A va B karton teoremalari - Cartans theorems A and B - Wikipedia

Yilda matematika, Kartan teoremalari A va B ikkita natijadir isbotlangan tomonidan Anri Kardan 1951 yil atrofida, a izchil sheaf $F$ a Stein manifold $X$ . Ular ikkalasiga ham tegishli bir nechta murakkab o'zgaruvchilar va umumiy rivojlanishida sheaf kohomologiyasi.

Teorema A.

F

bu uning global bo'limlari tomonidan kengaytirilgan.

Teorema B kohomologik nuqtai nazardan bayon etilgan (Kartan (1953, p. 51) J.-P.ga tegishli xususiyatlar Serre):

Teorema B.

H p (X, F) = 0

Barcha uchun

p > 0

.

Shunga o'xshash xususiyatlar tomonidan tashkil etilgan Serre (1957 ) izchil qistirmalar uchun algebraik geometriya, qachon $X$ bu afine sxemasi. Teorema B-ning analogi quyidagicha (Hartshorne 1977 yil, Teorema III.3.7):

Teorema B (Sxema teoretik analogi). Ruxsat bering

X

afine sxemasi bo'ling,

F

a kvazi-izchil sheaf ning

O X

- uchun modullar Zariski topologiyasi kuni

X

. Keyin

H p (X, F) = 0

Barcha uchun

p > 0

.

Ushbu teoremalar juda muhim dasturlarga ega. Masalan, ular yopiq kompleks submanifoldagi holomorf funktsiyani, $Z$ , Stein manifoldining $X$ barchasida holomorfik funktsiyaga qadar kengaytirilishi mumkin $X$ . Keyinchalik teoremalar ushbu teoremalardan foydalanilgan Jan-Per Ser isbotlash uchun GAGA teorema.

B teoremasi, agar shunday bo'lsa, aniq $H 1 (X, F) = 0$ barcha izchil chiziqlar uchun $F$ murakkab manifoldda $X$ (qariyb bir-biriga bog'laydigan qistirmalar $F$ noeteriya sxemasi bo'yicha $X$ ), keyin $X$ Stein (resp. affine); qarang (Serre 1956 yil ) (resp. (Serre 1957 yil ) va (Hartshorne 1977 yil, Teorema III.3.7)).

Shuningdek qarang Qarindosh bilan bog'liq muammolar

Adabiyotlar

Kartan, H. (1953), "Variétés analytiques complexes et cohomologie", Colloque tenu à Bruxelles: 41–55.
Gunning, Robert C.; Rossi, Gyugo (1965), Bir nechta murakkab o'zgaruvchilarning analitik funktsiyalari, Prentice Hall.
Xartshorn, Robin (1977), Algebraik geometriya, Springer-Verlag, ISBN 0-387-90244-9.
Serre, Jan-Per (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées, 36: 1–16.
Serre, Jan-Per (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802 / aif.59, ISSN 0373-0956, JANOB 0082175