Large Fields in Differential Galois Theory
Annette Bachmayr, David Harbater, Julia Hartmann, Florian Pop
aa r X i v : . [ m a t h . A C ] J a n LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY
ANNETTE BACHMAYR, DAVID HARBATER, JULIA HARTMANN AND FLORIAN POP
Abstract.
We solve the inverse differential Galois problem over differential fields with alarge field of constants of infinite transcendence degree over Q . More generally, we show thatover such a field, every split differential embedding problem can be solved. In particular, wesolve the inverse differential Galois problem and all split differential embedding problemsover Q p ( x ). Introduction
Large fields play a central role in field arithmetic and modern Galois theory, providing anespecially fruitful context for investigating rational points and extensions of function fieldsof varieties. In this paper we study differential Galois theory over this class of fields.Differential Galois theory, the analog of Galois theory for linear differential equations, hadlong considered only algebraically closed fields of constants; but more recently other con-stant fields have been considered (e.g. see [AM05], [And01], [BHH16], [CHvdP13], [Dyc08],[LSP17]). Results on the inverse differential Galois problem, asking which linear algebraicgroups over the constants can arise as differential Galois groups, have all involved constantfields that happen to be large. In this paper, we prove the following result (see Theorem 3.2):
Theorem A. If k is any large field of infinite transcendence degree over Q , then every linearalgebraic group over k is a differential Galois group over the field k ( x ) with derivation d/dx . In differential Galois theory (as in usual Galois theory), authors have considered embeddingproblems, which ask whether an extension with a Galois group H can be embedded into onewith group G , where H is a quotient of G . (For example, see [MvdP03], [Hrt05], [Obe03],[Ern14], [BHHW16], [BHH17].) In order to guarantee solutions, it is generally necessary toassume that the extension is split (i.e., G → H has a section). In this paper we prove thefollowing result about split embedding problems over large fields (see Theorem 4.3): Theorem B. If k is a large field of infinite transcendence degree over Q , then every splitdifferential embedding problem over k ( x ) with derivation d/dx has a proper solution. Date : January 12, 2018.The first author was funded by the Deutsche Forschungsgemeinschaft (DFG) - grant MA6868/1-1 andby the Alexander von Humboldt foundation through a Feodor Lynen fellowship. The second and thirdauthor were supported on NSF collaborative FRG grant: DMS-1463733; additional support was providedby NSF collaborative FRG grant DMS-1265290 (DH) and a Simons Fellowship (JH). The fourth author wassupported by NSF collaborative FRG grant DMS-1265290.
Mathematics Subject Classification (2010): 12H05, 12E30, 20G15 (primary); 12F12, 14H25 (secondary).
Key words and phrases.
Picard-Vessiot theory, large fields, inverse differential Galois problem, embeddingproblems, linear algebraic groups. he class of large fields, which was introduced by Pop [Pop96], includes in particular R , Q p , k (( t )), k (( s, t )), algebraically closed fields, and pseudo-algebraically closed fields. Resultsabout this class of fields thus have broad applications. Moreover, in usual Galois theory, thekey properties of large fields turn out to be just what is needed in order to carry out proofsof the inverse problem over function fields of arithmetic curves and to prove that finite splitembedding problems over such function fields have proper solutions. We refer the reader to[Pop14] for a further discussion.Theorem A generalizes a number of known results on the differential inverse Galois problem(e.g., in the cases of k being algebraically closed, or real, or a field of Laurent series in onevariable), as well as yielding other results (e.g., the cases of PAC fields, Laurent series inmore than one variable, and the p -adics). Moreover, we generalize this result further from k ( x ) to all differential fields with field of constants k that are finitely generated over k (Corollary 3.4). In particular, we solve the inverse differential Galois problem over Q p ( x )and more generally, over all differential fields that are finitely generated over Q p and havefield of constants Q p .In fact, our proof shows somewhat more. Given a differential field k of characteristiczero, and a linear algebraic group G over k , there exists an integer n such that for anylarge overfield k/k of transcendence degree at least n , there is a Picard-Vessiot ring over k ( x ) with differential Galois group G k (see Theorem 3.2(a)). A similar assertion holds in thesituation of Theorem B; see Theorem 4.3(a). In both cases, the integer n could in principlebe computed from the input data.Theorems A and B provide differential analogs of results in usual Galois theory about largefields; and our strategy here, like the one there, relies on reducing to the case of Laurentseries fields. On the one hand, Laurent series fields are large; on the other hand, any largefield k is existentially closed in the Laurent series field k (( t )). In usual Galois theory, theproof that every finite split embedding problem for k ( x ) has a proper solution if k is large(see [Pop96, Main Theorem A], [HJ98, Theorem C], [HS05, Theorem 4.3]) involved firstproving such a result for large fields of the form k = k (( t )); and a result in that case(see [Pop96, Lemma 1.4], [HJ98, Proposition B], [HS05, Theorem 4.1]) can be proven bymeans of patching, due to such fields being complete. In the current manuscript (wherewe restrict to fields of characteristic zero, as is common in differential Galois theory), webuild on [BHH17, Theorem 4.2], where it was shown that proper solutions exist to everysplit differential embedding problem over k (( t ))( x ) that is induced from a split embeddingproblem over k ( x ). (That assertion in turn built on results in [BHH16] and [BHHW16],which relied on patching methods.) Since Laurent series fields are large, the main result inthis current paper also yields a new result over Laurent series fields, namely that in [BHH17]the hypothesis on the embedding problem being induced from k ( x ) can be dropped.As in the case of embedding problems over large fields in usual Galois theory, it is necessaryin