On the definition of the Galois group of linear differential equations
aa r X i v : . [ m a t h . AG ] D ec On the definition of the Galois group of lineardifferential equations
Katsunori Saito
Graduate School of Mathematics, Nagoya University e-mail: [email protected]
Abstract
Let us consider a linear differential equation Y ′ = AY over a dif-ferential field K , where A ∈ M n ( K ). For a differential field extension L/K generated by a fundamental system F of the equation, we showthat Galois group according to the general Galois theory of Umemuracoincides with the Picard-Vessiot Galois group. This conclusion gen-eralized the comparision theorem of Umemura and Casale. There are two ways of understanding the Galois theory of algebraic equa-tions. In the first way, when we are given an algebraic equation, according toGalois’s original paper, the Galois group is described as a permutation groupof solutions of the equation.In the second way, Dedekind introduced a nice idea, an algebraic equa-tion over field k determines the normal extension F/k of fields. The Galoisgroup is attached to the normal extension. Namely the Galois group is theautomorphism group Aut(
F/k ) of the field extension. He replaced algebraicequations by field extension.The similar situation arises in Picard-Vessiot theory. We consider a lineardifferential equation Y ′ = AY (1)over differential field K with algebraically closed field of constants C . Inthe first approach, as Dedekind did, we define the normal extension L P V /K
1f the system of linear differential equation (1) that is uniquely determined,called the Picard-Vessiot extension of (1). Then the field of constants of L P V is equal to C and the Galois group Gal( L P V /K ) of system (1) is thedifferential automorphism group Aut ∂ ( L P V /K ) that has an algebraic groupstructure over C .In the second approach, linear differential equation (1) generates a neutraltannakian category of differential modules. The Galois group of system (1)is the affine group scheme of the neutral tannakian category.In this note, we consider a differential field L over K generated by asystem of solutions of linear differential equation (1). The field L satisfies thefollowing conditions. (1) There exists a fundamental matrix F ∈ GL n ( L ) sothat F ′ = AF . (2) L is generated over K by the entries f ij of F . We considerGalois group Inf-gal ( L/K ) of the differential field extension
L/K in generalGalois theory of Umemura, and compare to Galois group Gal( L P V /K ) of thePicard-Vessiot extension L P V /K for equation (1). Then we get the followingmain theorem (Theorem 4.19). Theorem 1.1.
We assume that the field K is algebraically closed. Let S bea differential domain generated over K by a system of solutions of a lineardifferential equation Y ′ = AY and R = K [ Z P V , (det Z P V ) − ] be a Picard-Vessiot ring for the equation Y ′ = AY . The field L = Q ( S ) is the field offractions of S and the field L P V = Q ( R ) is the field of fractions of R . Thenwe have an isomorphism Lie (Inf - gal ( L/K )) ≃ Lie (Gal( L P V /K )) ⊗ C L ♮ . Theorem also holds for a G -primitive extension.In the case of the field L is Picard-Vessiot field, that is C L = C K ,Umemura [4] prove similar theorem. Casale seems to have proved the theo-rem for tr . d .L/K = n , where A and F are of n × n degree matrices.We are inspired of [4] but we considerably simplified the argument there.This paper is organized as follows. In §
2, we give some definitions andresults on Picard-Vessiot theory. In §
3, we give definitions of general Galoistheory. In § Picard-Vessiot theory
In this section, we recall the results in Picard-Vessiot theory that is Galoistheory of linear differential equations. For more details, we refer to [1]. Allthe ring that we consider except for Lie algebras are commutative and unitary Q -algebra. Definition 2.1.
A derivation on a ring R is a map ∂ : R → R satisfyingthe following properties.(1) ∂ ( a + b ) = ∂ ( a ) + ∂ ( b ) (2) ∂ ( ab ) = ∂ ( a ) b + a∂ ( b ) for all a, b ∈ R .We call a ring R equipped with a derivation ∂ on R a differential ringand similarly a field K equipped with a derivation ∂ a differential field. We say differential ring S is a differential extension of the differential ring R or a differential ring over R if the ring S is an over ring of R and thederivation ∂ S of S restricted on R coincide with the derivation ∂ R of R . Wewill often denote a differential ring equipped with derivation ∂ by ( R, ∂ ) and ∂ ( a ) by a ′ . A derivation ∂ will be sometimes called a differentiation. Example 2.2.
The following rings are differential rings.1. The polynomial ring Q [ x ] over Q with derivation f f ′ = df /dx .2. The ring of formal power series C [[ x ]] over C with derivation f f ′ = df /dx .3. The ring of holomorphic functions O ( U ) on an open connected subset U ⊂ C with derivation f f ′ = df /dz . Example 2.3.
The following fields are differential fields.1. The rational function field Q ( x ) over Q with derivation f f ′ = df /dx .2. The field of formal Laurent series C [[ x ]][ x − ] over C with derivation f f ′ = df /dx .3. The field of meromorphic functions M ( U ) on an open connected subset U ⊂ C with derivation f f ′ = df /dz .In general the field of fractions of a differential domain is a differential field. Definition 2.4.
Let ( R, ∂ ) is a differential ring. An element c ∈ R is calleda constant if c ′ = 0 and a set C R denotes the set of all constants of R .
