Sheaf Cohomology and Dimension of Ordered Sets

E. E. Skurikhin

Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 289-317.

Résumé

The general concept of flabbiness and flabby dimension in abelian categories and, as particular cases, flabby and soft dimensions of quasiordered sets are considered. The sheaf theory technique is developed to the level that allows one to obtain the basic theorem of the cohomological theory of dimension, including flabby and Bredon's dimensions.

Bibliographie

[1] Kuzminov V. I., “Gomologicheskaya teoriya razmernosti”, UMN, 23:5 (1968), 3–49 | MR

[2] Bredon G., Teoriya puchkov, Nauka, M., 1988, 312 pp. | MR

[3] Skurikhin E. E., “Normalnye puchki i kogomologicheskaya razmernost vpolne regulyarnykh prostranstv”, DAN SSSR, 265:3 (1982), 541–544 | MR | Zbl

[4] Kashivara M., Shapira P., Puchki na mnogoobraziyakh, Mir, M., 1997, 656 pp.

[5] Verdier J.-L., “Categories derivees. Quelques resultats (Etat O)”, Lect. Notes Math., 569, 1977, 262–311 | MR | Zbl

[6] Hartshorn R., Residues and duality, Lect. Notes Math., 20, Springer-Verl., Berlin, 1966 | MR | Zbl

[7] Iversen B., Cohomology of sheaves, Springer, Berlin, New York, Heidelberg, 1986 | MR

[8] Gelfand S. I., Manin Yu. I., Metody gomologicheskoi algebry, T. 1, Nauka, M., 1988, 416 pp. | MR

[9] Skurikhin E. E., Puchki na normalnykh i parakompaktnykh reshetkakh, Dalnauka, Vladivostok, 1998, 145 pp.

[10] Skurikhin E. E., Puchkovye kogomologii i polnye brauerovy reshetki, Dalnauka, Vladivostok, 1993, 218 pp.

[11] M. Artin, A. Grothendieck, J.-L. Verdier (eds.), Theorie de topos et cohomologie etale de schemas, Seminaire geometrie algebrique 4, SGA4, Lect. Notes Math., 269, 270, Springer-Verl., Heidelberg, 1972 | MR | Zbl

[12] Skurikhin E. E., “Lokalnaya konechnost i normalnaya vlozhennost v distributivnykh reshetkakh”, Dalnevost. mat. sb., 2 (1996), 177–186 | MR | Zbl

[13] Skurikhin E. E., “Normalnaya raspolozhennost zamknutykh elementov uporyadochennykh mnozhestv”, Dalnevost. mat. sb., 1 (1995), 18–27 | Zbl

[14] Skurikhin E. E., “Algebraicheskaya normalnaya raspolozhennost i puchki na dualnoi reshetke”, Dalnevost. mat. sb., 3 (1997), 3–10

[15] Sitnikov K. A., Skurikhin E. E., “Gomologii na predpuchkakh mnozhestv”, Tr. MIAN, 196, 1991, 156–160 | MR

[16] Skurikhin E. E., “Puchkovye kogomologii predpuchkov mnozhestv i nekotorye ikh prilozheniya”, Tr. MIAN, 193, 1992, 169–173 | MR | Zbl

[17] Skurikhin E. E., “Vyalye puchki na predpuchkakh mnozhestv”, UMN, 53:6 (1998), 263–264 | MR | Zbl

[18] Skurikhin E. E., “Lebegovskaya razmernost kak razmernost Bredona”, UMN, 54:2 (1999), 187–188 | MR | Zbl

[19] Khirtsebrukh F., Topologicheskie metody v algebraicheskoi geometrii, Mir, M., 1973, 280 pp.

[20] Kuzminov V. I., Shvedov I. A., “O kogomologicheskoi razmernosti Bredona parakompaktnykh prostranstv”, DAN SSSR, 231:1 (1976), 24–27 | MR

[21] Skurikhin E. E., “Kogomologii i razmernost kvaziuporyadochennykh mnozhestv”, UMN, 56:1 (2001), 179–180 | MR | Zbl