Triviality of differential Galois cohomologies of linear differential algebraic groups
aa r X i v : . [ m a t h . AG ] J u l TRIVIALITY OF DIFFERENTIAL GALOIS COHOMOLOGIES OFLINEAR DIFFERENTIAL ALGEBRAIC GROUPS
ANDREI MINCHENKO AND ALEXEY OVCHINNIKOV
Abstract.
We show that the triviality of the differential Galois cohomologiesover a partial differential field K of a linear differential algebraic group isequivalent to K being algebraically, Picard–Vessiot, and linearly differentiallyclosed. This former is also known to be equivalent to the uniqueness up toan isomorphism of a Picard–Vessiot extension of a linear differential equationwith parameters. Introduction
Galois theory of linear differential equations with parameters [4] (also known asthe parameterized Picard–Vessiot theory) provides theoretical and algorithmic toolsto study differential algebraic dependencies of solutions of a linear ODE with one orseveral parameters. A parameterized Picard–Vessiot extension is a differential fieldgenerated by a complete set of solutions of the ODE and also satisfying additionaltechnical conditions. How to decide whether such an extension is unique is an openproblem in this theory. We study this question in the present paper as follows.Let F be a differential field of characteristic zero with commuting derivations { ∂ x , ∂ , . . . , ∂ m } . One can show using [6] that the uniqueness up to an isomorphismof a Picard–Vessiot extension of any parameterized linear differential equation withcoefficients in F is equivalent to the triviality of differential Galois (also known asconstrained) cohomologies [10] over K , the ∂ x -constants of F , of all linear differ-ential algebraic groups [1]. We show in our main result, Theorem 3.1, that thelatter triviality holds if and only if K is algebraically, Picard–Vessiot, and linearlydifferentially closed (the terminology is explained in Section 2).Such a question for m = 1 was settled in [14], in which case the “linearly differ-entially closed” condition does not play a role. This was extended to m > Definitions and notation
Definition 2.1. A differential ring is a ring R with a finite set ∆ = { δ , . . . , δ m } of commuting derivations on R . A differential ideal of ( R, ∆) is an ideal of R stableunder any derivation in ∆.For any derivation δ : R → R , we denote R δ = { r ∈ R | δ ( r ) = 0 } , which is a δ -subring of R and is called the ring of δ -constants of R . If R is a field and ( R, ∆) isa differential ring, then ( R, ∆) is called a differential field . The notion of differentialalgebra over ( R, ∆) is defined analogously. Definition 2.2.
An ideal I of R is called a differential ideal of ( R, ∆) if, for all δ ∈ ∆ and r ∈ I , δ ( r ) ∈ I .The ring of ∆-differential polynomials K { x , . . . , x n } in the differential indeter-minates x , . . . , x n and with coefficients in a ∆-field ( K, ∆) is the ring of polynomialsin the indeterminates formally denoted (cid:8) δ i · . . . · δ i m m x i (cid:12)(cid:12) i , . . . , i m > , i n (cid:9) with coefficients in K . We endow this ring with a structure of K -∆-algebra bysetting δ k (cid:0) δ i · . . . · δ i m m x i (cid:1) = δ i · . . . · δ i k +1 k · . . . · δ i m m x i . Definition 2.3 (see [12, Corollary 1.2(ii)]) . A differential field ( K, ∆) is said tobe differentially closed if, for every n > F ⊂ K { x , . . . , x n } , if the system of differential equations F = 0 has a solutionwith entries in some ∆-field extension L , then it has a solution with entries in K .Let U be a differentially closed ∆-field of characteristic 0 and K ⊂ U be itsdifferential subfield. Definition 2.4. A Kolchin-closed set W ⊂ U n over K is the set of common zeroesof a system of differential polynomials with coefficients in K , that is, there exists S ⊂ K { y , . . . , y n } such that W = { a ∈ U n | f ( a ) = 0 for all f ∈ S } . More generally, for a differential algebra R over K , W ( R ) = { a ∈ R n | f ( a ) = 0 for all f ∈ S } . Definition 2.5. If W ⊂ U n is a Kolchin-closed set defined over K , the differentialideal I ( W ) = { f ∈ K { y , . . . , y n } | f ( w ) = 0 for all w ∈ W ( U ) } is called the defining ideal of W over K . Conversely, for a subset S of K { y , . . . , y n } ,the following subset is Kolchin-closed in U n and defined over K : V ( S ) = { a ∈ U n | f ( a ) = 0 for all f ∈ S } . Definition 2.6.
