Triviality of differential Galois cohomologies of linear differential algebraic groups
Andrei Minchenko Alexey Ovchinnikov
math.AG 12H05, 20G07, 20G20, 34M15









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thmintroTheorem
corintro[thmintro]Corollary
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lemma[theorem]Lemma
proposition[theorem]Proposition
conjdefn[theorem]Conjectural definition
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conj[theorem]Conjecture



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remark[theorem]Remark
definition[theorem]Definition


example[theorem]Example

notation[theorem]Notation
assumption[theorem]Assumption

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    𝒜ℬ𝒞𝒟ℰℱℳ𝒩ℋℐ𝒥 ℒ𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒴𝒵𝒳𝐂δ∂𝐊ℑ𝔤𝔯𝗊Δ
    ∂𝕍𝔸HomGLGLEndAdSLSLMatPSL𝒢StabGalGal^δSpanAutR_uKerDiffLietr.deg𝕀idQuotAlgordSoldiagStab^δ∂-∂-trdegG_aG_mVectδImTriviality of cohomologies of  differential algebraic groups]Triviality of differential Galois cohomologies of linear differential algebraic groupsDepartment of Mathematics, University of Vienna, Austriaan.minchenko@gmail.comhttp://www.mat.univie.ac.at/ minchenko/CUNY Queens College, Department of Mathematics,
    65-30 Kissena Blvd, Queens, NY 11367 and
    CUNY Graduate Center, Ph.D. programs in Mathematics and Computer Science, 365 Fifth Avenue,
    New York, NY 10016aovchinnikov@qc.cuny.eduhttp://qc.edu/ aovchinnikov/We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.
    [
        Alexey Ovchinnikov
        Received: date / Accepted: date
    ===================================
    
    
    
    § INTRODUCTION
     
    Galois theory of linear differential equations with parameters <cit.> (also known as the parameterized Picard–Vessiot theory) provides theoretical and algorithmic tools to study differential algebraic dependencies of solutions of a linear ODE with one or several parameters. A parameterized Picard–Vessiot extension is a differential field generated by a complete set of solutions of the ODE and also satisfying additional technical conditions. How to decide whether such an extension is unique is an open problem 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,∂_1,…,∂_m}. One can show using <cit.> that the uniqueness up to an isomorphism of a Picard–Vessiot extension of any parameterized linear differential equation  with coefficients in F is equivalent to  the triviality of differential Galois (also known as constrained) cohomologies <cit.> over K, the ∂_x-constants of F, of all linear differential algebraic groups <cit.>. We show in our main result, Theorem <ref>, that the latter triviality holds if and only if K is algebraically, Picard–Vessiot, and linearly differentially closed (the terminology is explained in Section <ref>).
    
    Such a question for m=1 was settled in  <cit.>, in which case the “linearly differentially closed” condition does not play a role. This was extended to m>1 in <cit.> in terms of generalized strongly normal extensions. Our characterization is different and our arguments can be viewed as more transparent.
    
    
    § DEFINITIONS AND NOTATION
     A  differential ring is a ring R with a finite set Δ={δ_1,…,δ_m} of commuting derivations on R. A differential ideal of (R,Δ)
    is an ideal of R stable under any derivation in Δ. 
    
     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,Δ) is a differential ring, then (R,Δ) is called a  differential field.  The notion of  differential algebra over (R,Δ) is defined analogously. 
     
     
     An ideal I of R is called a differential ideal of (R,Δ) if, for all δ∈Δ and r ∈ I, δ(r)∈ I.
    
    
    The ring of Δ-differential polynomials K{x_1,…,x_n} in the differential indeterminates  x_1,…,x_n and with coefficients in a Δ-field (K,Δ) is the ring of polynomials in the indeterminates formally denoted 
        {δ_1^i_1·…·δ_m^i_m x_i |  i_1,…,i_m⩾ 0,  1⩽ i⩽ n}
     with coefficients in K. We endow this ring with a structure of K-Δ-algebra by setting 
    
        δ_k (δ_1^i_1·…·δ_m^i_m x_i )= δ_1^i_1·…·δ_k^i_k+1·…·δ_m^i_m x_i.
    
    A differential field (K,Δ) is said to be  differentially closed if, for every n⩾ 1 and every finite set of Δ-polynomials F ⊂ K{x_1,…,x_n }, if the system of differential equations F=0 has a solution  with entries in some Δ-field extension L, then it has a solution with entries in K.
    