our main result to assume that the embedding problem is split. In usual Galois theory,this is because in order for all finite embedding problems over k ( x ) to have proper solutions,it is necessary by [Ser02, I.3.4, Proposition 16] for k ( x ) to have cohomological dimensionat most one; and hence for k to be separably closed (not merely large). In differentialGalois theory, every finite regular Galois extension of k ( x ) is a Picard-Vessiot ring for a nite constant group, and so the same reason applies. On the other hand, in usual Galoistheory, every finite embedding problem over k ( x ) (even if not split) has a proper solution if k is algebraically closed, and in fact has many such solutions in a precise sense; this impliesthat the absolute Galois group of k ( x ) is free of rank card( k ) (see [Pop95] and [Har95]).In the differential situation, it was shown in [BHHW16, Theorem 3.7] that all differentialembedding problems over C ( x ) have proper solutions. The main theorem of the currentpaper combined with Proposition 3.6 of [BHHW16] implies that for any algebraically closedfield k of infinite transcendence degree over Q , every differential embedding problem over k ( x ) has a proper solution (Corollary 4.5).This manuscript is organized as follows. Section 1 begins with a short summary of ex-amples and properties of large fields. Proposition 1.3 in that section, which was proven byArno Fehm, states that the function field of a smooth connected variety over a subfield of alarge field can be embedded into that large field under certain hypotheses. This propositionand its corollary are key to reducing to the case of Laurent series fields in Sections 3 and 4.Section 2 reviews Picard-Vessiot theory over arbitrary fields of constants and discusses whatit means for a Picard-Vessiot ring to descend to a subfield of the given differential field.Sections 3 and 4 give the solutions to differential inverse problem and differential embeddingproblems, respectively. In each case, the main ingredient is a proposition proving that allinput data is defined over a rational function field over a finitely generated subfield of thefield of constants.We thank William Simmons and Henry Towsner for helpful discussions.1. Embeddings into large fields
The aim of this section is to prove that certain subfields of the Laurent series field k (( t ))can be embedded into k if k is a large field. We begin by recalling the definition andbasic properties of large fields. For field extensions l/k , we write td( l/k ) to abbreviate thetranscendence degree. Definition 1.1.
A field k is large if for every smooth k -curve the existence of one k -rationalpoint implies the existence of infinitely many such points.Basic examples of large fields include algebraically closed fields (or more generally, PACfields) and fields that are complete with respect to a nontrivial absolute value (see e.g.[Pop14]).In particular, C , R , Q p for p a prime, and the Laurent series field k (( t )) over an arbitraryfield k are all large.More generally, fraction fields of domains that are Henselian with respect to a non-trivialideal are large by [Pop10], Theorem 1.1. This includes Henselian valued fields (with re-spect to non-trivial valuations), for example Puiseaux series fields, as well as fraction fields k (( t , . . . , t n )) of power series rings in several variables. In Remark 1.5 we give more exam-ples of large fields.There are a number of characterizations of large fields (see [Pop14]). We list some of themhere for ease of citation. Proposition 1.2.
A field k is large if and only if it satisfies one of the following properties: a) Every smooth k -curve with a rational point has card( k ) rational points.(b) Every smooth k -variety X satisfies either X ( k ) = ∅ or X ( k ) is dense in X .(c) The field k is existentially closed in its Laurent series field k (( t )) . In particular, we will use that if k is a large field and X/k is a smooth variety which hasa k (( t ))-point, then X has a k -point.The following result was proven in [Feh11]; see Theorem 1 and Lemma 4 there. Below wegive another proof, using a different strategy. Proposition 1.3.
Let k be a large field, l ⊆ k be a subfield, and V be a smooth connected l -variety with function field L = l ( V ) and V ( k ) non-empty. Suppose that td( k/l ) > dim( V ) .Then the canonical embedding of fields l ֒ → k can be prolonged to an embedding of fields L ֒ → k . Equivalently, there exist k -rational points dominating the generic point of V .Proof. Since V is smooth and connected, it is also integral. Hence the given k -rational pointis contained in a nonempty (dense) affine open subvariety which is smooth and integral,and we may replace V by that subvariety (which we again call V ). Let R := l [ V ] be itscoordinate ring; then L = Frac( R ). Given any k -point of Spec( R ) (i.e., a point x ∈ Spec( R )together with an l -algebra map ı : κ ( x ) ֒ → k ), let d x := td (cid:0) κ ( x ) /l (cid:1) . Choose ( x, ı ) as abovesuch that d x is maximal; hence d x ≤ dim( V ). It suffices to show that d x = dim( V ), sincethen x is the generic point of V .Suppose to the contrary that d x < dim( V ). Let u := ( u , . . . , u r ) be a system of func-tions in R such that its image ˜ u = (˜ u , . . . , ˜ u r ) under the reduction map R → κ ( x ) is atranscendence basis of κ ( x ) over l . The composition l [ u ] → R → R/I x = κ ( x ) is injective,hence l [ u ] ∩ I x = { } , where I x ⊳ R is the prime ideal defining x . Let l = l ( u ) = Frac( l [ u ])and R := R ⊗ l [ u ] l . The l -embedding R ֒ → R defines a dominant morphism of schemes V := Spec R ֒ → Spec R = V , with V a smooth l -variety. Since κ ( x ) is an algebraic fieldextension of l , x ∈ V is the image of a closed point of V . Hence ı : κ ( x ) → k defines a k -point x ∈ V ( k ). Let ˜ l be the algebraic closure of l in k . Since td( l /l ) < td( L/l ) =dim( V ) td( k/l ), it follows that ˜ l is strictly contained in k . Hence by Theorem 3.1, 2) from[Pop14], V has a k -point that is not an ˜ l -point. The associated point z ∈ V = Spec( R )is equipped with an l -embedding ı : κ ( z ) ֒ → k whose image is thus not algebraic over l .Viewing z as a point of V via V ֒ → V , we obtain a contradiction because d z = td (cid:0) κ ( z ) /l (cid:1) = td (cid:0) κ ( z ) /l (cid:1) + td( l /l ) > td( l /l ) = td (cid:0) κ ( x ) /l (cid:1) = d x . (cid:3) Corollary 1.4.