Let W ⊂ U n be a Kolchin-closed set defined over K . The coordi-nate ring K { W } of W over K is the differential algebra over KK { W } = K { y , . . . , y n } (cid:14) I ( W ) . If K { W } is an integral domain, then W is said to be irreducible . This is equivalentto I ( W ) being a prime differential ideal. Definition 2.7.
Let W ⊂ U n be a Kolchin-closed set defined over K . Let I ( W ) = p ∩ . . . ∩ p q be a minimal differential prime decomposition of I ( W ), that is, the p i ⊂ K { y , . . . , y n } are prime differential ideals containing I ( W ) and minimal with thisproperty. This decomposition is unique up to permutation (see [8, Section VII.29]).The irreducible Kolchin-closed sets W i = V ( p i ) are defined over K and called the irreducible components of W . We have W = W ∪ . . . ∪ W q . Definition 2.8.
Let W ⊂ U n and W ⊂ U n be two Kolchin-closed sets definedover K . A differential polynomial map (morphism) defined over K is a map ϕ : W → W , a ( f ( a ) , . . . , f n ( a )) , a ∈ W , RIVIALITY OF COHOMOLOGIES OF DIFFERENTIAL ALGEBRAIC GROUPS 3 where f i ∈ K { x , . . . , x n } for all i = 1 , . . . , n .If W ⊂ W , the inclusion map of W in W is a differential polynomial map. Inthis case, we say that W is a Kolchin-closed subset of W .Let W be an irreducible Kolchin-closed set and P ⊂ K { x , . . . , x n } be its definingdifferential ideal, which is prime. It is shown in [9, Section II.12] that there existsa non-negative integer H such that, for all h > H ,dim (cid:0) P ∩ K (cid:2) δ i · . . . · δ i m m x i | i n, i j > , j m, i + . . . + i m h (cid:3)(cid:1) coincides with a polynomial in h . The degree of this polynomial is denoted by τ ( W ) and called the differential type of W (if W is a single point and so the abovepolynomial is 0, we set τ ( W ) = − Example 2.9.
Let GL n ⊂ U n be the group of n × n invertible matrices withentries in U . One can see GL n as a Kolchin-closed subset of U n × U defined over K , defined by the equation x · det( X ) − K (cid:8) U n × U (cid:9) = K { X, x } , where X is an n × n -matrix of differential indeterminates over K and x a differential indeterminateover K . One can thus identify the differential coordinate ring of GL n over K with F { X, / det( X ) } , where X = ( x i,j ) i,j n is a matrix of differential indeterminatesover K . We also denote the special linear group that consists of the matrices ofdeterminant 1 by SL n ⊂ GL n . Definition 2.10 ([1, Chapter II, Section 1, page 905]) . A linear differential al-gebraic group (LDAG) G ⊂ U n defined over K is a subgroup of GL n that is aKolchin-closed set defined over K . If G ⊂ H ⊂ GL n are Kolchin-closed subgroupsof GL n , we say that G is a Kolchin-closed subgroup of H . Definition 2.11.
Let G be an LDAG defined over F . The irreducible componentof G containing the identity element e is called the identity component of G anddenoted by G ◦ . The LDAG G ◦ is a δ -subgroup of G defined over F . The LDAG G is said to be connected if G = G ◦ , which is equivalent to G being an irreducibleKolchin-closed set [1, page 906]. Definition 2.12 ([3, Definition 2.6]) . Let G be an LDAG over K . The strong iden-tity component G of G is defined to be the smallest differential algebraic subgroup H of G defined over U such that τ ( G/H ) < τ ( G ).By [3, Remark 2.7(2)], G is a normal subgroup of G defined over K . Definition 2.13 ([3, Definition 2.10]) . An infinite LDAG G defined over K is almost simple if, for any normal proper differential algebraic subgroup H of G defined over K , we have τ ( H ) < τ ( G ). Definition 2.14.