    Let  be a differentially closed Δ-field  of characteristic 0 and  K⊂ be its differential subfield.
    
    
     A  Kolchin-closed  set W ⊂^n over K   is the set of common zeroes
    of a system of differential polynomials with coefficients in K, that is, there exists  S ⊂ K{y_1,…,y_n} such that
    
        W = { a ∈^n |  f(a) = 0  f ∈ S }.
    
     More generally, for a differential algebra R over K, 
        W(R) = { a ∈ R^n |    f(a) = 0  f ∈ S }.
    
    If W ⊂^n is a  Kolchin-closed set defined  over K,  the differential ideal  
        (W) = { f∈ K{y_1,  … , y_n} |  f(w) = 0  w∈ W()}
    
    is called the  defining ideal of W over K.
    Conversely, for a  subset  S of K{y_1,…,y_n}, the following subset  is Kolchin-closed in  ^n and defined over K:
    
        V(S)=  { a ∈^n |  f(a)= 0  f ∈ S }.
    
    Let W ⊂^n be a Kolchin-closed set defined over K. The  coordinate ringK{W} of 
    W over K is the differential algebra over K
        K{W} = K{y_1,…,y_n}/(W).
    
    If K{W} is an integral domain, then W is said to be  irreducible. This  is equivalent to (W) being a prime  differential ideal.
    
    Let W ⊂^n be a Kolchin-closed set defined over K. 
    Let  (W) = 𝔭_1∩…∩𝔭_q be  a minimal differential prime decomposition of  (W), that is, the 𝔭_i ⊂ K{y_1,…,y_n} are prime differential ideals containing (W) and minimal with this property. This decomposition is unique  up to permutation (see <cit.>).  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_1∪…∪ W_q. 
    
    Let W_1 ⊂^n_1 and W_2 ⊂^n_2 be two Kolchin-closed sets defined over K. 
    A   differential polynomial map (morphism) defined over K is a map  
        φ : W_1→ W_2,   a ↦(f_1(a),…,f_n_2(a)),   a ∈ W_1 ,
     where f_i ∈ K{x_1,…,x_n_1} for all i=1,…,n_2.  
    
    If W_1 ⊂ W_2, the inclusion map of W_1 in W_2 is a differential polynomial map. In this case, we say that W_1 is 
    a  Kolchin-closed subset of W_2.
    
    
    Let W be an irreducible Kolchin-closed set and P ⊂ K{x_1,…,x_n} be its defining differential ideal, which is prime. It is shown in <cit.> that there exists a non-negative integer H such that, for all h⩾ H,
    
        (P∩ K[δ_1^i_1·…·δ_m^i_mx_i |  1⩽ i⩽ n, i_j⩾ 0, 1⩽ j⩽ m, i_1+… +i_m ⩽ h])
    
    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 above polynomial is 0, we set τ(W)=-1). 
    
    
    Let _n ⊂^n be the group of n × n invertible matrices   with entries in . One can see
    _n as a Kolchin-closed subset of ^n^2× defined over K, defined by the equation x·(X)-1 in K{^n^2×}=K{X,x}, where X is an n × n-matrix of  differential indeterminates over K and x a  differential indeterminate over K. One can thus identify the differential coordinate ring of _n over K with  F{X,1/(X)}, where X=(x_i,j)_1 ⩽ i,j ⩽ n is
    a matrix of differential indeterminates over K. We also denote  the special linear group that consists of the matrices  of determinant 1 by _n ⊂_n.
     A  linear differential algebraic group (LDAG) G ⊂^n^2 defined  over K
    is a subgroup of   _n that is a Kolchin-closed set defined over K. If G ⊂ H ⊂_n are Kolchin-closed subgroups of 
    _n, we say that G is a Kolchin-closed subgroup of H.
    
    Let G be an LDAG  defined over F. The irreducible component of G containing the identity element e  is called the  identity component of G and denoted 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 irreducible Kolchin-closed set <cit.>.
    
    Let G be an LDAG over K. The  strong identity componentG_0 of G is defined to be the smallest  differential algebraic subgroup H of G defined over  such that τ(G/H) < τ(G). 
    