3y definition, the set of constants C R of ( R, ∂ ) forms a ring. Similarly C K is a field for a differential field ( K, ∂ ). We sometimes say the ring ofconstants C R or the field of constants C R . Example 2.5.
1. The set of constants of differential ring ( Q [ x ] , d/dx ) is Q .2. The set of constants of differential field ( C ( z, exp z ) , d/dz ) is C .3. We consider a differential ring ( C ( x, y ) , ∂ ). The derivation ∂ = ∂/∂x + ∂/∂y acts as follows. ∂ ( C ) = 0 , ∂ ( x ) = 1 , ∂ ( y ) = 1. Since ∂ ( x − y ) = ∂ ( x ) − ∂ ( y ) = 1 − C ( x, y ) is C ( x − y ). Definition 2.6.
A differential ideal I of a differential ring ( R, ∂ ) is an idealof R satisfying f ′ ∈ I for all f ∈ I . A simple differential ring is a differentialring whose differential ideals are only (0) and R .From now on let K be a differential field with derivation ∂ and we assumethat the field C K = C of constants of the base field K is algebraically closed. We consider a linear differential equation Y ′ = AY, A ∈ M n ( K ) , where Y = ( y ij ) is a n × n matrix of variables y ij and Y ′ = ( y ′ ij ). Definition 2.7.
A Picard-Vessiot ring ( R, ∂ ) over K for the equation Y ′ = AY with A ∈ M n ( K ) is a differential ring R over K satisfying:(1) R is a simple differential ring.(2) There exists a fundamental matrix F = ( f ij ) ∈ GL n ( R ) for Y ′ = AY so that F ′ = ( f ′ ij ) = AF .(3) R is generated as a ring over K by the entries f ij of F and the inverseof the determinant of F , i.e., R = K [ f ij , (det F ) − ] . Lemma 2.8 (Lemma 1.17 [1]) . Let R be a simple differential ring over K .(1) R has no zero divisors.(2) Suppose that R is finitely generated over K , then the field of fractionsof R has C as a set of constants. By Lemma 2.8, Picard-Vessiot ring R is a domain and the field Q ( R ) offractions of R has C as the constants field. The following Proposition saysexistence and uniqueness of a Picard-Vessiot ring.4 roposition 2.9 (Proposition 1.20 [1]) . Let Y ′ = AY be a matrix differentialequation over K .(1) There exists a Picard-Vessiot ring R for the equation.(2) Any two Picard-Vessiot rings for the equation are isomorphic.(3) The field of constants of the quotient field Q ( R ) of a Picard-Vessiotring is C . We also consider a Picard-Vessiot field.
Definition 2.10.
A Picard-Vessiot field for the equation Y ′ = AY over K is the field of fractions of Picard-Vessiot ring for this equation. The following Proposition characterizes a Picard-Vessiot field.
Proposition 2.11 (Proposition 1.22 [1]) . Let Y ′ = AY be a matrix differen-tial equation over K and let L ⊃ K be an extension of differential fields. Thefield L is a Picard-Vessiot field for this equation if and only if the followingconditions are satisfied.(1) The field of constants of L is C .(2) There exist a fundamental matrix F ∈ GL n ( L ) for the equation, and(3) the field L is generated over K by the entries of F . We give the following examples of Picard-Vessiot ring and field.
Example 2.12.
We work over (
K, ∂ ) = ( C ( x ) , d/dx ). We consider a lineardifferential equation dydx = y. (2)We take a differential extension ring R := C ( x )[exp x, (exp x ) − ] over K andthe field of fractions L := Q ( R ) = C (exp x ) of R . Since the field L satisfiesthe three conditions of Proposition 2.11, the field L is a Picard-Vessiot fieldand the ring R is a Picard-Vessiot ring for equation (2).5 xample 2.13. We work over (
K, ∂ ) = ( C ( x ) , d/dx ). We consider a lineardifferential equation y ′′ + 1 x y ′ = 0 . (3)We denote this equation by the matrix equation Y ′ = (cid:18) − x (cid:19) Y. (4)We take two solutions y = log x and y = 1 of equation (3) and set F := (cid:18) y y y ′ y ′ (cid:19) = (cid:18) log x x (cid:19) . Since det F = 1 /x = 0, the matrix F is a fundamental matrix of equation(4). So we take a differential ring R := K [ F, (det F ) − ] = C ( x )[log x, x, , , x ] = C ( x )[log x ]and the field of fractions L := Q ( R ) = C ( x )(log x )of R . Then the field L satisfies the three conditions of Proposition 2.11. So L is a Picard-Vessiot field and the ring R is a Picard-Vessiot ring for equation(3).We define the Galois group for Picard-Vessiot extension. Definition 2.14.
The differential Galois group of an equation Y ′ = AY over K is defined as the group Gal(
L/K ) of differential K -automorphism ofa Picard-Vessiot field L for the equation. We can think of the differential Galois group Gal(
L/K ) as a subgroup ofGL n ( C ) in the following manner. Let F ∈ GL n ( L ) be a fundamental matrixso that F ′ = AF . The image of σ ( F ) for σ ∈ Gal(
L/K ) is also fundamentalmatrix for the equation. In fact, σ ( F ) ′ = σ ( F ′ ) = σ ( AF ) = σ ( A ) σ ( F ) = Aσ ( F ) .
