In this paper, we generalize the definition of the differential Galois group
and the Galois correspondence theorem established previously for Picard-Vessiot
extensions of real differential fields with real closed field of constants to
any Picard-Vessiot extension.Comment: arXiv admin note: substantial text overlap with arXiv:1207.189

Crespo, Teresa

Hajto, Zbigniew

Sowa-Adamus, Elzbieta

English

arXiv

Sowa-Adamus, Elżbieta

Jagiellonian Univeristy Repository

Arnold Math J. (2016) 2:21–27DOI 10.1007/s40598-015-0029-zRESEARCH CONTRIBUTIONGalois Correspondence Theorem for Picard-VessiotExtensionsTeresa Crespo1 · Zbigniew Hajto2 ·Elz˙bieta Sowa-Adamus3Received: 16 April 2015 / Revised: 23 September 2015 / Accepted: 23 October 2015 /Published online: 3 November 2015© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2015Abstract For a homogeneous linear differential equation defined over a differentialfield K , a Picard-Vessiot extension is a differential field extension of K differentiallygenerated by a fundamental system of solutions of the equation and not adding con-stants. When K has characteristic 0 and the field of constants of K is algebraicallyclosed, it is well known that a Picard-Vessiot extension exists and is unique up toK -differential isomorphism. In this case the differential Galois group is defined as thegroup of K -differential automorphisms of the Picard-Vessiot extension and a Galoiscorrespondence theorem is settled. Recently, Crespo, Hajto and van der Put haveproved the existence and unicity of the Picard-Vessiot extension for formally real(resp. formally p-adic) differential fields with a real closed (resp. p-adically closed)field of constants. This result widens the scope of application of Picard-Vessiot theorybeyond the complex field. It is then necessary to give an accessible presentation ofT. Crespo and Z. Hajto acknowledge support of grant MTM2012-33830, Spanish Science Ministry. E.Sowa-Adamus acknowledges support of the Polish Ministry of Science and Higher Education.B Teresa Crespoteresa.crespo@ub.eduZbigniew Hajtozbigniew.hajto@uj.edu.plElz˙bieta Sowa-Adamusesowa@agh.edu.pl1 Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes585, 08007 Barcelona, Spain2 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Prof. S. Łojasiewicza6, 30-348 Kraków, Poland3 Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza30, 30-059 Kraków, Poland12322 T. Crespo et al.Picard-Vessiot theory for arbitrary differential fields of characteristic zero which easesits use in physical or arithmetic problems. In this paper, we give such a presentationavoiding both the notions of differential universal extension and specializations usedby Kolchin and the theories of schemes and Hopf algebras used by other authors.More precisely, we give an adequate definition of the differential Galois group as alinear algebraic group and a new proof of the Galois correspondence theorem for aPicard-Vessiot extension of a differential field with non algebraically closed field ofconstants, which is more elementary than the existing ones.Keywords Differential field · Picard-Vessiot extension · Linear algebraic group ·Galois correspondenceMathematics Subject Classification 12H05 · 12F101 IntroductionFor a homogeneous linear differential equation L(Y ) = 0 defined over a differentialfield K with field of constants C , a Picard-Vessiot extension is a differential field L ,differentially generated over K by a fundamental system of solutions of L(Y ) = 0and with constant field equal to C . A classical result states that the Picard-Vessiotextension exists and is unique up to K -differential isomorphism in the case C alge-braically closed (see Kolchin 1948, Crespo and Hajto 2011 or Put and Singer 2003).Recently, an existence and uniqueness result for Picard-Vessiot extensions has beenestablished in the case when the differential field K is a formally real field (resp. aformally p-adic field) with real closed (resp. p-adically closed of the same rank thanK ) field of constants C (see Crespo et al. 2015). In Crespo et al. (2013) we presenteda Galois correspondence theorem for Picard-Vessiot extensions of formally real dif-ferential fields with real closed field of constants. In this paper we establish a Galoiscorrespondence theorem for general Picard-Vessiot extensions, i.e. without assumingthe field of constants of the base field to be algebraically closed, valid in particular forPicard-Vessiot extensions of formally p-adic differential fields with p-adically closedfield of constants.Kolchin