    By <cit.>, G_0 is a normal subgroup of G defined over K.
      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).
     For a system  of Δ-differential equations over K, 
    K is said to be -closed, or closed w.r.t. , if the consistency of  (i.e., the existence of a solution in ) implies the existence of a solution in K.
    K is said to be PV closed if, for all r, 1⩽ r⩽ m, for all  sets {D_1,…,D_r}⊂ KΔ of commuting derivations, for all n⩾ 1, and for all A_1,…,A_r∈_n× n(K), K is closed w.r.t. 
    
        {D_i(Z)=A_i· Z, z· Z=1}_i=1^r,
     where Z and z are unknown matrices of sizes n× n and 1× 1, respectively (see <cit.> for a coordinate-free definition).
    K is said to be Δ-linearly closed  if it is closed w.r.t. any system of linear (not necessarily homogeneous) Δ-differential equations in one unknown over K.
     A  principal homogeneous space (PHS) over an LDAG G over K is a Kolchin-closed  X defined over K together with a differential algebraic isomorphism
    X× G → X× X over K.
    
    The set of equivalence classes of PHS of G over K is denoted by H^1_Δ(K,G).
    We write H^1_Δ(K,G)={1} if all principal homogeneous spaces of G are isomorphic over K. For example, H^1_Δ(,G)={1}.
    
    
    
    § MAIN RESULT
    
    The following are equivalent:
    
      (1)K is algebraically closed, PV closed, and Δ-linearly closed;
    
      (2) for any linear differential algebraic group G, H^1_Δ(K,G)={1}.
    
    Let us show the implication ⟸. If K is not algebraically closed, then there exists 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 since there are no K-points of X, that is, homomorphisms E→ K over K. Hence,  H^1(K,G)≅ H^1_Δ(K,G)≠{1} (with the isomorphism following from <cit.>). 
    
    Suppose that  K is not PV closed. Hence, there exists a set D={D_1,…,D_r}⊂ KΔ of commuting derivations and a system (<ref>) with no solutions in _n(K). 
    We claim that H^1_Δ(K,_n^D)≠{1}. Indeed, let 
    
        J:={(B_1,…,B_r)∈𝔤𝔩_n()^r : D_iB_j-D_jB_i=[B_j,B_i]}
    
    and
    
        ℓ: _n→ J,     x↦(x^-1D_1(x),…,x^-1D_r(x)).
      We have (ℓ)=_n^D. 
    Moreover, by <cit.>, ℓ is surjective. Hence, the sequence
    
        {1}@>>> _n^D@>>> _n@>ℓ>> J@>>> {0}
    
    is exact. By assumption, ℓ is not surjective on K-points. Let (A_1,…,A_n)∉ℓ(_n(K)). By <cit.>, ℓ^-1(A_1,…,A_n) is a non-trivial torsor for _n^D(K).
    
    
    If  K is not Δ-linearly closed, then there exist a positive integer r, 
    a Δ-subgroup B⊂^r defined over K, and a surjective Δ-linear map Λ: → B over K that is not surjective on K-points. By <cit.>, H^1_Δ(K,Λ)≠{1}.
    
    Let us prove the implication ⟹. By <cit.>, given a short exact sequence
    
        {1}@>>> G'@>>> G@>>> G”@>>>{1}
    
    of LDAGs over K in which G'⊂ G is normal <cit.>, 
    
        H^1_Δ(K,G')=H^1_Δ(K,G”)={1} H^1_Δ(K,G)={1}.
     
    This is called the inductive principle <cit.>. As in <cit.>, the problem reduces to the following three cases:
    
      * G is finite;
    
      * G⊂;
    
      * G=()— the group of constants of ;
    
      * G=H^P, where H is a 
    linear algebraic group (LAG) over , P⊂ KΔ is a Lie subspace, and H^P is the functor of taking constant points with respect to P: H^P(L):=H(L^P) for a Δ-ring extension L of K.   Note that case (<ref>) is included into this case, but we have separated case (<ref>) for the clarity of the presentation.
    Let us explain the the reduction and then show that H^1_Δ(K,G)=1 for any G of types (<ref>)–(<ref>). The exact sequence
    
