, ∇ acts on them via ∇(C i ) = −constr(A)C i and thus ∇(C i ) = − j f j (x)constr(N j )C i . Now, by hypothesis W is stable under g Katz ⊗ k k ′ so that constr(N j )C i is a linear combination (over k ′ ) of the C l . It follows that ∇(C i ) is in W as announced.
Each space in Constr(V ) which is invariant under g is in (Tannakian) correspondence with a submodule of Constr(M ), invariant under ∇ and hence under g Katz . A reciprocal property would be to characterize, among all subspaces in any Constr(M) that are stable under g Katz , which ones are stable under ∇ (and hence are in Tannakian correspondence with a g-module in Constr(V ). Proposition 4.8 shows that this it is possible to do that at the cost of extending scalars. When we study the subspaces of Constr(M )⊗ k k ′ which are invariant under g Katz ⊗ k k ′ , the ones which are invariant under ∇ (and hence in tannakian correspondence with a g-module) are exactly those which admit a constant basis.