Jean Pierre Serre Géométrie Algébrique et Géométrie Analytique (GAGA) Annales de l institut Fourier, Tome 6 (1956), p. 1-42.
2 0 X X X X X Kähler 1 X X X Chow X n 12 1 H. Cartan [3] H. Cartan W-L. Chow 2 X O x H x X x O x H x 4 3 [18] [3], exp. XVIII- XIX A B 4 2 Betti Chow 0 EGA, SGA EGA, SGA 2015.8.27 1 =courant(distribution) 2 [3], exp. XX
3 1 n 1 n 0 C n n U C n U x U x W W f 1,...,f k U W f i (z) = 0, i = 1,...,k z W U C n U X C X X C [18], n 3 H C n H C C n x U ε x : C C n,x C U,x H x ε x C U,x H U,x H U,x C U H U U A (U) x ε x : H x H U,x H U,x A (U) x f H x U x H U,x H x /A (U) x U [3], exp. VI [18], n 32 U V C r C s ϕ : U V f H V,ϕ(x) f ϕ H U,x x U ϕ(x) s x r H U ϕ : U V ϕ ϕ 1 ϕ ϕ U V H U H V U U C r C r U U C r+r [18], n 33 ϕ : U V ϕ : U V ϕ ϕ : U U V V U U U U
n 2 4 n 2 1. 3 X C X H X (H I ) X {V i } V i U i n 1 (H II ) X n 1 X H X X X Y ϕ : X Y f H Y,ϕ(x) f ϕ H X,x N. Bourbaki V X V U V (H II ) X Y X Y ϕ : V U ϕ(y V ) U Y X X [18], n 35 X X X X ϕ : V U ϕ : V U ϕ ϕ : V V U U V V X X X X X X X X [18], 34-35 n 3 [2], exp. XV X H X H X [18], n 6 Y X x X A (Y ) x H X,x Y x f H X,x A (Y ) x H X A (Y ) A (Y ) H X /A (Y ) Y Y H Y H Y [18], n 5 1. a) H X [18], n 15 b) Y X A (Y ) H X [18], n 12 3
n 4 5 C n X K. Oka H. Cartan [1], 1 2 [2], exp. XV-XVI X C n U H X = H U /A (X) H U A (X) H U H X [18], n 16 b) n 20 ([3], exp. XX) n 4 X x X H x X x C m x f H x /m C H x C X = C n H x n C{z 1,...,z n } H x C{z 1,...,z n } X C n H x Noether H. Cartan ( [3], exp. VII) H x X x [3], X x C n H x C{z 1,...,z n } H x n [16] x X n X X H x 0 [15], IV, 2 0 = p i p i H x X i X x p i = A (X i ) x H x /p i = H Xi,x X X i X x X i H x (Krull) X x H x r [14], 4 [3], exp. VIIIH x C{z 1,...,z r } C{z 1,...,z r } C[[z 1,..., z r ]] r H x [16], p. 18 2 C Zariski
n 5 6 n 5 1. a)c n Zariski b) C n Zariski c) U U C n C n Zariski f : U U f d) c) f C n Zariski a) b) c) U = C U c) f 1 d) X C [18], n 34 V X Zariski ϕ : V U U Zariski 1, b) U 2. X ϕ : V U Zariski V ϕ V X U n 2 X Zariski V ϕ : V U U ϕ 1 V ϕ : V U 1, d) V V V V 1, a) V V V V X H X (H I ) (H II ) [18], n 34 (VA II ) U i U j T ij U i U j Zariski X X ϕ : V U X Zariski H X,x O X,x C (X) x [3], exp. VIII X an X X an X X an X an