        {1}@>>> G^∘@>>> G@>>> G/G^∘@>>>{1}
    
    reduces the problem to case (<ref>) and  to showing that, for a connected G, H^1_Δ(K,G)=1. 
    To show the latter equality, let us use induction on the differential type τ(G), the case τ(G)=-1 (that is, G={1}, because G is assumed to be connected) being trivial. Let G_0⊂ G be the strong identity component. 
     Suppose τ(G)⩾ 0. One has the following exact sequence:
    
        {1}@>>> G_0@>>> G@>>> G/G_0@>>>{1},
     
    and τ(G/G_0)<τ(G). By the induction, this reduces the problem to the case G=G_0. Moreover, by <cit.>, it suffices to assume that G is almost simple.  
    If G is almost simple non-commutative, it is simple by <cit.>. By <cit.>, it corresponds to case (<ref>) because K is PV closed. If G is commutative, G is also commutative, hence
    there are n_1,n_2⩾ 0 such that G is isomorphic over K (recall that K is algebraically closed) to a direct product of n_1 copies of  and n_2 copies of  (we will not use the almost simplicity in the commutative case). It follows by induction on n_1+n_2  that H^1_Δ(K,G)=1 if H^1_Δ(K,G)=1 for any connected Kolchin closed subgroup G of  or . Indeed, if n_1⩾ 1, one has a natural projection G⊂G→ whose kernel is contained in the direct product of n_1-1 copies of  and n_2 copies of . Similarly, if n_2⩾ 1, one considers a projection G⊂G→.
    
    
    The case G⊂ reduces to G=^KΔ=() and case (<ref>) by considering the logarithmic derivatives (defined on ) 
        ℓ_i:G→,   x↦ x^-1∂_ix,
      for i=1,…, m subsequently, as all infinite differential algebraic subgroups of  contain ^KΔ<cit.>. 
    
    In case (<ref>), H^1_Δ(K,G)={1} by <cit.>, because K is algebraically closed. In case (<ref>), we have the following exact sequence:
    
        {0}@>>> G@>ι>> @>π>>  @>>>{0},
    
    and we have H^1_Δ(K,G) = {1} by <cit.> since K is linearly Δ-closed.
    Case (<ref>) is included into case (<ref>), as noted before. 
    It remains to consider case (<ref>).  
    Choose a basis {D_1,…,D_r} of commuting derivations of P and let 
    
        J:={(B_1,…,B_r)∈( H)^r : D_iB_j-D_jB_i=[B_j,B_i]}
    
    and, since H is defined over , by <cit.>, we have:
    
        ℓ: H→ J,     x↦(x^-1D_1(x),…,x^-1D_r(x)).
      We have (ℓ)=H^P=G.  Let B_1,…,B_r ∈ J.
    Since <cit.> can be rewritten in a straightforward way for several commuting derivations, the surjectivity of  ℓ is implied by 
    
        p ∉ [D_1(x)-xB_1,…,D_r(x)-xB_r]
     
    for any order 0 non-zero differential polynomial p in x (which includes p= x), as in the proof <cit.>.
    Since (<ref>) is shown in the proof of <cit.> given the conditions in (<ref>), we conclude that ℓ is surjective. Hence, the sequence
    
        {1}@>>> G@>>> H@>ℓ>> J@>>> {0}
    
    is exact. Since K is algebraically closed, H^1_Δ(K,H)={1}. By <cit.>, this implies that all torsors for G are isomorphic to ℓ^-1(a), a∈ J(K). Moreover, all of them are isomorphic (that is, H^1_Δ(K,G)={1}) if ℓ is surjective on K-points. 
    
    Due to the PV closedness, H^1_Δ(K,_n^P)={1}. 
    Since H is defined over ℚ, taking P-constants, which can be viewed as applying the functor ( · ,^P), is exact, because ^P is algebraically closed and so the polynomial map π is surjective:
    
        {1}@>>>G=H^P@>ι>>_n^P@>π>>(_n/H)^P@>>> {1},
    
    Since K is algebraically closed, the map π is surjective on K-points. Therefore, by the corresponding exact sequence of cohomologies <cit.>, H^1_Δ(K,G)=H^1_Δ(K,_n^P)={1}. 
    
    
    § ACKNOWLEDGMENTS
    
    The authors are grateful to Anand Pillay and Michael F. Singer for the discussions and comments. This work has been partially supported by the NSF grants CCF-0952591 and DMS-1413859, and   by the Austrian Science Foundation FWF, grant P28079.
    abbrvnat