Let k be a large field, k ⊆ k and k ⊆ k (( t )) be subfields with k ⊆ k , td( k /k ) td( k/k ) and k /k finitely generated. Then there exists a k -embedding k ֒ → k .In particular, if k ⊆ k are fields such that k is large and td( k/k ) is infinite, then for everyfinitely generated field extension k /k with k ⊆ k (( t )) there is a k -embedding k ֒ → k .Proof. Let k be as in the statement of the corollary. Since K := k (( t )) is separably gener-ated over k and k is relatively algebraically closed in K (that is, K /k is a regular fieldextension), it follows that k is separably generated over k and k is relatively algebraicallyclosed in k as well. Equivalently, there exists a geometrically integral smooth k -variety V with k ( V ) = k . For such a V , V ( k ) is non-empty (because it contains the generic pointof V ) and thus V ( K ) is non-empty as well since K ⊇ k . Therefore, so is V ( K ), since = k (( t )) ⊇ k (( t )) = K . By Proposition 1.2(c), k is existentially closed in K = k (( t ));and so V ( k ) is also non-empty. An application of Proposition 1.3 yields a k -embedding k ֒ → k (with l of loc.cit. replaced by k ). (cid:3) We conclude this section by providing some examples of large fields of prescribed tran-scendence degree:
Remark 1.5.
Let κ be a Henselian valued field with respect to a non-trivial valuation v .Every relatively algebraically closed subfield k ⊆ κ is henselian with respect to the restriction v | k of v to k . Thus if v | k is non-trivial, then k is large. This gives a recipe to construct largesubfields k ⊆ κ of any positive transcendence degree d bounded by that of κ .Explicit examples are ( d = 0) the algebraic p -adics (i.e., the relative algebraic closure of Q in Q p ) and ( d = 1) the algebraic Laurent series over Q (i.e., the algebraic closure of Q ( t )in Q (( t ))). An example of countable infinite transcendence degree is the algebraic closure of Q ( t, x , x , . . . ) in Q ( x , x , . . . )(( t )). By taking a bigger set of variables x i , we obtain largefields of any given uncountable transcendence degree.2. Picard-Vessiot theory
Let F be a differential field of characteristic zero with field of constants K . Classically,differential Galois theory over F is set up under the assumption that K is algebraicallyclosed; we refer to [vdPS03] for this case. We start this section by recapitulating differentialGalois theory over differential fields with arbitrary fields of constants. Details can be foundin [Dyc08] and [BHH16].If R is a differential ring, we let C R denote its ring of constants. The field of constants K = C F is relatively algebraically closed in F . Consider a matrix A ∈ F n × n and thecorresponding linear differential equation ∂ ( y ) = Ay . A fundamental solution matrix for thisequation is a matrix Y ∈ GL n ( R ) with entries in some differential ring extension R/F suchthat ∂ ( Y ) = A · Y ; i.e., the columns of the matrix Y form a fundamental set of solutions. A Picard-Vessiot ring for ∂ ( y ) = Ay is a simple differential ring extension R/F with C R = K such that R is generated by the entries of a fundamental solution matrix Y ∈ GL n ( R )together with the inverse of its determinant. For short, we write R = F [ Y, det( Y ) − ]. Itfollows from the differential simplicity that R is an integral domain with C Frac( R ) = C F : Lemma 2.1.
Let R be a simple differential ring containing Q . Then R is an integral domain,its field of constants is a field and Frac( R ) has the same field of constants.Proof. As in [vdPS03, Lemma 1.17.1], it can be shown that every zero divisor is nilpotentand that the radical ideal is a differential ideal (see also [Dyc08, Lemma 2.2]). Hence R isan integral domain. If x ∈ Frac( R ) is constant, then I = { a ∈ R | ax ∈ R } is a non-zerodifferential ideal in R and thus 1 ∈ I and x ∈ R . Hence C Frac( R ) = C R and in particular, C R is a field. (cid:3) Conversely, if R is generated by a fundamental solution matrix, then differential simplicityfollows from being an integral domain together with C Frac( R ) = F . If C F is algebraicallyclosed, this is a well-known criterion. For arbitrary fields of constants it was proven in[Dyc08, Cor. 2.7]: roposition 2.2. Let
L/F be an extension of differential fields with C L = C F and considera matrix A ∈ F n × n . Assume that there exists a matrix Y ∈ GL n ( L ) with ∂ ( Y ) = AY . Then R = F [ Y, det( Y ) − ] ⊆ L is a Picard-Vessiot ring for the differential equation ∂ ( y ) = Ay . The differential Galois group
Aut ∂ ( R/F ) of a Picard-Vessiot ring
R/F is defined as thefunctor G from the category of K -algebras to the category of groups, defined by G ( S ) :=Aut ∂ ( R ⊗ K S/F ⊗ K S ), the group of ( F ⊗ K S )-linear automorphisms of ( R ⊗ K S ). Here, weequip the K -algebra S with the trivial derivation. It can be shown that Aut ∂ ( R/F ) is repre-sented by the K -Hopf algebra C R ⊗ F R = K [( Y − ⊗ Y ) , det( Y − ⊗ Y ) − ], where ( Y − ⊗ Y ) isan abbreviation for the matrix product ( Y − ⊗ · (1 ⊗ Y ). Hence the functor Aut ∂ ( R/F ) isan affine group scheme of finite type over K , and thus (since char( K ) = 0) a linear algebraicgroup over K . Note that if K is algebraically closed, then Aut ∂ ( R/F ) is determined by its K -rational points Aut ∂ ( R/F )( K ) = Aut ∂ ( R/F ), the group of F -linear differential automor-phisms of R , which is classically the definition of the differential Galois group. However,if K is not algebraically closed, then Aut ∂ ( R/F ) does not contain enough information onAut ∂ ( R/F ) in general.In the remainder of this section, we study the behavior of Picard-Vessiot rings underextensions of the constants. The following is a well-known statement in differential algebra;see for example [Mau10, Lemma 10.7] for a proof.