n 6 7 X an X Y (X Y ) an = X an Y an Y X Zariski Y an X an Y an X an Y f : X Y X Y f X an Y an n 6 X x X X x O x X an x H x f O x x θ(f) θ : O x H x O x H x θ : Ô x Ĥx n 24 3. θ : Ô x Ĥx Y X Zariski I (Y ) x I (Y,X) x X O x Y x Zariski f [18], n 39 I (Y ) x θ H x A (Y ) x n 3 4. A (Y ) x θ(i (Y ) x ) X C n 3 4 3 Ôx Ĥx n C[[z 1,...,z n ]] 4 a I (Y ) x H x O x H x θ H x X x [1], n 3 [3], exp. VI, p. 6 a Y f A (Y ) x H x [14], p. 278 [2], exp. XIV, p. 3 [3], exp. VIII, p. 9 r 0 f r a f r a.ĥx = I (Y ) x.ĥx = I (Y ) x.ôx I (Y ) x Y x Chevalley [16], p. 40 [17], p. 67 I x.ôx f r I (Y ) x.ôx f I (Y ) x.ôx H x Noether a.ĥx H x = a [15], IV 27 f I (Y ) x 4 X U
n 7 Zariski 8 O x = O U,x /I (X, U) x H x = H U,x /A (X, U) x θ : O x H x θ : O U,x H U,x θ : Ô U,x ĤU,x A (X, U) x = θ(i (X, U) x ).H U,x 3 29 4 A (Y ) x A (Y,U) x A (Y,U) x θ(i (Y,U) x ) 3 θ : O x H x O x H x 1. (O x, H x ) 4 3 28 2. O x H x Noether [16], p. 26 n 4 X 3. X r X an r n 7 Zariski 5. X U X U X Zariski Zariski U X Y U X X Zariski x X x U x Y = X A (Y ) x =0 n 6 A (Y ) x θ(i (Y ) x ) θ 3 I (Y ) x =0 x Zariski Y = X U X Zariski 5 θ : O x H x 3 5 6. X X an Chow [7] [19], n 4 X Y Y Zariski Zariski U f : U X f T X Y Zariski U = Y X
n 8 9 X X = f(y ) X X T X Y U T Y U Y 5 U = Y Chevalley 2. f : X Y X Y f(x) Y Zariski U f(x) Y Zariski Zariski X Y [4], exp. 3 [17], p. 15 X i (i I) X Y i f(x i ) Y Zariski Y i Y = Y i J I Y j (j J) Y j J Y j Zariski Zariski U j f(x j ) U j U j Y k (k J, k j) U = U j Y j J 7. f : X Y X Y f(x) Y Zariski T f(x) Y Zariski 2 f : X T U f(x) T Zariski Zariski 5 U T f(x) T T f(x) n 8 n 19 8. X Y f : X Y X Y f T X Y Zariski f p =pr X T p x (x, f(x)) p p f =pt Y p 1 9. T X p : T X p T X p Zariski F T Zariski p p(f ) X 7
n 8 10 p : F X p(f ) X Zariski p X O X T O T t T x = p(t) p p 4 p : O X,x O T,t pzariski p O X,x O T,t A = O X,x,A = O T,t A A B B H X,x H T,t A A B B 3 p B = B X i X x X i A p i = I (X i ) x A i = A/p i X i x K i A i X i p i A 0= p i A p i S A A S K i 3 T i = p 1 (X i ) p Zariski T i T t A p i A i = A /p i K i A i A A S K i p i A = p i A i A i,k i K i A S A S K i = K i p T i X i p : T i X i Zariski T i X i K i K i C n i =[K i : K i ] 5 X i Zariski U i U i T i n i p n i =1K i = K i A S A S K i K i A S = A S f A f A S g A s S g = sf g sa g sb g sb 3 1 (A, B) sb A = sa n 22 g sa f A g = sf s(f f )=0 s A f = f A = A 3. A p i K i A/p i S A p i A S K i A S A S [15], 4 P. Samuel 5 [17], p. 16