For a system S of ∆-differential equations over K , K is saidto be S -closed , or closed w.r.t. S , if the consistency of S (i.e., the existence of asolution in U ) implies the existence of a solution in K . Definition 2.15. K is said to be PV closed if, for all r , 1 r m , for allsets { D , . . . , D r } ⊂ K ∆ of commuting derivations, for all n >
1, and for all A , . . . , A r ∈ Mat n × n ( K ), K is closed w.r.t.(2.1) { D i ( Z ) = A i · Z, z · det Z = 1 } ri =1 , where Z and z are unknown matrices of sizes n × n and 1 ×
1, respectively (see [7,page 51] for a coordinate-free definition).
ANDREI MINCHENKO AND ALEXEY OVCHINNIKOV
Definition 2.16. K is said to be ∆ -linearly closed if it is closed w.r.t. any systemof linear (not necessarily homogeneous) ∆-differential equations in one unknownover K . Definition 2.17. A principal homogeneous space (PHS) over an LDAG G over K is a Kolchin-closed X defined over K together with a differential algebraic isomor-phism X × G → X × X over K .The set of equivalence classes of PHS of G over K is denoted by H ( K, G ). Wewrite H ( K, G ) = { } if all principal homogeneous spaces of G are isomorphic over K . For example, H ( U , G ) = { } .3. Main result
Theorem 3.1.
The following are equivalent:(1) K is algebraically closed, PV closed, and ∆ -linearly closed;(2) for any linear differential algebraic group G , H ( K, G ) = { } .Proof. Let us show the implication ⇐ =. If K is not algebraically closed, then thereexists a non-trivial Galois extension E/K given by an irreducible polynomial f .The set X of its roots is a K -torsor for G = Gal ( E/K ). It is non-trivial sincethere are no K -points of X , that is, homomorphisms E → K over K . Hence, H ( K, G ) ∼ = H ( K, G ) = { } (with the isomorphism following from [10, p. 177,Theorem 4]).Suppose that K is not PV closed. Hence, there exists a set D = { D , . . . , D r } ⊂ K ∆ of commuting derivations and a system (2.1) with no solutions in GL n ( K ).We claim that H ( K, GL Dn ) = { } . Indeed, let J := { ( B , . . . , B r ) ∈ gl n ( U ) r : D i B j − D j B i = [ B j , B i ] } and ℓ : GL n → J, x ( x − D ( x ) , . . . , x − D r ( x )) . We have Ker( ℓ ) = GL Dn . Moreover, by [10, Proposition 14, p. 26], ℓ is surjective.Hence, the sequence { } −−−−→ GL Dn −−−−→ GL n ℓ −−−−→ J −−−−→ { } is exact. By assumption, ℓ is not surjective on K -points. Let ( A , . . . , A n ) / ∈ ℓ (GL n ( K )). By [10, p. 192, Proposition 8], ℓ − ( A , . . . , A n ) is a non-trivial torsorfor GL Dn ( K ).If K is not ∆-linearly closed, then there exist a positive integer r , a ∆-subgroup B ⊂ G ra defined over K , and a surjective ∆-linear map Λ : G a → B over K that isnot surjective on K -points. By [10, p. 192, Proposition 8], H ( K, Ker Λ) = { } .Let us prove the implication = ⇒ . By [10, p. 170, Theorem 2], given a shortexact sequence { } −−−−→ G ′ −−−−→ G −−−−→ G ′′ −−−−→ { } of LDAGs over K in which G ′ ⊂ G is normal [10, p. 169],(3.1) H ( K, G ′ ) = H ( K, G ′′ ) = { } = ⇒ H ( K, G ) = { } . This is called the inductive principle [14]. As in [14], the problem reduces to thefollowing three cases:(1) G is finite; RIVIALITY OF COHOMOLOGIES OF DIFFERENTIAL ALGEBRAIC GROUPS 5 (2) G ⊂ G a ;(3) G = G m ( C ) — the group of constants of G m ;(4) G = H P , where H is a linear algebraic group (LAG) over Q , P ⊂ K ∆ is aLie subspace, and H P is the functor of taking constant points with respectto P : H P ( L ) := H ( L P ) for a ∆-ring extension L of K . Note that case (3)is included into this case, but we have separated case (3) for the clarity ofthe presentation.Let us explain the the reduction and then show that H ( K, e G ) = 1 for any e G oftypes (1)–(4). The exact sequence { } −−−−→ G ◦ −−−−→ G −−−−→ G/G ◦ −−−−→ { } reduces the problem to case (1) and to showing that, for a connected G , H ( K, G ) = 1. To show the latter equality, let us use induction on the differ-ential type τ ( G ), the case τ ( G ) = − G = { } , because G is assumedto be connected) being trivial. Let G ⊂ G be the strong identity component.Suppose τ ( G ) >
