OUTLINE O F APPROACH
XI
Using the normality of Picard-Vessiot extensions, we can prove their uniqueness:
THEOREM
3. Any two Picard-Vessiot extensions of F for L are isomorphic over F.
PROOF.
W e can assume that one of the extensions is constructed as in Theorem 2,
namely E\ = Q(S/P)\ le t E2 b e the other one. Th e differential rin g T = (S/P) ®
F
E
2
is a finitel y generate d F-algebra , henc e Noetherian . Le t M b e a maxima l prim e
differential idea l in T. The n the differential fiel d E = Q(T/M) ha s no new constants
(corollary t o Theore m 1) an d ther e ar e differentia l injection s 0 7 : E\ E induce d
by th e tenso r inclusion s 5 ^ 5 0 1 and e n l ^ e . The n b y th e abov e remar k o n
normality o a2G^x i s a n isomorphis m fro m E\ t o E
2
.
In fact , th e sam e metho d o f proo f use d i n Theore m 3 als o prove s a stron g
assertion abou t th e automorphism s o f a Picard-Vessio t extension :
THEOREM
4 . Let E be a Picard-Vessiot extension of F for L, and let x be an
element of E not in F. Then there is a differential automorphism a of E over F with
o{x) ^ x.
PROOF.
I n th e proo f o f Theore m 3 , assum e E\ = E
2
= Q(S/P). On e check s
that z=xg)l-lg)xi s no t nilpotent in T, an d we localize T a t the multiplicativel y
closed se t generate d b y z , callin g th e resul t T also . The n procee d a s i n Theore m
3. B y construction , z ha s non-zer o imag e i n Q(T/M) fro m whic h i t follow s tha t
o\{x) 7 ^ o2(x) an d henc e tha t x ^ a(x).
The se t o f differentia l automorphism s o f E ove r F form s a group , whic h i s
denoted G(E/F). Th e nex t resul t i s tha t thi s i s a n algebrai c grou p ove r th e fiel d
C o f constant s o f F.
THEOREM
5 . Let E be a Picard-Vessiot extension of F for L. Then G(E/F)
is a linear algebraic group over C. In fact, if E = Q(S/P) as in 'Theorem 2 , and
V = £ Cy/.o , then GL(V) acts naturally on S andG(E/F) = {a GL(V)\P° = P}.
Of course one needs to verify that the right-hand sid e of the equation in Theorem
5 is Zariski closed .
Theorem 4 is really half the Galois correspondence theorem. Th e other half will
be provided b y the following theorem , whic h is the version o f Kolchin's theorem fo r
which w e have bee n aiming :
THEOREM
6 . Let E be a Picard-Vessiot extension of F for L. Let G G(E/F)
be a Zariski closed subgroup. Let T be the set of all f in E which satisfy a linear
homogeneous differential equation over
EG.
Then T is a finitely generated G-stable
EG-algebra
with fraction field E, and if F denotes the algebraic closure of F, then
there is a G-isomorphism F ®p T F ® c C[G], (Here C[G] is the coordinate ring
of the algebraic group G.) In fact T consists of all the elements of E such that the
C-span of their G-orbit is finite dimensional
PROOF.
I n th e proof , w e ca n assum e tha t E G = F (se e th e remar k afte r th e
definition o f Picard-Vessio t extensions) . B y Theore m 3 w e ca n assum e tha t E =
Q(S/P) a s i n Theore m 2 . Le t S denot e th e se t o f Cj-semi-invariant s i n E. The n
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