11 IV, 3 m i = p i A S m i A S m i A = p i[15], A S /m i A/p i K i ϕ : A S A S /m i = K i p i = 0 m i = 0 ϕ b i m j, j i A S b = b i b m i A S x i b i x i =1 x i 1(mod. m i ) x i 0(mod. m j ), j i ϕ(a S ) K i (1, 0,...,0),...,(0,...,0, 1) A S K i ϕ 3 n 9 X X an n 5 F X F X an π : F X F X F f (π(f),f) X an F X an F F F X an F x X F x = F x F F F F X an X X O n 6 3 O X an H 2. F X F X an F an F an = F O H F an F H F an H O H α : F F an O ϕ : F G
n 10 12 ϕ an : F an G an F an F 10. a) F an b) F α : F F an c) F F an F 1 F 2 F 3 F 1 F 2 F 3 3 1 F 1 O H F 2 O H F 3 O H a) b) c) O an = H F x X x Zariski U O q O p F 0 a) U H q H p F an 0 U x H n 3, 1 F an [18], n 15 I O I an I H n 10 Y X Zariski F Y F X F X Y [18], n 5 F X X (F X ) an X an F an Y an X an Y an (F an ) X 11. (F an ) X (F an ) X Y an Y an x Y A = O X,x, A = O Y,x, B = H X,x, B = H Y,x, E = F x
n 11 13 (F an ) X x = E A B (F X ) an x = E A B n 6 4 A A a B = B/aB = B A A θ x : E A B = E A A A B E A B x 11 F an F F X n 11 n 9 X F X F an F U X Zariski s F U s F X an U an s α(s ) = s 1 F an = F O H U an s α(s ) ε : Γ(U, F ) Γ(U an, F an ) U = {U i } X Zariski Ui an X an U an i 0,...,i q ε : Γ(U i0 U iq, F ) Γ(U an i 0 U an i q, F an ) [18], n 18 ε : C (U, F ) C (U an, F an ) d ε : H q (U, F ) H q (U an, F an ) U ε : H q (X, F ) H q (X an, F an ) ϕ : F G 0 A B C 0
n 12 14 A H q (X, C ) ε H q (X, C an ) δ H q+1 (X, A ) ε δ H q+1 (X an, A an ) U [18] n 12 X P r (C) Zariski 1. X F q 0 n 11 ε : H q (X, F ) H q (X an, F an ) q =0 Γ(X, F ) Γ(X an, F an ) 2. F G X F an G an F G 3. X an M X F F an M F 1. X an X X 2. ε H q (X, F ) H q (X an, F ) H q (X an, F an ) 1 H q (X, F ) H q (X an, F ) F X K = C(X) [19], 2 q>0 H q (X, K) =0 H q (X an,k) K X an q Betti n 13 1 X P r (C) F X
n 13 1 15 [18], n 26 H q (X, F ) = H q (P r (C), F ) H q (X an, F an ) = H q (P r (C) an, F an ) F an 11 ε : H q (P r (C), F ) H q (P r (C) an, F an ) X = P r (C) 4. 1 O q =0 H 0 (X, O) H 0 (X an, O an ) q>0 H q (X, O) = 0 [18], n 65, 8 Dolbeault [8] H q (X an, O an ) X (0,q) 6 5. 1 O(n) O(n) [18], n 54 n 16 r =dimx r =0 t t 0,...,t r E t =0 0 O( 1) O O E 0 O O E O( 1) O t [18], n 81 n Z 0 O(n 1) O(n) O E (n) 0 n 11 H q (X, O(n 1)) H q (X, O(n)) H q (E,O E (n)) H q+1 (X, O(n 1)) ε ε H q (X an, O(n 1) an ) H q (X an, O(n) an ) H q (E an, O E (n) an ) H q+1 (X an, O(n 1) an ) ε ε q 0 n Z ε : H q (E,O E (n)) H q (E an, O E (n) an ) 1 O(n) O(n 1) 4 n =0 n 6 n 16 H q (X, O) Laurent J. Frenkel Kähler