6e consider the matrix C ( σ ) = F − σ ( F ) ∈ GL n ( L ). The matrix C ( σ ) isalso in GL n ( C ). Indeed C ( σ ) ′ = ( F − σ ( F )) ′ = ( F − ) ′ σ ( F ) + F − σ ( F ) ′ = − F − F ′ F − σ ( F ) + F − Aσ ( F )= − F − ( AF ) F − σ ( F ) + F − Aσ ( F ) = O. So we get the map Gal(
L/K ) → GL n ( C ) , σ C ( σ ) which is an injectivegroup homomorphism. Following Theorem says the Galois group has a linearalgebraic group structure over the constants field C . Theorem 2.15 (Theorem 1.27 [1]) . Let Y ′ = AY be a linear differentialequation over K , having Picard-Vessiot field L ⊃ K and differential Galoisgroup G = Gal( L/K ) . Then(1) G , considered as a subgroup of GL n ( C ) , is an algebraic group definedover C .(2) The Lie algebra of G coincides with the Lie algebra of the derivationsof L/K that commute with the derivation ∂ L on L .(3) The field L G of G -invariant elements of L is equal to K . Example 2.16.
The Picard-Vessiot extension C ( x, exp x ) / C for y ′ = y inExample 2.12. We take a fundamental solution y = exp x and an element ofthe Galois group σ ∈ Gal( C ( x, exp x ) / C ). We have y − σ ( y ) = c ∈ GL( C ) ≃ C × ≃ G m ( C ) . The image σ ( y ) = yc = c exp x is again a fundamental solution of y ′ = y .Then we get Gal( C ( x, exp x ) / C ) ≃ G m . So the Galois group of the Picard-Vessiot extension C ( x, exp x ) / C is the multiplicative group G m ( C ) Example 2.17.
The Picard-Vessiot extension C ( x )(log x ) / C ( x ) for y ′′ +(1 /x ) y ′ = 0 . in Example 2.13. We take a fundamental matrix F = (cid:18) log x x (cid:19) . and an element of the Galois group σ ∈ Gal( C ( x )(log x ) / C ( x )). Then F − σ ( F ) = C ( σ ) = (cid:18) a bc d (cid:19) ∈ GL( C )7o we have σ ( F ) = (cid:18) log x x (cid:19) (cid:18) a bc d (cid:19) . Since entries 1 /x, , F are in C ( x ), these entries are invariant under σ .Hence we get a = d = 1 and b = 0.Gal( C ( x )(log x ) / C ( x )) ≃ (cid:26)(cid:18) c (cid:19)(cid:27) ≃ G a ( C )Then the Galois group of the Picard-Vessiot extension C ( x )(log x ) / C ( x ) isthe additive group G a ( C ).Following Proposition says there exists a Galois correspondence. Proposition 2.18 (Proposition 1.34 [1]) . Let Y ′ = AY be a differential equa-tion over K with Picard-Vessiot field L with Galois group G := Gal( L/K ) .We consider the two sets S := { closed subgroups of G } and L := { differential subfields M of L , containing K } . We define the map α : S → L , H L H = { a ∈ L | σ ( a ) = a, for every σ ∈ H } , and β : L → S , M Gal(
L/M ) = { σ ∈ G | σ | M = Id M } . Then(1) The maps α and β are mutually inverse.(2) If the subgroup H ∈ S is a normal subgroup of G then M = L H isinvariant under G . Conversely, if M ∈ L is invariant under G then H = Gal ( L/M ) is a normal subgroup of G .(3) If H ∈ S is normal subgroup of G then the canonical map G → Gal(
M/K ) is surjective and has kernel H . Moreover, M is a Picard-Vessiot field for some linear differential equation over K with Galoisgroup G/H .
4) Let G o denote the identity component of the algebraic group G . Then L G o ⊃ K is a finite Galois extension with Galois group G/G o and isthe algebraic closure of K in L . For a differential ring (
R, ∂ ), we will denote the abstract ring R by R ♮ .Let ( R, { ∂ , ∂, · · · , ∂ d } ) be a partial differential ring. So ∂ i are mutuallycommutative derivations of R such that we have[ ∂ i , ∂ j ] = ∂ i ∂ j − ∂ j ∂ i = 0 , for 1 ≤ i, j ≤ d For example, the ring of power series (cid:18) S [[ X , X , · · · , X d ]] , (cid:26) ∂∂X , ∂∂X , · · · , ∂∂X d (cid:27)(cid:19) is a partial differential ring for a Q -algebra S .We call a morphism( R, { ∂ , ∂, · · · , ∂ d } ) −→ (cid:18) S [[ X , X , · · · , X d ]] , (cid:26) ∂∂X , ∂∂X , · · · , ∂∂X d (cid:27)(cid:19) (5)of differential ring by a Taylor morphism. Among Taylor morphisms (5),there exists the universal one ι R . Definition 3.1.