Lemma 2.3.
Let R be a simple differential ring with field of constants K and let S be a K -algebra which we equip with the trivial derivation. Then there is a bijection between thedifferential ideals in R ⊗ K S and the ideals in S , given by I I ∩ S for differential idealsin R ⊗ K S and J R ⊗ K J for ideals in S . In particular, if S is a field then R ⊗ K S is asimple differential ring. If G is a linear algebraic group over a field K and K ′ /K is a field extension, we let G K ′ denote the base change of G from K to K ′ . If K ′ /K is algebraic and R/F is a Picard-Vessiotring with differential Galois group G , then F ′ = F ⊗ K K ′ is a differential field extension of F and R ⊗ K K ′ is a Picard-Vessiot ring over F ′ with differential Galois group G K ′ . Indeed,since K is algebraically closed in F , F ⊗ K K ′ is an integral domain and as K ′ /K is algebraic,it is an algebraic field extension of F ; thus the derivation extends uniquely to F ′ (with fieldof constants C F ′ = K ′ ). Moreover, R ⊗ K K ′ is generated over F ′ by the same fundamentalsolution matrix as R/F , C R ⊗ K K ′ = K ⊗ K K ′ = C F ′ and R ⊗ K K ′ is a simple differential ringby Lemma 2.3. Finally, Aut ∂ ( R ⊗ K K ′ /F ′ ) = Aut ∂ ( R/F ) K ′ is immediate from the definition.If K ′ /K is a non-algebraic field extension, F ⊗ K K ′ is not a field but merely an integraldomain and it is slightly more complicated to extend the constants from K to K ′ . Weconsider the differential field extension F ′ = Frac( F ⊗ K K ′ ) of F . The following propositionshows that if R/F is a Picard-Vessiot ring with differential Galois group G , R ⊗ F F ′ is aPicard-Vessiot ring over F ′ with differential Galois group G K ′ . Note that this generalizes theconstruction in the case when K ′ /K is algebraic: If K ′ /K is algebraic, then Frac( F ⊗ K K ′ ) = F ⊗ K K ′ and R ⊗ F F ′ ∼ = R ⊗ K K ′ . Proposition 2.4.
Let F be a field of characteristic zero with field of constants K and let R/F be a Picard-Vessiot ring with differential Galois group G . Let K ′ /K be a field extension nd define F ′ = Frac( F ⊗ K K ′ ) and R ′ = R ⊗ F F ′ . Then F ′ is a differential field extensionof F with C F ′ = K ′ and R ′ is a Picard-Vessiot ring over F ′ with Galois group G K ′ .Proof. The derivation on F extends canonically to F ⊗ K K ′ and hence to F ′ by consideringelements in K ′ as constants. By Lemma 2.3, F ⊗ K K ′ is a simple differential ring and thuswe can apply Lemma 2.1 to obtain C F ′ = C F ⊗ K K ′ = K ′ .Since R/F is a Picard-Vessiot ring, there exists a differential equation ∂ ( y ) = Ay over F and a fundamental solution matrix Y ∈ GL n ( R ) with R = F [ Y, det( Y ) − ]. We identify R with a subring of R ′ and obtain R ′ = F ′ [ Y, det( Y ) − ]. Let S denote the set of non-zeroelements in F ⊗ K K ′ . Then R ′ = R ⊗ F F ′ = R ⊗ F S − ( F ⊗ K K ′ ) = S − ( R ⊗ F ( F ⊗ K K ′ )).Hence R ′ = S − ( R ⊗ K K ′ ) and Frac( R ′ ) = Frac( R ⊗ K K ′ ) , where we identified R ⊗ K K ′ with the subring R ⊗ F ( F ⊗ K K ′ ) of R ′ . As R is simple, R ⊗ K K ′ is simple by Lemma 2.3 and has field of constants K ′ . It follows from Lemma 2.1that Frac( R ′ ) = Frac( R ⊗ K K ′ ) has field of constants K ′ = C F ′ . Therefore, R ′ is a Picard-Vessiot ring over F ′ by Proposition 2.2 (applied to L = Frac( R ′ )).Let G ′ denote the differential Galois group of R ′ /F ′ . We claim that G ′ = G K ′ . For every K ′ -algebra S , there is an injective group homomorphism G K ′ ( S ) = Aut ∂ ( R ⊗ K S/F ⊗ K S ) → G ′ ( S ) = Aut ∂ ( R ′ ⊗ K ′ S/F ′ ⊗ K ′ S )using that R ′ ⊗ K ′ S is a localization of R ⊗ K S . Conversely, every γ ∈ G ′ ( S ) restricts toan injective differential homomorphism R ⊗ K S → R ′ ⊗ K S . The matrix B = Y − γ ( Y ) ∈ GL n ( R ′ ⊗ K S ) has constant entries and is thus contained in GL n ( S ). Therefore, γ ( Y ) = Y B is contained in R ⊗ K S . Since R = F [ Y, det( Y ) − ], we conclude that γ ( R ⊗ K S ) = R ⊗ K S .Thus γ restricts to an element in G K ′ ( S ). Hence the homomorphism G K ′ ( S ) → G ′ ( S ) is abijection and it defines an isomorphism of linear algebraic groups G K ′ → G ′ . (cid:3) We record a special case here for later use.