n 14 2 16 1 q q>2r H q (X, F ) H q (X an, F an ) [18], n 55, 1 0 R L F 0 L O(n) 5 1 L H q (X, R) H q (X, L ) H q (X, F ) H q+1 (X, R) H q+1 (X, L ) ε 1 ε 2 ε 3 ε 4 ε 5 H q (X an, R an ) H q (X an, L an ) H q (X an, F an ) H q+1 (X an, R an ) H q+1 (X an, L an ) ε 4 ε 5 ε 2 ε 3 R ε 1 ε 3 n 14 2 A = H om(f, G ) F G [18], n 11 n 14 f A x x F G F an G an f an f f an A A n 9 B = H om(f an, G an ) O ι : A an B 6. ι : A an B x X F [18], n 14 A x = Hom Ox (F x, G x ) A an x = Hom Ox (F x, G x ) Ox H x F an B x = Hom Hx (F x Ox H x, G x Ox H x ) ι x : Hom Ox (F x, G x ) Ox H x Hom Hx (F x Ox H x, G x Ox H x ) (O x, H x ) 21
n 15 3 17 2 H 0 (X, A ) ε H 0 (X an, A an ) ι H 0 (X an, B) H 0 (X, A ) H 0 (X an, B) F G F an G an f H 0 (X, A ) ι ι ε(f) =f an 2 ι ε 1 ε A [18], n 14 6 ι n 15 3 F 2 F G X g : F an G an 2 f : F G g = f an f A B 0 A F f G B 0 10 a) 0 A an F an g G an B an 0 g A an = B an = 0 10 b) A = B =0 f F X P r (C) Y X = P r (C) M Y an Y an M X X an 3 X X G G an M X I = I (Y ) Y f I x f G x ϕ M Y an Gx an = Mx X ϕ an 10 b) ϕ I.G =0YFG = F X [18], n 39, 3 11 (F an ) X (F X ) an = G an M X Y F an M n 16 3 M (n) X = P r (C) r r =0 n Z M (n) t 0,...,t r X U i t i 0 M i M U i t n j /tn i M j M i U i
n 16 3 M (n) 18 U j M i M (n) [18], n 54 M (n) M M M (n) =M H H (n) F F an (n) =F (n) an 7. E P r (C) A E q>0 n H q (E an, A (n)) = 0 [3], exp. XVIII B E F A = F an A (n) = F (n) an 1 H q (E an, A (n)) H q (E,F (n)) 7 [18], n 65, 7 8. M X = P r (C) n(m ) n n(m ) x X H x M (n) x H 0 (X an, M (n)) [3], exp. XVIII A H 0 (X an, M (n)) M (n) x m n k x U k i θ i M i (t k /t i ) m n θ i M (n) M (m) θ : M (n) M (m) θ U k H 0 (X an, M (n)) M (n) x [18], n 12 x y M (n) y X an x X n x M H 0 (X an, M (n)) M (n) x x H t =0 A (E) E n 3 0 A (E) H H E 0 A (E) H ( 1) H ( 1) A (E) t 5 M M H A (E) M M H H E 0 B M H H E C M H A (E) M C = T or1 H (M, H E )A (E) H ( 1) M H A (E)
n 16 3 M (n) 19 M ( 1) (1) 0 C M ( 1) M B 0 M (n) (1) (2) 0 C (n) M (n 1) M (n) B(n) 0 P n M (n) B(n) (2) (3) 0 C (n) M (n 1) P n 0 (4) 0 P n M (n) B(n) 0 (5) H 1 (X an, M (n 1)) H 1 (X an, P n ) H 2 (X an, C (n)) (6) H 1 (X an, P n ) H 1 (X an, M (n)) H 1 (X an, B(n)) B C A (E).B =0 A (E).C =0 B C E 7 n 0 n n 0 H 1 (X an, B(n)) = 0 H 2 (X an, C (n)) = 0 (5) (6) (7) dim H 1 (X an, M (n 1)) dim H 1 (X an, P n ) dim H 1 (X an, M (n)) [5] [3], exp. XVII n n 0 dim H 1 (X an, M (n)) n n 1 n 0 dim H 1 (X an, M (n)) n n 1 n>n 1 (8) dim H 1 (X an, M (n)) = dim H 1 (X an, P n ) = dimh 1 (X an, M (n)) n 1 n 0 H 1 (X an, B(n)) = 0 (6) H 1 (X an, P n ) H 1 (X an, M (n)) (8) (4) 7 n>n 1 (9) H 0 (X an, M (n)) H 0 (X an, B(n)) 7 Kodaira-Spencer Lefschetz [12]