The universal Taylor morphism ι R is a differential mor-phism ( R, { ∂ , ∂, · · · , ∂ d } ) −→ (cid:18) R ♮ [[ X , X , · · · , X d ]] , (cid:26) ∂∂X , ∂∂X , · · · , ∂∂X d (cid:27)(cid:19) (6) by setting ι R ( a ) = X n ∈ N d n ! ∂ n ( a ) X n for an element a ∈ R , where we use the standard notation for multi-index. Proposition 3.2 (Umemura [2] Proposition (1.4)) . (i) The universal Taylormorphism is a monomorphism.(ii) The universal Taylor morphism is universal among the Taylor mor-phisms. L / K Let (
L, ∂ L ) / ( K, ∂ K ) be a differential field extension. We assume that theabstract field L ♮ is finitely generated over the abstract field K ♮ . We have theuniversal Taylor morphism ι L : L → L ♮ [[ X ]] . (7)We choose a mutually commutative basis { D , D , · · · , D d } of the L ♮ -vectorspace Der( L ♮ /K ♮ ) of K ♮ -derivations of the abstract field L ♮ . We introducepartial differential field L ♯ := ( L ♮ , { D , D , · · · , D d } ). Similarly the deriva-tions { D , D , · · · , D d } operate on coefficients of the ring L ♮ . Then we in-troduce { D , D , · · · , D d } -differential structure on the ring L ♮ [[ X ]]. So thering L ♮ [[ X ]] has the differential structure defined by the differentiation d/dX and the set { D , D , · · · , D d } of derivations. We denote this differential ringby L ♯ [[ X ]] = (cid:18) L ♮ [[ X ]] , ddx , D , D , · · · , D d (cid:19) . We replace the target space L ♮ [[ X ]] of universal Taylor morphism (7) by L ♯ [[ X ]] so that we have ι L : L → L ♯ [[ X ]] . In the definition below, we work in the differential ring L ♯ [[ X ]]. We denote L ♯ by partial differential field of constant power series so that L ♯ := ( ∞ X i =0 a i X i ∈ L ♯ [[ X ]] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i = 0 for i ≥ ) . Therefore L ♯ is a partial differential sub-field of L ♯ [[ X ]]. Definition 3.3.
The Galois hull L / K is a partial differential algebra exten-sion in the partial differential ring L ♯ [[ X ]] , where L is the partial differential ub-algebra generated by the image ι ( L ) and L ♯ in L ♯ [[ X ]] . And K is thepartial differential sub-algebra generated by the image ι ( K ) and L ♯ in L ♯ [[ X ]] so that K = ι ( K ) .L ♯ . F L/K of infinitesimal deformations
For the partial differential field L ♯ , we have the universal Taylor morphism ι L ♯ : L ♯ → L ♮ [[ W , W , · · · , W d ]] = L ♮ [[ W ]] , (8)where we denoted the variables X i in (8) by variables W i . The morphism (8)gives a differential ring morphism (cid:18) L ♯ [[ X ]] , (cid:26) ddX , D , D , · · · , D d (cid:27)(cid:19) −→ (cid:18) L ♮ [[ W , W , · · · , W d ]][[ X ]] , (cid:26) ddX , ∂∂W , ∂∂W , · · · , ∂∂W d (cid:27)(cid:19) . (9)Restricting the differential morphism (9) to the differential sub-algebra L ,we get a differential morphism ι : L → L ♮ [[ W, X ]] . (10)Similarly, for an L ♮ -algebra A , we have the partial differential morphism L ♮ [[ W, X ]] → A [[ W, X ]] (11)induced of the morphism L ♮ → A . We get the differential morphism ι : L → A [[ W, X ]] (12)by composing (10) and (11).
Definition 3.4.
We define the infinitesimal deformation functor F L/K : (
Alg/L ♮ ) → ( Sets ) from the category ( Alg/L ♮ ) of L ♮ -algebra to the category ( Sets ) of sets as theset of infinitesimal deformations of the morphism (10) . So F L/K ( A ) = { f : L → A [[ W, X ]] | f is a partial differential morphismcongruent to the morphism ι modulo nilpotent elementssuch that f | K = ι } . .4 Group functor Inf - gal ( L/K ) of infinitesimal auto-morphisms Definition 3.5.
The Galois group in general Galois Theory is the groupfunctor
Inf - gal ( L/K ) : (
Alg/L ♮ ) → ( Grp ) associating to an L ♮ -algebra A the automorphism group Inf - gal ( L/K )( A ) = { f : L ˆ ⊗ L ♯ A [[ W ]] → L ˆ ⊗ L ♯ A [[ W ]] | f is a differential K ˆ ⊗ L ♯ A [[ W ]] -automorphism continuous with respect to the W -adictopology and congruent to the identity modulo nilpotent elements } . Then the group functor Inf-gal (
L/K ) operates on the functor F L/K . Theoperation (Inf-gal (
L/K ) , F L/K ) is a principal homogeneous space. See The-orem (5.11) [3].