0. One has the following exact sequence: { } −−−−→ G −−−−→ G −−−−→ G/G −−−−→ { } , and τ ( G/G ) < τ ( G ). By the induction, this reduces the problem to the case G = G . Moreover, by [3, Theorem 2.27 and Remark 2.28(2)], it suffices to assumethat G is almost simple. If G is almost simple non-commutative, it is simpleby [13, Theorem 3]. By [2, Theorems 9 and 17], it corresponds to case (4) because K is PV closed. If G is commutative, G is also commutative, hence there are n , n > G is isomorphic over K (recall that K is algebraically closed)to a direct product of n copies of G a and n copies of G m (we will not use thealmost simplicity in the commutative case). It follows by induction on n + n that H ( K, G ) = 1 if H ( K, e G ) = 1 for any connected Kolchin closed subgroup e G of G a or G m . Indeed, if n >
1, one has a natural projection G ⊂ G → G a whosekernel is contained in the direct product of n − G a and n copies of G m . Similarly, if n >
1, one considers a projection G ⊂ G → G m .The case G ⊂ G m reduces to G = G K ∆ m = G m ( C ) and case (2) by consideringthe logarithmic derivatives (defined on G m ) ℓ i : G → G a , x x − ∂ i x, for i = 1 , . . . , m subsequently, as all infinite differential algebraic subgroups of G m contain G K ∆ m [1, Proposition 31].In case (1), H ( K, G ) = { } by [10, p. 177, Theorem 4], because K is alge-braically closed. In case (2), we have the following exact sequence: { } −−−−→ G ι −−−−→ G a π −−−−→ G a −−−−→ { } , and we have H ( K, G ) = { } by [10, p. 193, Corollary 1] since K is linearly ∆-closed. Case (3) is included into case (4), as noted before. It remains to considercase (4). Choose a basis { D , . . . , D r } of commuting derivations of P and let(3.2) J := { ( B , . . . , B r ) ∈ (Lie H ) r : D i B j − D j B i = [ B j , B i ] } and, since H is defined over Q , by [1, p. 924, Corollary], we have: ℓ : H → J, x ( x − D ( x ) , . . . , x − D r ( x )) . ANDREI MINCHENKO AND ALEXEY OVCHINNIKOV
We have Ker( ℓ ) = H P = G . Let B , . . . , B r ∈ J . Since [11, Lemma 1] can berewritten in a straightforward way for several commuting derivations, the surjec-tivity of ℓ is implied by(3.3) p / ∈ [ D ( x ) − xB , . . . , D r ( x ) − xB r ]for any order 0 non-zero differential polynomial p in x (which includes p = det x ),as in the proof [11, Proposition 6]. Since (3.3) is shown in the proof of [10, Proposi-tion 14, p. 26] given the conditions in (3.2), we conclude that ℓ is surjective. Hence,the sequence { } −−−−→ G −−−−→ H ℓ −−−−→ J −−−−→ { } is exact. Since K is algebraically closed, H ( K, H ) = { } . By [10, p. 192, Propo-sition 8], this implies that all torsors for G are isomorphic to ℓ − ( a ), a ∈ J ( K ).Moreover, all of them are isomorphic (that is, H ( K, G ) = { } ) if ℓ is surjectiveon K -points.Due to the PV closedness, H ( K, GL Pn ) = { } . Since H is defined over Q , taking P -constants, which can be viewed as applying the functor Hom( · , U P ), is exact,because U P is algebraically closed and so the polynomial map π is surjective: { } −−−−→ G = H P ι −−−−→ GL Pn π −−−−→ (GL n /H ) P −−−−→ { } , Since K is algebraically closed, the map π is surjective on K -points. Therefore,by the corresponding exact sequence of cohomologies [10, p. 170, Theorem 2], H ( K, G ) = H ( K, GL Pn ) = { } . (cid:3) Acknowledgments
The authors are grateful to Anand Pillay and Michael F. Singer for the discus-sions and comments. This work has been partially supported by the NSF grantsCCF-0952591 and DMS-1413859, and by the Austrian Science Foundation FWF,grant P28079.
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