n 17 3 20 n>n 1 H 0 (X an, B(n)) B(n) x B E G an 1 H 0 (X an, B(n)) = H 0 (X, G (n)) n H 0 (X, G (n)) G (n) x [18], n 55, 1 n A = H x,m= M (n) x, p = A (E) x N H 0 (X an, M (n)) M A B(n) x = M (n) x Hx H E,x = M A A/p = M/pM N M/pM M/pM M = N + pm M = N + mmm A M = N, 24, 8 n 17 3 M X = P r (C) 8 n M (n) H p M H ( n) p L 0 O( n) p R 0 R L an 0 M 0 R L 1 L an 1 R L an 1 g L an 0 M 0 2 f : L 1 L 0 g = f an F f f L 0 F 0 L 1 10 L an 1 g L an 0 F an 0 M F an 3 4 n 18 Betti σ C P r (C) t 0,...,t r x x σ t σ 0,...,tσ r σ Pr (C)
n 19 Chow 21 X P r (C) Zariski σ X σ P r (C) Zariski X X σ Jacobi 12. X X X σ Betti b n (X) X n Betti Ω p (X) an X p h p,q (X) = dimh q (X an, Ω p (X) an ) Dolbeault ( [8]) b n (X) = h p,q (X) p+q=n b n (X σ ) = p+q=n h p,q (X σ ) 1 h p,q (X) =dimh q (X, Ω p (X)) Ω p (X) X p h p,q (X σ )=dimh q (X σ, Ω p (X σ )) ω X Zariski U ω σ X σ Zariski U σ X Zariski U σ C(U, Ω p (X)) C(U σ, Ω p (X σ )) H q (U, Ω p (X σ )) H q (U σ, Ω p (X σ )) H q (X, Ω p (X)) H q (X σ, Ω p (X σ )) h p,q (X) =h p,q (X σ ) 12 A. Weil. V K V K C X Betti K C C X X σ 1 n 19 Chow ( [6]) 13. 3 X Y X an H. Cartan (no 3, 1) H Y = H X /A (Y ) X an X F ( 3) H Y = F an 10, b) F an F
n 20 22 [18], no 81 F F x 0 x X Zariski F F an = H Y Y Zariski Chow 14. X X X 6 Y U Y Y Zariski Zariski f : U X T X Y Zariski T = T (X Y ) X Y T T T Y Y Y = f 1 (X ) Y U Y Chow Y Y Zariski 7 f : Y X X = f(y ) X Zariski 15. X Y f T f X Y f T X Y 14 T 8 f. n 20 G X X G Abel G A H 0 (X, A ) H 1 (X, A ) [9] [10], V H 1 (X, G ) X G A. Weil [20] H 1 (X, O X ) C G an X G H 1 (X an, G an ) X G E E an no 11 ε : H 1 (X, G ) H 1 (X an, G an ) 16. X ε E E X G 16 E E ϕ : E E
n 20 23 E E X X G G (E,E ) X X X E E G G G (g, g ).h = ghg 1 T (E,E ) G T E E ϕ T s 15 s : X T s ϕ X ε :H 1 (X, G ) H 1 (X an, G an ) 16 G G Abel G 17. G C ε G = O G an = O an 1 18. G GL n (C) ε GL n (C) C n [18], no 41 M X an H n X F O n F an M 3 X F x X H x Fx an = F x Ox H x Hx n 30 A = O x,a = H x E = F x F x Ox n F F O n 1. n =1 GL n (C) C X H 1 (X, G ) ( [20], 3) 18 X C X Kodaira-Spencer [12] Lefschetz 2. 18 Kodaira [11] 7 8 G H G [13] G/H G H G G G/H H