L/K generated by a lineardifferential equation over K in general Ga-lois theory Let (
K, ∂ K ) be a differential field of characteristic 0 with an algebraicallyclosed field of constants C . We consider a differential domain ( S, ∂ S ) that isan over ring of a differential field ( K, ∂ K ) satisfying the following conditions.1. There exist n × n matrices Z = ( z ij ) ∈ GL n ( S ) and A ∈ M n ( K ) suchthat Z is a solution of a linear differential equation Y ′ = AY .2. S is generated as a ring by the entries z ij and the inverse (det Z ) − ofthe determinant Z over k , i.e., S = k [ z ij , (det Z ) − ] ≤ i,j ≤ n .We will say the differential domain ( S, ∂ S ) is generated over K by a system ofsolutions of the linear differential equation Y ′ = AY or generated by a lineardifferential equation over K . Similarly we say that the field of fractions L := Q ( S ) is generated by a linear differential equation over K .A Picard-Vessiot ring is a special case of above ring. We know that thePicard-Vessiot ring R over K is a domain and the constants of R is equal to C . There is no increase of constants, more over this is also true that the fieldof fractions Q ( R ) so that the field of constants equals C . However, it is notalways true that the ring of constants of S equals C .12 xample 4.1. We work over a differential field ( ¯ Q ( e t ) , ∂ t ). We consider adifferential domain ¯ Q ( e t )[ πe t , ( πe t ) − ] = ¯ Q ( e t )[ π, π − ]with derivation ∂ t . Hence the differential ring ¯ Q ( e t )[ π, π − ] is generated bya linear differential equation Y ′ = Y over ¯ Q ( e t ). Since C ¯ Q ( e t )[ π, π − ] = ¯ Q [ π, π − ] ) ¯ Q = C ¯ Q ( e t ) , the over-ring ¯ Q ( e t )[ π, π − ] has more constant than the base field ¯ Q ( e t ).So we can not treat the differential domain S generated by a linear differ-ential equation over K in Picard-Vessiot theory. We will compare the Galoisgroup of generated over k by a system of solutions of the linear differentialequation Y ′ = AY, A ∈ M n ( K ) , (13)and the Galois group of Picard-Vessiot extension for the equation (13) ingeneral Galois theory. If the ring S is a Picard-Vessiot ring, by Proposition2.18 (4) in Section 2 we get an isomorphismLie(Gal( Q ( S ) /K )) ≃ Lie(Gal( Q ( S ) G o /K )) . As we interested in the Lie algebra of the Galois group, replacing the basefield K by its algebraic closure in Q ( S ).Let S be generated by linear differential equation (13). And the field L = Q ( S ) is the field of fractions of S . Then there exists n × n matrix Z = ( z ij ) ∈ GL n ( L ) such that Z ′ = AZ (14)and L = k ( z ij ). In the following, we work in the differential ring L ♯ [[ X ]].Since the universal Taylor morphism ι is a differential morphism, the imageof the matrix Z by the ι satisfies ddX ( ι ( Z )) = ι ( A ) ι ( Z ) (15)by (14). Lemma 4.2.
If we set B := ι ( Z )( Z ♯ ) − ∈ GL n ( L ♯ [[ X ]]) , where Z ♯ = ( z ♯ij ) ,then the matrix B is in GL n ( K ♯ [[ X ]]) . roof. We write ι ( A ) = X k ! A k X k , A k ∈ M n ( K ♯ )and B = X k ! B k X k , B k ∈ M n ( L ♯ ) . It is sufficient to show that B k is in M n ( K ♯ ) for k ∈ N . We show this byinduction on k . For k = 0, indeed B = I n ∈ M n ( K ♯ ) by definition of B .Assume B l ∈ M n ( K ♯ ) for l < k . Since Z ♯ is constant matrix with respect to d/dX , it is follows from (15) ddx B = ι ( A ) B. (16)We rewrite (16) ddx (cid:18)X k ! B k X k (cid:19) = (cid:18)X k ! A k X k (cid:19) (cid:18)X k ! B k X k (cid:19) . (17)Comparing coefficients of X k − of (17), we get B k = X l + m = k − ( k − l ! m ! A l B m ∈ M n ( K ♯ ) . In the construction of Galois hull L in general differential Galois theory,we consider a differential field extension L/K . However, we replace thedifferential field L by the differential ring S = k [ Z, (det Z ) − ]. We considerthe restriction of the universal Taylor morphism ι to the differential sub-algebra S = K [ Z, (det Z ) − ] of L . And we replace the Galois hull L by thesub-ring S := ι ( S ) .L ♯ of L ♯ [[ X ]]. Lemma 4.3.
In the differential ring L ♯ [[ X ]] , the differential sub-ring ι ( K )[ B, (det B ) − ] .L ♯ coincides with the differential sub-ring S = ι ( K [ Z, (det Z ) − ]) .L ♯ . roof. From B = ι ( Z )( Z ♯ ) − , S = ι ( K [ Z, (det Z ) − ]) .L ♯ = ι ( K )[ ι ( Z ) , (det ι ( Z )) − ]) .L ♯ = ι ( K )[ BZ ♯ , (det BZ ♯ ) − ]) .L ♯ = ι ( K )[ B, (det B ) ] .L ♯ Lemma 4.4.