Corollary 2.5.
Let k ( x ) be a rational function field of characteristic zero equipped withthe derivation d/dx and let K/k be a field extension. If
R/k ( x ) is a Picard-Vessiot ringwith differential Galois group G then R ⊗ k ( x ) K ( x ) is a Picard-Vessiot ring over K ( x ) withdifferential Galois group G K .Proof. Since K ( x ) = Frac( k ( x ) ⊗ k K ), the claim follows from Proposition 2.4. (cid:3) Definition 2.6.
For F and F ′ as in Proposition 2.4, we say that a Picard-Vessiot ring R ′ /F ′ descends to a Picard-Vessiot ring over F if there exists a Picard-Vessiot ring R/F togetherwith an F ′ -linear differential isomorphism R ⊗ F F ′ ∼ = R ′ .In particular, a Picard-Vessiot ring R over K ( x ) descends to a Picard-Vessiot ring over k ( x ) if there exists a Picard-Vessiot ring R /k ( x ) together with a K ( x )-linear differentialisomorphism R ∼ = R ⊗ k ( x ) K ( x ). . The inverse differential Galois problem
The aim of the next proposition is to show that the data associated to a Picard-Vessiotring over a rational function field over some field k is in fact already given over the rationalfunction field over a finitely generated subfield of k . This is technical but not surprising sinceall related objects (the linear algebraic group as well as the Picard-Vessiot ring itself) arefinitely generated. An analogous result for embedding problems can be found in the nextsection (Proposition 4.2 below). Proposition 3.1.
Let F = K ( x ) be a rational function field of characteristic zero withderivation ∂ = d/dx and let R/F be a Picard-Vessiot ring with differential Galois group G . Let further k ⊆ K be a subfield and let G be a linear algebraic group over k with ( G ) K = G . Then there is a finitely generated field extension k /k with k ⊆ K such that R/K ( x ) descends to a Picard-Vessiot ring R /k ( x ) with differential Galois group ( G ) k .Proof. As R is a finitely generated F -algebra, we can write R as a quotient of a polynomialring F [ X , . . . , X r ] by an ideal J . We fix generators g , . . . , g m of J : R = K ( x )[ X , . . . , X r ] / ( g , . . . , g m ) . We fix an extension of ∂ from F to F [ X , . . . , X r ] such that this derivation induces the givenderivation on R . In particular, J is a differential ideal in K ( x )[ X , . . . , X r ]. We can nowchoose a finitely generated field extension k/k with k ⊆ K such that(1) g i ∈ k ( x )[ X , . . . , X r ] for all i = 1 , . . . , m , and(2) ∂ ( X i ) ∈ k ( x )[ X , . . . , X r ] for all i = 1 , . . . , r and(3) R = K ( x )[ Y, det( Y ) − ] for a fundamental solution matrix Y ∈ GL n ( R ) with the propertythat all entries of Y have representatives in k ( x )[ X , . . . , X r ], and(4) the element in R represented by X i can be written as a polynomial expression over k ( x )in the entries of Y and det( Y ) − for all i = 1 , . . . , r .Property (2) implies that k ( x )[ X , . . . , X r ] is a differential subring of K ( x )[ X , . . . , X r ]. Set I = J ∩ k ( x )[ X , . . . , X r ]. Then I is a differential ideal in k ( x )[ X , . . . , X r ] and it contains g , . . . , g m by (1). As K ( x ) /k ( x ) is faithfully flat, I is thus generated by g , . . . , g m . Wedefine R = k ( x )[ X , . . . , X r ] /I . Hence R = k ( x )[ X , . . . , X r ] / ( g , . . . , g m )is a differential ring and as K ( x ) is flat over k ( x ), there is a K ( x )-linear isomorphism ofdifferential rings R ⊗ k ( x ) K ( x ) ∼ = R. Let c ∈ C R . As C R = K , there exists an a ∈ K such that we have c ⊗ ⊗ a in R ⊗ k ( x ) K ( x ). Thus a ∈ k ( x ) and c = a ∈ k . Hence C R = k .Next, consider a non-zero differential ideal I ⊆ R . Then J = I ⊗ k ( x ) K ( x ) is a non-zerodifferential ideal in R ⊗ k ( x ) K ( x ) ∼ = R , and as R is a simple differential ring, we conclude1 ∈ J . As K ( x ) /k ( x ) is faithfully flat, R ⊗ k ( x ) K ( x ) is faithfully flat over R and therefore I = J ∩ R . Hence 1 ∈ I and we conclude that R is a simple differential ring.Finally, (3) implies that the matrix Y has entries in the subring R of R . Its determinantdet( Y ) ∈ R is a unit when considered as an element in R ⊗ k ( x ) K ( x ) and thus det( Y ) s invertible in R , so Y ∈ GL n ( R ). Set A = ∂ ( Y ) Y − . As Y is a fundamental solutionmatrix for R/K ( x ), A has entries in K ( x ). On the other hand, Y ∈ GL n ( R ) impliesthat the entries of A are contained in R . Hence A has entries in R ∩ K ( x ) = k ( x ) andthus Y is a fundamental solution matrix for a differential equation over k ( x ). Furthermore, R = k ( x )[ Y, det( Y ) − ] by (4). Hence R is a Picard-Vessiot ring over k ( x ).Let G be the differential Galois group of R /k ( x ). Then G is a linear algebraic groupover k and ( G ) K = G by Corollary 2.5. Therefore, ( G ) K = (( G ) k ) K , and hence thereexists a finite extension k /k with ( G ) k = ( G ) k and we conclude that R descends tothe Picard-Vessiot ring R ⊗ k ( x ) k ( x ) over k ( x ) with differential Galois group ( G ) k byCorollary 2.5. (cid:3) Theorem 3.2. (a) Let k be a field of characteristic zero, and let G be a linear algebraic group over k .Then there exists a constant c G ∈ N , depending only on G , with the following property:For all large fields k with k ⊆ k and td( k/k ) > c G , G k is a differential Galois groupover ( k ( x ) , ddx ) .