n 20 24 G/H G A. Grothendieck 19. X P H X P P H G P H G P H G G P 0 h : P 0 P H G E E 0 P H G P 0 G/H G E 0 = P 0 G G/H E = (P H G) G G/H = P H G/H h f : E 0 E E = P H G/H s H G/H G f s E 0 s 0 = f 1 s s 0 15 G P 0 H P 0 /H E 0 P 0 H E 0 G G/H H P 1 = s 1 0 (P 0 ) P 0 s 0 : X E 0 P 1 X H P 1 P s 0 = f 1 s f P 1 = s 1 0 (P 0) P H G H s : X E P P P H G s X E = P H G/H 18 19 20. G GL n (C) (R) GL n (C)/G GL n (C) X ε : H 1 (X, G ) H 1 (X an, G an )
n 21 25 (R) a) G Rosenlicht [13] b) G = SL n (C) c) G = Sp n (C) n =2m GL n (C)/G i<j a ij x i x j (R) u ij x i x j C(u ij ) i<j m x 2i 1 x 2i i=1 (R) G G = O + n (C) (R) n 3 ε :H 1 (X, G ) H 1 (X an, G an ) n 21 3. B A B A A E F G E A B F A B G A B Tor A Q Tor A 1 (B,Q) = 0 Tor Q Tor Q B A A a Tor A 1 (B,A/a) =0 a A B B 1. A B A B 2. S A A S A [18], no 48, 1 A B θ : A B A B B
n 22 26 A E F A E A B F A B B f : E F f 1 E A B F A B B A B Hom A (E,F) Hom B (E A B,F A B) ι : Hom A (E,F) A B Hom B (E A B,F A B) 21. ι A Noether E A B A A F T (E) = Hom A (E,F) A B T (E) = Hom B (E A B,F A B) ι T (E) T (E) E = A T (E) =T (E) =F A B ι E A Noether E L 1 L 0 E 0 L 0 L 1 0 T (E) T (L 0 ) T (L 1 ) ι ι 0 ι 1 0 T (E) T (L 0 ) T (L 1 ) B A Hom ι 0 ι 1 ι n 22 4. A B A (A, B) B/A A 22. (A, B) B A
n 22 27 a)a ) A A E E E A B a ) A a ab A = a E A 0 A B B/A 0 Tor A 1 (A, E) Tor A 1 (B,E) Tor A 1 (B/A,E) A A E B A E A A E = E Tor A 1 (A, E) =0 0 Tor A 1 (B,E) Tor A 1 (B/A,E) E E A B Tor A 1 (B/A,E) 0 Tor A 1 (B,E) 0 E E A B a ) A/a A/a A B 23. A B C (A, C) (B,C) (A, B) B A A 0 E F 0 E A B F A B N E A B F A B C B 0 N B C (E A B) B C (F A B) B C (E A B) B C E A C (F A B) B C F A C C A E A C F A C N B C =0 22 (B,C) N =0 B A E A E E A B E A C (A, C) E E A B (A, B) 22 (A, B) (B,C) (A, C) (A, B) (A, C) (B,C)
n 23 28 n 23 A Noether 8 m 24. A E E = me E =0 [15], p. 138 [4], exp. I E 0 e 1,...,e n E e n me e n = x 1 e 1 + + x n e n x i m (1 x n )x n = x 1 e 1 + + x n 1 e n 1 1 x n A e 1,...,e n 1 E n. E A E F E = F + me E = F E/F = m(e/f) A E m m n E 0 ( [15], p. 153) 25. E A a) E F E m F m b) E E m E ( [15], [3], exp. VIII bis) a) (Krull, [15]) r n r F m n E = m n r (F m r E) (Artin, Rees, [4], exp. 2) E a) 0 E F F = mf F =0 24 E b) E A Ê Â E A m A E E Â Ê Ê Ê Â E Ê ε : E A Â Ê 26. A E ε 8 Zariski ( [15], p. 157)