introduced the concept of strongly normal differential field extension andobtained a satisfactory Galois correspondence theorem for this class of extensionswithout assuming the field of constants of the differential base field to be algebraicallyclosed (seeKolchin 1973, ChapterVI). Note that, for a strongly normal extension L|K ,in the case when the constant field of K is not algebraically closed, the differentialGalois group is no longer the group DAutK L of K -differential automorphisms of L ,rather one has to consider as well K -differential morphisms of L in larger differentialfields. It is worth noting that a Picard-Vessiot extension is always strongly normal.The following simple example illustrates the necessity to change the definition of thedifferential Galois group when the field of constants is not algebraically closed. Letq be an odd prime number and C a field not containing the q-th roots of unity (e.g.C = Qp, with q  p − 1). Let k = C(t) be the field of rational functions over C inthe variable t endowed with the derivation determined by t ′ = 1. Let L = k(y) be123Galois Correspondence Theorem for Picard-Vessiot Extensions 23the field of rational functions over k in the variable y and extend the derivation to Lby y′ = y. Let K = k(yq). Then L|K is clearly a Picard-Vessiot extension for thedifferential equation Y ′ = Y with [L : K ] = q and DAutK L = AutK L = {I d}.In the case of Picard-Vessiot extensions, we can adopt a definition of the differentialGalois group inspired by Kolchin’s but simpler than his one. We obtain then a Galoiscorrespondence theorem which classifies intermediate differential fields of a Picard-Vessiot extension in terms of its differential Galois group. The fact that Kolchin usedthe notion of differential universal extension andWeil’s algebraic geometry, which hasbeen later replaced by Grothendieck’s one, has led several authors to redo Kolchin’stheory using the more modern language of schemes or Hopf algebras. However, as theknowledge of scheme and Hopf algebra theories is not extended to non-algebraists,we have chosen a simpler approach which will make the theory accessible to a widerrange of mathematicians.In this paper, we shall deal with fields of characteristic 0. For the sake of simplicityin the exposition we consider ordinary differential fields.2 Main ResultWe recall now the precise definition of Picard-Vessiot extension.Definition 1 Given a homogeneous linear differential equationL(Y ) := Y (n) + an−1Y (n−1) + . . . + a1Y ′ + a0Y = 0of order n over a differential field K , with field of constants C , a differential extensionL|K is a Picard-Vessiot extension for L if1. L = K 〈η1, . . . , ηn〉, where η1, . . . , ηn is a fundamental set of solutions ofL(Y ) =0 in L and K 〈η1, . . . , ηn〉 denotes the differential field generated by η1, . . . , ηnover K .2. Every constant of L lies in K , i.e. the field of constants of L is C .As mentioned in the introduction, a Picard-Vessiot extension is strongly nor-mal. Hence, the fundamental theorem established by Kolchin (1973, chapter VI )applies to Picard-Vessiot extensions. However, for a strongly normal extension L|K ,Kolchin defines the differential Galois group DGal(L|K ) by means of differentialK -isomorphisms of L in the differential universal extension U of L . He obtains thenthat DGal(L|K ) has the structure of an algebraic group defined over the field of con-stants CU of U . Afterwards, by using the notion of specialization in Weil’s algebraicgeometry, he proves that there exists an algebraic group G defined over the field ofconstants C of K such that G(CU ) = DGal(L|K ). It is worth noting that Kovacicin Kovacic (2003, 2006) established a theory of strongly normal extensions whichavoids the differential universal extension by using tensor products. He defines thedifferential Galois group using a differential scheme. Umemura (1996) introducedthe notion of automorphic differential field extensions, which allows a finite exten-sion of the constant field, and includes both strongly normal extensions and almost12324 T. Crespo et al.classically Galois extensions (see Umemura 1996 Corollary (2.26)), a special classof Hopf-Galois finite extensions introduced in Greither and Pareigis (1987). Here afinite extension of fields is considered as a differential field extension using the trivialderivation. The differential Galois group of an automorphic differential field extensionis defined as a group functor. Using Kovacic’s theory of strongly normal extensions,Wibmer (2014) gave an intrinsic characterization