(b) If k is a large field of infinite transcendence degree over Q , then every linear algebraic k -group is a differential Galois group over k ( x ) endowed with ∂ = d/dx .Proof. Let K := k (( t )) be the Laurent series field over k . Then ∂ = d/dx extends from k ( x ) to K ( x ) and by [BHH16, Thm. 4.5], there exists a Picard-Vessiot ring R/K ( x ) withdifferential Galois group G K . Then by Proposition 3.1, there exists a finitely generated fieldextension k /k with k ⊆ K such that R/K ( x ) descends to a Picard-Vessiot ring R /k ( x )with differential Galois group G k . Set c G := td( k /k ).Let k be a large field with k ⊆ k and td( k/k ) > c G . Then by Corollary 1.4, thereexists a k -embedding k ֒ → k . To conclude the proof of (a), we can now base change R to R ⊗ k ( x ) k ( x ), and obtain a Picard-Vessiot ring over k ( x ) with differential Galois group( G k ) k = G k by Corollary 2.5.The proof of assertion (b) follows easily from (a), by noticing that every linear algebraic k -group G descends to a subfield k ⊆ k , which is finitely generated over Q . (cid:3) Remark 3.3.
In principle, the bound c G in Theorem 3.2 above can be computed from theinput data.By [BHH16, Cor. 4.14] (this is an adaption of a trick due to Kovacic), this result extendsfrom the rational function field k ( x ) to all finitely generated field extensions with arbitraryderivations that have field of constants k : Corollary 3.4.
Let k be large field of infinite transcendence degree over Q . Let F be adifferential field with a non-trivial derivation and field of constants k . If F/k is finitelygenerated, then every linear algebraic group over k is a differential Galois group over F . This result in particular applies if the field of constants k is Q p (or, more generally, aHenselian valued field of infinite transcendence degree) or if k is the fraction field k (( t , . . . , t n ))of a power series ring in several variables. . Differential embedding problems
In this section, we solve split differential embedding problems over k ( x ) for large fields k of infinite transcendence degree. To this end, we work with differential torsors, which wereintroduced in [BHHW16]. Let F be a differential field of characteristic zero with field ofconstants K and let G be a linear algebraic group over K . We equip its coordinate ring K [ G ] with the trivial derivation, hence F [ G F ] = F ⊗ K K [ G ] is a differential ring extension of F . We write F [ G ] = F [ G F ]. A differential G F -torsor is a G F -torsor X = Spec( R ) such that R is a differential ring extension of F and such that the co-action ρ : R → R ⊗ F F [ G ] is adifferential homomorphism. A morphism of differential G F -torsors ϕ : X → Y is a morphismof G F -torsors (i.e., a G F -equivariant morphism of varieties) such that the correspondinghomomorphism F [ Y ] → F [ X ] is a differential homomorphism.If Spec( R ) is a differential G F -torsor and H is a closed subgroup of G , the ring of invariantsis defined as R H F = { r ∈ R | ρ ( r ) = r ⊗ } . If N is a normal closed subgroup of G , thenSpec( R N F ) is a differential ( G/N ) F -torsor and the co-action R N F → R N F ⊗ F F [ G/N ] = R N F ⊗ F F [ G ] N F is obtained from restricting the co-action ρ : R → R ⊗ F F [ G ] (see Prop. 1.17together with Prop. A.6(b) in [BHHW16]).By Kolchin’s theorem, if R/F is a Picard-Vessiot ring with differential Galois group G ,then Spec( R ) is a G F -torsor. The co-action ρ : R → R ⊗ F F [ G ] can be described explicitlyas follows. Let Y ∈ GL n ( R ) be a fundamental solution matrix, i.e., R = F [ Y, det( Y ) − ].Recall that K [ G ] = C R ⊗ F R is generated by the entries of the matrix Y − ⊗ Y and its inverse.Then ρ is determined by setting ρ ( Y ) = Y ⊗ ( Y − ⊗ Y ). Conversely, if X = Spec( R ) is adifferential G F -torsor with the property that R is a simple differential ring and C R = K ,then R is a Picard-Vessiot ring over F with differential Galois group G ([BHHW16, Prop.1.12]). Lemma 4.1.