The sub-ring S of L ♯ [[ X ]] is a differential sub-ring.Proof. We show that the ring S is closed under the derivations d/dX and D i for 1 ≤ i ≤ d . Since both ι ( K [ Z, (det Z ) − ]) and L ♯ are closed under thedifferentiation d/dX , the ring S is closed under the differentiation. To showthat the ring S closed under the derivations D i , by Lemma 4.3, we will showthe ring ι ( K )[ B, (det B ) − ] .L ♯ is closed under the derivations. The sub-ring L ♯ is closed obviously. By Lemma 4.2, the ring ι ( K )[ B, (det B ) − ] is in thering K ♯ [[ X ]] so that derivations D i act trivially on ι ( K )[ B, (det B ) − ]. Then ι ( K )[ B, (det B ) − ] .L ♯ closed under the derivations D i .From Lemma 4.2 we get ι ( K )[ B, (det B ) − ] ⊂ K ♯ [[ X ]]. So we have C ⊂ C ι ( K )[ B, (det B ) − ] ⊂ K ♯ . Example 4.5.
We consider a differential ring C ( x )[exp x, (exp x ) − ]. Thering C ( x )[exp , (exp x ) − ] is a Picard-Vessiot ring over C ( x ) for a linear differ-ential equation Y ′ = Y with the fundamental matrix Z = exp x . The imageof fundamental matrix ι ( Z ) is ι ( Z ) = exp x + (exp x ) X + 12! (exp x ) X + · · · = exp( x + X )Then the matrix B is B = ι ( Z )( Z ♯ ) − = (exp( x + X ))(exp x ) − = exp( X )So ι ( K ) ♯ [ B, (det B ) − ] = ι ( C ( x ))[exp X, (exp X ) − ]In this case C ι ( C ( x ))[exp X, (exp X ) − ] = C . Example 4.6.
We consider a differential ring C ( x )[log x ]. The ring C ( x )[log x ]is a Picard-Vessiot ring over C ( x ) for a linear differential equation Y ′ = (cid:18) − x (cid:19) Y Z = (cid:18) log x x (cid:19) . The image of fundamental matrix ι ( Z ) is ι ( Z ) = (cid:18) log x + x X − x X + · · · x − x X + x X + · · · (cid:19) = (cid:18) log( x + X ) 1 x + X (cid:19) . So the matrix B is B = ι ( Z )( Z ♯ ) − = (cid:18) log( x + X ) 1 x + X (cid:19) (cid:18) x − x log x (cid:19) = (cid:18) x (log(1 + Xx )0 xx + X (cid:19) . Then we get, x = (cid:18) xx + X (cid:19) ( x + X ) ∈ ι ( C ( x ))[ B, (det B ) − ]Since x is a constant with respect to derivations d/dX and D i , the ring ofconstants C ( x ) ♯ [ B, (det B ) − ] is larger than the field C .Then we consider the sub-ring ι ( K )[ B, (det B ) − ] .K ♯ of S . The sub-ringis also a differential sub-ring. The following lemma is a famous result calledlinear disjointness theorem. Lemma 4.7 (Kolchin) . Let ( R, ∂ ) be a differential ring and M be a differ-ential sub-field of R . Then the field M and the ring of constants C R of R arelinearly disjoint over the field of constants C M of M .Proof. See [2] Lemma (1.1)The lemma above as well as the lemma below are quite useful.
Lemma 4.8.
Let M be a field and ( M [[ X ]] , d/dX ) be the differential ringof power series with coefficients in M . Let R be a differential sub-ring of M [[ X ]] containing the field M . Then the ring R is a domain and the field offractions Q ( R ) has a differential field structure and we have C Q ( R ) = M. roof. Since M [[ X ]] is a domain, it is clear that the sub-ring R is a domain.The field of fractions Q ( R ) is a sub-field of Q ( M [[ X ]]) = M [[ X ]][ X − ] andcontains M . So, M ⊂ C Q ( R ) ⊂ C M [[ X ]][ X − ] = M. Remark 4.9.
Lemma 4.7 and Lemma 4.8 are also true if rings are partialdifferential ring.Applying Lemma 4.8 to ι ( K )[ B, (det B ) − ] .K ♯ ⊂ K ♯ [[ X ]] and S ⊂ L ♯ [[ X ]],we have following corollaries. Corollary 4.10.
The field of constants C Q ( ι ( K )[ B, (det B ) − ] .K ♯ ) of the field offractions Q ( ι ( K )[ B, (det B ) − ] .K ♯ ) of ι ( K )[ B, (det B ) − ] .K ♯ is K ♯ Corollary 4.11.
The field of constants of the differential field ( Q ( S ) , d/dX ) is L ♯ . Lemma 4.12.
The sub-ring ι ( K )[ B, (det B ) − ] .K ♯ of ( S , d/dX ) and thesub-field L ♯ are linearly disjoint over K ♯ . So we have a d/dX -differentialisomorphism ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ L ♯ ≃ S Proof.