n 24 29 0 R L E 0 A L A Noether R 25 R m L m E m L 0 R L Ê 0 R A Â L A Â E A Â 0 ε ε ε R L Ê 0 ε ε Ê = Â.E [15], p. 153, 1 A R ε ε n 24 Noether 27. A Â (A, Â) Â A E F E A Â F A Â E F 26 E A Â Ê F A Â F Ê F E Ê E (A, Â) 22 a ) A B θ A B θ A B θ θ : Â B 28. θ : Â B A θ B (A, B) A B B = Â (A, Â) (B, B) 23 (A, B) 29. A B a A θ A B θ 28 θ A/a B/θ(a)B (A/a,B/θ(a)B)
30 26 A/a Â/a B/θ(a)B B/θ(a) B 30. A A θ A A 28 E A A E = E A A A n E A n A θ A m m A A m m m 0 A A A A = m + A A/m = A /m E/mE = E /m E A E n A /m E /m E E n e 1,...,e n E/mE E/mE A/m e i f : A n E 24 f N f (A, A ) ( 28) 0 N A n f E 0 0 N A n f E 0 E N A n 0 N /m N A n /m A n E /m E 0 f A n /m A n E /m E N /m N =0 N =0( 24) N =0 (A, A ) [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, p. 29-64. [2] H. Cartan. Séminaire E. N. S., 1951-1952. [3] H. Cartan. Séminaire E. N. S., 1953-1954. [4] H. Cartan et C. Chevalley. Séminaire E. N. S., 1955-1956. [5] H. Cartan et J.-P. Serre. Un théorème de finitude concernant les variétés analytiques compactes. C. R., 237, 1953, p. 128-130.
31 [6] W-L. Chow. On compact complex analytic varieties. Amer.J.ofMaths., 71, 1949, p. 893-914. [7] W-L. Chow. On the projective embedding of homogeneous varieties. Lefschetz s volume, Princeton, 1956. [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, p. 175-177. [9] J. Frenkel. Cohomologie à valeurs dans un faisceau non abélien. C. R., 240, 1955, pp. 2368-2370. [10] A. Grothendieck. A general theory of fibre spaces with structure sheaf. Kansas Univ., 1955. [11] K. Kodaira. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Maths., 60, 1954, p. 28-48. [12] K. Kodaira and D. C. Spencer. Divisor class groups on algebraic varieties. Proc. Nat. Acad. Sci. U. S. (1., 39, 1953, p. 972-877. [13] M. Rosenlicht. Some basic theorems on algebraic groups. Amer.J.ofMaths., 78, 1956, p. 401-443. [14] W. Rückert. Zum Eliminationsproblem der potenzreihenidéale. Math. Ann., 107, 1933, pp. 259-281. [15] P. Samuel. Commutative Algebra (Notes by D. Herzig). Cornell Univ., 1953. [16] P. Samuel. Algèbre locale. Mém. Sci. Math., 123, Paris, 1953. [17] P. Samuel. Méthodes d algèbre abstraite en géométrie algébrique. Ergebn. der Math., Springer, 1955. [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, p. 197-278. [19] J.-P. Serre. Sur la cohomologie des variétés algébriques J. de Math. Pure et Appl., 35, 1956. [20] A. Weil. Fibre-spaces in algebraic geometry (Notes by A. Wallace). Chicago Univ., 1952.