Let
K/k be a field extension in characteristic zero and let F be a differentialfield with field of constants k . We equip K with the trivial derivation and set F = Frac( F ⊗ k K ) . Let further G be a linear algebraic group over k . Assume that we are given a Picard-Vessiot ring R/F with differential Galois group G K which descends to a Picard-Vessiot ring R /F with differential Galois group G . Then the following holds.(a) The co-action ρ : R → R ⊗ F F [ G ] restricts to the co-action ρ : R → R ⊗ F F [ G ] .(b) For every closed subgroup H of G , the isomorphism R ⊗ F F ∼ = R restricts to anisomorphism R H F ⊗ F F ∼ = R H F .Proof. Let Y ∈ GL n ( R ) be a fundamental solution matrix, i.e., R = F [ Y, det( Y ) − ]. As R descends to R , there is a differential isomorphism R ⊗ F F ∼ = R over F . Hence afteridentifying R with a subring of R , we obtain an equality R = F [ Y, det( Y ) − ]. Define Z = Y − ⊗ Y ∈ GL n ( R ⊗ F R ) ⊆ GL n ( R ⊗ F R ). Recall that F [ G ] = F [ Z, det( Z ) − ]and the co-action ρ : R → R ⊗ F F [ G ] is given by Y Y ⊗ Z . Similarly, the co-action ρ : R → R ⊗ F F [ G ] is given by Y Y ⊗ Z . Hence ρ = ρ ⊗ F F and (a) follows.The H -invariants are defined as R H = { f ∈ R | ρ ( f ) = f ⊗ } and so the equality ρ = ρ ⊗ F F implies (b). (cid:3) split differential embedding problem ( N ⋊ H, S ) over F consists of a semidirect product N ⋊ H of linear algebraic groups over K together with a Picard-Vessiot ring S/F withdifferential Galois group H . A proper solution of ( N ⋊ H, S ) is a Picard-Vessiot ring
R/F with differential Galois group N ⋊ H and an embedding of differential rings S ⊆ R such thatthe following diagram commutes: N ⋊ H ∼ = (cid:15) (cid:15) / / / / H ∼ = (cid:15) (cid:15) Aut ∂ ( R/F ) res / / / / Aut ∂ ( S/F )Equivalently, R is a Picard-Vessiot ring with differential Galois group N ⋊ H such thatthere exists an isomorphism of differential H F -torsors Spec( S ) ∼ = Spec( R N F ) ([BHHW16,Lemma 2.8]). Proposition 4.2.
Let F = K ( x ) be a rational function field of characteristic zero withderivation ∂ = d/dx and let k ⊆ K be a subfield. Let ( N ⋊ H , S ) be a split differentialembedding problem over k ( x ) and assume that there exists a proper solution R of the induceddifferential embedding problem (( N ) K ⋊ ( H ) K , S ⊗ k ( x ) K ( x )) over K ( x ) . Then there existsa finitely generated field extension k /k with k ⊆ K such that the following holds: R/K ( x ) descends to a Picard-Vessiot ring R /k ( x ) that is a proper solution of the split differentialembedding problem (( N ) k ⋊ ( H ) k , S ⊗ k ( x ) k ( x )) over k ( x ) .Proof. We define N = ( N ) K , H = ( H ) K , S = S ⊗ k ( x ) K ( x ) and further G = N ⋊ H and G = N ⋊ H , hence ( G ) K = G . By Proposition 3.1, there exists a finitely generatedextension k /k with k ⊆ K such that R descends to a Picard-Vessiot ring R /k ( x ) withdifferential Galois group ( G ) k . Therefore, we can write R = K ( x )[ X , . . . , X r ] /I and R = k ( x )[ X , . . . , X r ] /I , for some polynomial ring K ( x )[ X , . . . , X r ] with a suitable derivationthat restricts to k ( x )[ X , . . . , X r ] and some differential ideal I that is generated by itscontraction I = I ∩ k ( x )[ X , . . . , X r ]. Similarly, we can write S = k ( x )[ Y , . . . , Y s ] /J , S = K ( x )[ Y , . . . , Y s ] /J with J = J ⊗ k ( x ) K ( x ). We define S = S ⊗ k ( x ) k ( x ). Then S = k ( x )[ Y , . . . , Y s ] /J with J = J ⊗ k ( x ) k ( x ). Since K ( x ) /k ( x ) is faithfully flat, J = J ∩ k ( x )[ Y , . . . , Y s ]. Let ϕ : S → R N K ( x ) be the given isomorphism of H K ( x ) -torsors. After passing from k to a finitely generatedextension, we may assume that(1) ϕ maps the elements in S = K ( x )[ Y , . . . , Y s ] /J represented by Y , . . . , Y s to elements in R = K ( x )[ X , . . . , X r ] /I that are represented by elements in k ( x )[ X , . . . , X r ](2) R N K ( x ) is generated as a K ( x )-algebra by finitely many elements α , . . . , α m ∈ R = K ( x )[ X , . . . , X r ] /I with the property that all α , . . . , α m are represented by elements in k ( x )[ X , . . . , X r ](3) for i = 1 , . . . , m , α i = ϕ ( β i ) for an element β i ∈ S = K ( x )[ Y , . . . , Y s ] /J that is repre-sented by an element in k ( x )[ Y , . . . , Y s ]. or the sake of simplicity, we will write expressions such as N k ( x ) , H k ( x ) meaning ( N ) k ( x ) ,( H ) k ( x ) . We will also write expressions such as k [ G ], k [ H ] meaning k [ G ] and k [ H ],respectively.Property (1) implies ϕ ( S ) ⊆ R ∩ R N K ( x ) and as R ∩ R N K ( x ) = R N k x ) by Lemma 4.1.