We work in the differential field Q ( S ). To apply Lemma 4.7 to thedifferential sub-field Q ( ι ( K )[ B, (det B ) − ] .K ♯ )of Q ( S ), the field Q ( ι ( K )[ B, (det B ) − ] .K ♯ ) and C Q ( S ) are linearly disjointover C Q ( ι ( K )[ B, (det B ) − ] .K ♯ ) . So the differential sub-ring ι ( K )[ B, (det B ) − ] .K ♯ and C Q ( S ) also linearly disjoint over C Q ( ι ( K )[ B, (det B ) − ] .K ♯ ) . Now Lemma fol-lows from Corollary 4.10 and Corollary 4.11.From now on, we work in the partial differential ring L ♮ [[ W, X ]] and iden-tify a sub-ring R of L ♯ [[ X ]] with its image of the universal Taylor morphism ι L ♯ . Proposition 4.13.
In the partial differential ring (cid:18) L ♮ [[ W , W , · · · , W d ]][[ X ]] , (cid:26) ddX , ∂∂W , ∂∂W , · · · , ∂∂W d (cid:27)(cid:19) , he sub-ring S .L ♮ is a partial differential sub-ring. So we have a partialdifferential isomorphism S .L ♮ ≃ ( ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ L ♯ ) ⊗ K ♮ L ♮ . Proof.
Since the universal Taylor morphism ι L ♯ is differential morphism, thesub-ring S is closed under the differentiations d/dX and ∂/∂W i by Lemma4.4. And the constants power series L ♮ is clearly closed under the differen-tiations. So the sub-ring S .L ♮ is a partial differential sub-ring. In the sameway as the proof of Lemma 4.12, we have { ∂/∂W i } -differential isomorphism S .L ♮ ≃ S ⊗ K ♮ L ♮ . (18)Isomorphism (18) is also { d/dX, ∂/∂W i } -differential isomorphism becausethe sub-ring S and the sub-field L ♮ are closed under differentiations d/dX and ∂/∂W i . By Lemma 4.12, we have { d/dX } -differential isomorphism S ≃ ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ L ♯ . (19)Since the sub-ring ( ι ( K )[ B, (det B ) − ] .K ♯ and L ♯ also closed under differen-tiations d/dX and ∂/∂W i , isomorphism (19) is { d/dX, ∂/∂W i } -differentialisomorphism. So we get { d/dX, ∂/∂W i } -differential isomorphism S ⊗ K ♯ L ♮ ≃ ( ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ L ♯ ) ⊗ K ♮ L ♮ . (20)Then the Proposition follows from (18) and (20). Corollary 4.14. K .L ♮ ≃ ι ( K ) .K ♯ ⊗ K ♯ L ♯ ⊗ K ♮ L ♮ . Corollary 4.15.
For an L ♮ -algebra A , S .A ≃ ι ( K ) .K ♯ ⊗ K ♯ L ♯ ⊗ K ♮ A Proof.
The proof in Proposition 4.13 works also in these cases.We take a subset ( ι ( K ) .K ♯ ) ∗ of the ring ι ( K )[ B, (det B ) − ] .K ♯ . The set( ι ( K ) .K ♯ ) ∗ is a multiplicative set. The localization of ι ( K )[ B, (det B ) − ] .K ♯ by ( ι ( K ) .K ♯ ) ∗ is equal to Q ( ι ( K ) .K ♯ )[ B, (det B ) − ]. Lemma 4.16.
The field of constants C Q ( ι ( K ) .K ♯ ) of Q ( ι ( K ) .K ♯ ) and the fieldof constants C Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] of Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] are equal to K ♯ . roof. We can apply Lemma 4.8 to ι ( K ) .K ♯ ⊂ K ♯ [[ X ]] then C Q ( ι ( K ) .K ♯ ) = K ♯ . K ♯ ⊂ Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] ⊂ K ♯ [[ X ]][ X − ] . We have C Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] = K ♯ . Lemma 4.17. Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] is a Picard-Vessiot ring over thefield Q ( ι ( K ) .K ♯ ) for the equation Y ′ = ι ( A ) Y if the field K is algebraicallyclosed.Proof. We denote Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] by T . By the proof of Lemma4.16, T in K ♯ [[ X ]][ X − ], them the field of fractions Q ( T ) of the ring T hasalso the field of constants K ♯ . And Q ( T ) has the fundamental matrix B ofthe equation Y ′ = ι ( A ) Y and Q ( T ) is generated over Q ( ι ( K ) .K ♯ ) by theentries of B . Then by Proposition 2.11 of Section 2 the field Q ( T ) is aPicard-Vessiot field and then T is a Picard-Vessiot ring. Lemma 4.18.
We assume that the field K ♯ is algebraically closed. Let R bea Picard-Vessiot ring for the equation Y ′ = AY over the field Q ( K ⊗ C K ♯ ) .Then R and Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] are differentially isomorphic.Proof. By the differential isomorphism Q ( ι ( K ) .K ♯ ) ≃ Q ( K ⊗ C K ♯ ) , the ring Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] is a differential over ring of Q ( K ⊗ C K ♯ )Then by Lemma 4.17 we consider Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] is a Picard-Vessiot ring over the field Q ( K ⊗ C K ♯ ) for the equation Y ′ = AY . So thislemma follows from Proposition 2.9 (2) of Section 2.The following theorem is the main result. Theorem 4.19.