of a Picard-Vessiot extension L|Kin terms of K -differential embeddings of L in differential L-algebras.In this section, we give a more natural definition of the differential Galois groupof a Picard-Vessiot extension, which avoids the use both of the differential univer-sal extension and of scheme theory. We endow the differential Galois group with alinear algebraic group structure over the field of constants C and establish a Galoiscorrespondence theorem in our setting.2.1 Differential Galois GroupFor K a differential field with field of constants C , we fix an algebraic closure Kcof K and let C denote an algebraic closure of C contained in Kc. We shall denoteby K the composition field of K and C inside Kc. Whenever F is a differential fieldextension of K , we fix Fc containing Kc, so that K ⊂ F . Let {αi }i∈I be a C-basis ofC . If the field of constants of the differential field extension F of K is equal to C , theextensions C |C and F |C are linearly disjoint over C and then {αi }i∈I is an F-basis ofF .Remark 1 If K is a differential field with field of constants C , then K ⊗C C is a fieldby Jacobson (1976, Chapter IV, Theorem 21 (2)). Since K and C are linearly disjointover C , we have K  K ⊗C C . Therefore we can also define K as K ⊗C C .For a Picard-Vessiot extension L|K , we shall consider the set DHomK (L , L) ofK -differential morphisms from L into L . We shall see that we can define a groupstructure on this set and we shall take it as the differential Galois group DGal(L|K )of the Picard-Vessiot extension L|K . We shall prove that it is a C-defined (Zariski)closed subgroup of some C-linear algebraic group.We observe that we can define mutually inverse bijectionsDHomK (L , L) → DAutK Lσ → σ̂ ,DAutK L → DHomK (L , L)τ → τ|L ,where σ̂ is the extension of σ to L . For an element∑λiαi in L , where λi ∈ L ,we define σ̂ (∑λiαi ) = ∑ σ(λi )αi . We may then transfer the group structure fromDAutK L to DHomK (L , L). Let us note that DAutK L is the differential Galois groupof the Picard-Vessiot extension L|K .Let now η1, . . . , ηn be C-linearly independent elements in L such that L =K 〈η1, . . . , ηn〉 and σ ∈ DHomK (L , L). We have then σ(η j ) = ∑ni=1 ci jηi ,1 ≤ j ≤ n, with ci j ∈ C . We may then associate to σ the matrix (ci j ) in GL(n,C).The proofs of Propositions 16 and 17 and Corollary 18 in Crespo et al. (2013) remainvalid in our present setting. We obtain then the following results.123Galois Correspondence Theorem for Picard-Vessiot Extensions 25Proposition 1 Let K be a differential field with field of constants C, L =K 〈η1, . . . , ηn〉 a Picard-Vessiot extension of K , where η1, . . . , ηn are C-linearly inde-pendent. There exists a set S of polynomials P({Xi j }1≤i, j≤n), with coefficients in Csuch that1) If σ ∈ DHomK (L , L) and σ(η j ) = ∑ni=1 ci jηi , then P(ci j ) = 0,∀P ∈ S.2) Given a matrix (ci j ) ∈ GL(n,C) with P(ci j ) = 0,∀P ∈ S, there exists a differ-ential K -morphism σ from L to L such that σ(η j ) = ∑ni=1 ci jηi .The preceding proposition gives that DGal(L|K ) is a C-defined closed subgroupof GL(n,C).Proposition 2 Let K be a differential field with field of constants C, L|K a Picard-Vessiot extension. For a ∈ L \ K, there exists a K -differential morphism σ : L → Lsuch that σ(a) = a.For a subset S of DGal(L|K ), we set LS := {a ∈ L : σ(a) = a, ∀σ ∈ S}.Corollary 1 Let K be a differential field with field of constants C, L|K a Picard-Vessiot extension. We have LDGal(L|K ) = K.As mentioned above, some authors have also considered Picard-Vessiot exten-sions over differential fields with non algebraically closed field of constants withdifferent approaches than ours. Takeuchi (1989) defines a Picard-Vessiot extensionof C-ferential fields (a notion which generalizes differential, partial differential anditerative differential fields) and the Hopf algebra of such an extension. If L|K is aPicard-Vessiot extension of differential fields with field of constants C and Hopf alge-bra H , then the group of C-points of the affine group scheme SpecH is isomorphicto the group of C-points of DGal(L|K ). Dyckerhoff (2008) and Maurischat (2010)define the differential Galois group as a group functor which is representable by anaffine group scheme coinciding with Takeuchi’s one.2.2 Fundamental TheoremLet K be a differential field with field of constants C and L|K a Picard-Vessiotextension. For a closed subgroup H of DGal(L|K ), LH is a differential subfield ofL containing K . If E is an intermediate differential field, i.e. K ⊂ E ⊂ L , thenL|E is a Picard-Vessiot extension