(a)we conclude that ϕ restricts to an injective differential homomorphism ϕ : S → R N k x ) . It remains to show that ϕ is an isomorphism of H k ( x ) -torsors.We claim that R N k x ) = k [ α , . . . , α m ]. Since R ∩ R N K ( x ) = R N k x ) , Property (2) impliesthat α i is contained in R N k x ) for all i , and hence R N k x ) ⊇ k [ α , . . . , α m ]. On the otherhand, α , . . . , α m generate R N K ( x ) , that is, R N K ( x ) = k [ α , . . . , α m ] ⊗ k ( x ) K ( x ) . By Lemma 4.1.(b), we also have an equality R N K ( x ) = R N k x ) ⊗ k ( x ) K ( x ) and thus R N k x ) ⊗ k ( x ) K ( x ) = k [ α , . . . , α m ] ⊗ k ( x ) K ( x )and we conclude R N k x ) = k [ α , . . . , α m ] . Therefore, Property (3) implies that ϕ is surjective. Finally, since ϕ is H K ( x ) -equivariant,we conclude that its restriction is H k ( x ) -equivariant, where we use Lemma 4.1.(a) togetherwith the fact that the co-action of H K ( x ) on R N K ( x ) is given by restricting R → R ⊗ F F [ G ]to R N F → R N F ⊗ F F [ G ] N F = R N F ⊗ F F [ H ]. (cid:3) Theorem 4.3 (Main theorem) . (a) Let k be a field of characteristic zero, and let E = ( N ⋊ H , S ) be a split differentialembedding problem over ( k ( x ) , ddx ) . Then there is a constant c E ∈ N , depending onlyon E , with the following property: For all large fields k with k ⊆ k and td( k/k ) > c E ,the induced differential embedding problem (( N ) k ⋊ ( H ) k , S ⊗ k ( x ) k ( x )) over thedifferential field ( k ( x ) , ddx ) has a proper solution.(b) If k is a large field of infinite transcendence degree over Q , then every split differentialembedding problem over the differential field ( k ( x ) , ddx ) has a proper solution.Proof. Set G = N ⋊ H . We define K = k (( t )) and endow K ( x ) with the derivation d/dx .Then ˆ S = S ⊗ k ( x ) K ( x ) is a Picard-Vessiot ring over K ( x ) with differential Galois group( H ) K by Corollary 2.5. By [BHH17], the split embedding problem (( N ) K ⋊ ( H ) K , ˆ S ) has aproper solution, i.e., there exists a Picard-Vessiot ring ˆ R/K ( x ) with differential Galois group( G ) K such that ˆ R ( N ) K ( x ) and ˆ S are isomorphic as differential H K ( x ) -torsors.Then by Proposition 4.2, there exists a finitely generated field extension k /k with k ⊆ K = k (( t )) with the property that ˆ R descends to a Picard-Vessiot ring R /k ( x ) withdifferential Galois group ( G ) k and such that R ( N ) k x ) and S ⊗ k ( x ) k ( x ) are isomorphicas differential ( H ) k ( x ) -torsors. Set c E := td( k /k ).Now suppose that k is a large field with k ⊆ k and td( k/k ) > c E . Set N = ( N ) k , H = ( H ) k , G = ( G ) k and S = S ⊗ k ( x ) k ( x ). We claim that the embedding problem N ⋊ H, S ) over k ( x ) has a proper solution. By Corollary 1.4, there exists a k -embedding k ֒ → k and hence we can define R = R ⊗ k ( x ) k ( x ). Then R is a Picard-Vessiot ring over k ( x ) with differential Galois group (( G ) k ) k = ( G ) k = G by Corollary 2.5. The isomor-phism R ( N ) k x ) ∼ = S ⊗ k ( x ) k ( x ) of differential ( H ) k ( x ) -torsors gives rise to an isomorphism R N k ( x ) ∼ = S ⊗ k ( x ) k ( x ) of differential H k ( x ) -torsors by base change from k ( x ) to k ( x ), wherethe equality R ( N ) k x ) ⊗ k ( x ) k ( x ) = R N k ( x ) follows from Lemma 4.1.(b) and H k ( x ) -equivariancefollows from Lemma 4.1.(a). As S ⊗ k ( x ) k ( x ) = S , we obtain an isomorphism of H k ( x ) -torsors R N k ( x ) ∼ = S . Hence R solves the embedding problem ( N ⋊ H, S ) over k ( x ) which concludesthe proof of (a).Assertion (b) follows from (a) as follows: Let ( N ⋊ H , S ) be a split differential embeddingproblem over k ( x ), i.e., G = N ⋊ H is a linear algebraic group over k and S/K ( x ) is agiven Picard-Vessiot ring with differential Galois group H . We fix a finitely generated fieldextension k / Q with k ⊆ k such that G and its structure of a semidirect product descends toa linear algebraic group G = N ⋊ H over k . By Proposition 3.1, we may in addition choose k such that S descends to a Picard-Vessiot ring S over k ( x ) with differential Galois group H , i.e., S ⊗ k ( x ) k ( x ) ∼ = S . We conclude the proof by applying part (a) of the theorem. (cid:3) Remark 4.4.
In principle, the bound c E in Theorem 4.3(a) can be computed from the inputdata. Corollary 4.5.
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Author information:Annette Bachmayr (n´ee Maier): Mathematisches Institut der Universit¨at Bonn, D-53115Bonn, Germany.email: [email protected]
David Harbater: Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104-6395, USA.email: [email protected]
Julia Hartmann: Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104-6395, USA.email: [email protected]
Florian Pop: Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104-6395, USA.email: [email protected]@math.upenn.edu