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