and DGal(L|E) is a C-defined closed subgroup ofDGal(L|K ).Theorem 1 Let L|K be a Picard-Vessiot extension,DGal(L|K ) its differential Galoisgroup.1. The correspondencesH → LH , E → DGal(L|E)define inclusion inverting mutually inverse bijective maps between the set of C-defined closed subgroups H ofDGal(L|K ) and the set of differential fields E withK ⊂ E ⊂ L.12326 T. Crespo et al.2. The intermediate differential field E is a Picard-Vessiot extension of K if and onlyif the subgroup DGal(L|E) is normal in DGal(L|K ). In this case, the restrictionmorphismDGal(L|K ) → DGal(E |K )σ → σ|Einduces an isomorphismDGal(L|K )/DGal(L|E)  DGal(E |K ).Proof 1. It is clear that both maps invert inclusion. If E is an intermediate differentialfield of L|K , we have LDGal(L|E) = E , taking into account that L|E is Picard-Vessiotand corollary 1. For H a C-defined closed subgroup of DGal(L|K ), let ̂H = {̂σ : σ ∈H}, for σ̂ the extension ofσ to L , as defined in section 2.1. Then ̂H is a closed subgroupof DGal(L|K ) and from Picard-Vessiot theory for differential fields with algebraicallyclosed field of constants (see e.g. Crespo and Hajto 2011, Theorem 6.3.8), we obtainthe equality ̂H = DGal(L|L ̂H ). Now, from the definition of σ̂ , the equality L ̂H = LHfollows and hence H = DGal(L|LH ).2. If E is a Picard-Vessiot extension of K , then E is a Picard-Vessiot extension ofK and so DGal(L|E) is normal in DGal(L|K ). Reciprocally, if DGal(L|E) is nor-mal in DGal(L|K ), then the subfield of L fixed by DGal(L|E) is a Picard-Vessiotextension of K . Now, this field is E . So, E is differentially generated over K by aC-vector space V of finite dimension. Let {v1, . . . , vn} be a C-basis of V and let{αi }i∈I be a C-basis of C , as above. We may write each v j , 1 ≤ j ≤ n as a lin-ear combination of the elements αi with coefficients in E . Now, there is a finitenumber of αi ’s appearing effectively in these linear combinations. Let ˜C be a finiteGalois extension of C containing all these αi ’s. We have then vi ∈ ˜E := ˜C · Eand ˜E is differentially generated over ˜K := ˜C · K by V . We may extend the actionof Gal(˜C |C) to ˜E and consider the transform c(V ) of V by c ∈ Gal(˜C |C). Let˜V = ⊕c∈Gal(˜C |C)c(V ). We have that ˜E is differentially generated over ˜K by ˜V and ˜Vis Gal(˜C |C)-stable, hence E is differentially generated over K by the C-vector space˜VGal(˜C |C) = {y ∈ V : c(y) = y, ∀c ∈ Gal(˜C |C)}. Wemay then conclude that E |K isa Picard-Vessiot extension. Finally, from the fundamental theorem of Picard-Vessiottheory in the case of algebraically closed fields of constants (Crespo and Hajto 2011theorem 6.3.8), we obtain an isomorphism DAutK L/DAutE L  DAutK E , inducedby restriction, which implies DGal(L|K )/DGal(L|E)  DGal(E |K ). unionsqReferencesCrespo, T., Hajto Z.: Algebraic Groups and Differential Galois Theory. Graduate Studies in Mathematics122, American Mathematical Society (2011)Crespo, T., Hajto, Z., van der Put, M.: Real and p-adic Picard-Vessiot fields. Math. Ann. (2015). doi:10.1007/s00208-015-1272-2Crespo, T., Hajto, Z., Sowa, E.: Picard-Vessiot theory for real fields. Israel J. Math. 198, 75–90 (2013)123Galois Correspondence Theorem for Picard-Vessiot Extensions 27Dyckerhoff T.: The Inverse Problem of Differential Galois Theory Over the Field R(z). arXiv:0802.2897(2008)Greither, C., Pareigis, B.: Hopf Galois theory for separable field extensions. J. Algebra 106, 239–258 (1987)Jacobson, N.: Lectures in Abstract Algebra, vol. 3. Springer, New York (1976)Kolchin, E.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)Kolchin, E.: Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differ-ential equations. Ann. Math. 49, 1–42 (1948)Kovacic, J.J.: The differential Galois theory of strongly normal extensions. Trans. Am. Math. Soc. 355,4475–4522 (2003)Kovacic, J.J.: Geometric characterization of strongly normal extensions. Trans. Am. Math. Soc. 358, 4135–4157 (2006)Maurischat, A.: Galois theory for iterative connections and nonreduced Galois groups. Trans. Am. Math.Soc. 362, 5411–5453 (2010)van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations. Springer, Berlin (2003)Takeuchi, M.: A Hopf algebraic approach to the Picard-Vessiot theory. J. Algebra 122, 481–509 (1989)Umemura, H.: Galois theory of algebraic and differential equations. Nagoya Math. J. 144, 1–58 (1996)Wibmer, M.: An intrinsic characterization of Picard-Vessiot extensions. J. Algebra 408, 183–188 (2014)123

Galois correspondence theorem for Picard-Vessiot extensions

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