We assume that the field K is algebraically closed. Let S bea differential domain generated over K by a system of solutions of a lineardifferential equation Y ′ = AY and R = K [ Z P V , (det Z P V ) − ] be a Picard-Vessiot ring for the equation Y ′ = AY . The field L = Q ( S ) is the field offractions of S and the field L P V = Q ( R ) is the field of fractions of R . Thenwe have an isomorphism Lie (Inf - gal ( L/K )) ≃ Lie (Gal( L P V /K )) ⊗ C L ♮ . roof. We have to showInf-aut( S ˆ ⊗ L ♯ A [[ W ]] / K ˆ ⊗ L ♯ A [[ W ]]) ≃ Inf-aut( K [ Z P V , (det Z P V )] ⊗ C A/K ⊗ C A )for A = L ♮ [ ε ] with ε = 0, where Inf-aut( A / B ) denotes the set of B -automorphisms of A congruent to the identity map of A modulo nilpotentelement. The following argument works for every L ♮ -algebra A . We have thefollowing isomorphisms K [ Z P V , (det Z P V ) − ] ⊗ C A ≃ ( K [ Z P V , (det Z P V ) − ] ⊗ C K ♮ ) ⊗ K ♮ A ≃ ( K [ Z P V , (det Z P V ) − ] .K ♮ ) ⊗ K ♮ A, (21)over K ⊗ C A . Given an infinitesimal automorphism f of L P V ⊗ C A over K ⊗ C A , by isomorphisms (21) defines an infinitesimal automorphism ˜ f of( K [ Z P V , (det Z P V ) − ] .K ♮ ) ⊗ K ♮ A over K ⊗ C A . The infinitesimal automorphism ˜ f extended to the localizationof ( K [ Z P V , (det Z P V ) − ] .K ♮ ) by the multiplicative system ( K ⊗ C K ♯ ) ∗ . ByLemma 4.18, we get an infinitesimal automorphism ¯ f of Q ( ι ( K ) .K ♯ )[ B, (det B ) − ] ⊗ K ♮ A over Q ( ι ( K ) .K ♯ ) ⊗ C A . Since B is a system of solutions of linear differ-ential equation, the infinitesimal automorphism ¯ f induces an infinitesimalautomorphism ¯ f ′ of ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♮ A over ι ( K ) .K ♮ ⊗ K ♯ A . Since the W i ’s are variable, ¯ f ′ defines an infinitesimalautomorphism of ι ( K )[ B, (det B ) − ] .K ♮ ⊗ K ♯ A [[ W ]]over ι ( K ) .K ♮ ⊗ K ♯ A [[ W ]]. Therefore an infinitesimal automorphism of S ⊗ L ♯ A [[ W ]] over K ⊗ L ♯ A [[ X ]] by Lemma 4.12. So consequently an infinitesimalautomorphism of S ˆ ⊗ L ♯ A [[ W ]] over K ˆ ⊗ L ♯ A [[ W ]].To prove the converse, we notice that we haveInf-aut( S ˆ ⊗ L ♯ A [[ W ]] / K ˆ ⊗ L ♯ A [[ W ]]) ≃ Inf-aut( ι ( K )[ B, (det B ) − ] K ♯ ⊗ K ♯ A/ι ( K ) ⊗ K ♯ A )20n fact, given an g ∈ Inf-aut( S ˆ ⊗ L ♯ A [[ W ]] / K ˆ ⊗ L ♯ A [[ W ]]) so that g : S ˆ ⊗ L ♯ A [[ W ]] → S ˆ ⊗ L ♯ A [[ W ]] . By corollary 4.15 the restriction ˜ g := g | ι ( K )[ B, (det B ) − ] .K ♯ .A to the subalgebra ι ( K )[ B, (det B ) − ] .K ♯ .A ≃ ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ A of S ⊗ L ♯ A [[ W ]] maps ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ A to ι ( K )[ B, (det B ) − ] .K ♯ .A ≃ ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ A. Therefore we have a commutative diagram ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ A ˜ g −−−→ ι ( K )[ B, (det B ) − ] .K ♯ ⊗ K ♯ A y y S ˆ ⊗ L ♯ A [[ W ]] −−−→ g S ˆ ⊗ L ♯ A [[ W ]] . Now by isomorphisms (21),Inf-aut( S ˆ ⊗ L ♯ A [[ W ]] / K ˆ ⊗ L ♯ A [[ W ]]) ≃ Inf-aut( K [ Z P V , (det Z P V )] ⊗ C A/K ⊗ C A )When the base field K is not algebraically closed, the above argumentallows us to prove the following result. Theorem 4.20.
Let S be a differential domain generated over K by a systemof solutions of the linear differential equation Y ′ = AY and R be a Picard-Vessiot ring for the equation Y ′ = AY . The field L = Q ( S ) is the field offractions of S and the field L P V = Q ( R ) is the field of fractions of R . Thereexists a finite field extension ˜ L of L ♮ , we have an isomorphism Lie (Inf - gal ( L/K )) ⊗ L ♮ ˜ L ≃ Lie (Gal( L P V /K )) ⊗ C ˜ L. Proof.
We replace the base field K by its algebraic closure ¯ K and L by L ⊗ K ¯ K . Then we can apply the argument of proof of Theorem 4.19.21 eferenceseferences