Quasicoherent sheaves over affine Hensel schemes

Greco, Silvio; Strano, Rosario

doi:10.1090/S0002-9947-1981-0632537-7

Quasicoherent sheaves over affine Hensel schemes

Authors: Silvio Greco and Rosario Strano
Journal: Trans. Amer. Math. Soc. 268 (1981), 445-465
MSC: Primary 14A20; Secondary 13J15
DOI: https://doi.org/10.1090/S0002-9947-1981-0632537-7
MathSciNet review: 632537
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following two theorems concerning affine Hensel schemes are proved. Theorem A. Every quasi-coherent sheaf over an affine Hensel scheme is generated by its global sections. Theorem B. ${H^p}(X, F) = 0$ for all positive $p$ and all quasi-coherent sheaves $F$ over an affine Hensel scheme $X$.

M. Artin, On the joins of Hensel rings, Advances in Math. 7 (1971), 282–296 (1971). MR 289501, DOI https://doi.org/10.1016/S0001-8708%2871%2980007-5
N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1293, Hermann, Paris, 1961 (French). MR 0171800
David A. Cox, Algebraic tubular neighborhoods. I, II, Math. Scand. 42 (1978), no. 2, 211–228, 229–242. MR 512271, DOI https://doi.org/10.7146/math.scand.a-11749

---, Algebraic tubular neighborhoods. II, Math. Scand. 42 (1978), 229-243.

James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978. Reprinting of the 1966 original; Allyn and Bacon Series in Advanced Mathematics. MR 0478089
Roger Godement, Topologie algébrique et théorie des faisceaux, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1252, Hermann, Paris, 1958 (French). Publ. Math. Univ. Strasbourg. No. 13. MR 0102797

A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique. I, Springer-Verlag, Berlin, 1971.

---, Eléments de géométrie algébrique. II, Inst. Hautes Étude Sci. Publ. Math., no. 8, 1961.

---, Eléments de géométrie algébrique. III, Inst. Hautes Étude Sci. Publ. Math., no. 11, 1961.

S. Greco, Anelli henseliani, Categories and commutative algebra (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1971) Edizioni Cremonese, Rome, 1973, pp. 41–71 (Italian). MR 0366915
Silvio Greco, Sulla nozione di preschema henseliano, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 50 (1971), 118–121 (Italian, with English summary). MR 313322
Silvio Greco, Sul sollevamento dei rivestimenti étale, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 67–80 (Italian). MR 0349678
Laurent Gruson, Une propriété des couples henséliens, Colloque d’Algèbre Commutative (Rennes, 1972) Publ. Sém. Math. Univ. Rennes, Année 1972, Univ. Rennes, Rennes, 1972, pp. 13 (French). MR 0412187
Silvio Greco and Rosario Strano, Theorems A and B for Henselian schemes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 67 (1979), no. 1-2, 95–98 (1980) (English, with Italian summary). MR 617282
Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI https://doi.org/10.2307/1970720

---, Algebraic geometry, Springer-Verlag, Berlin, 1977.

Heisuke Hironaka, Formal line bundles along exceptional loci, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 201–218. MR 0262249
H. Kurke, G. Pfister, and M. Roczen, Henselsche Ringe und algebraische Geometrie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1975 (German). Mathematische Monographien, Band II. MR 0491694
Fulvio Mora, Ideali di definizione e morfismi di schemi henseliani, Ann. Mat. Pura Appl. (4) 105 (1975), 191–204 (Italian, with English summary). MR 441959, DOI https://doi.org/10.1007/BF02414929
Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
Marko Roczen, Henselsche Äquivalenz von Singularitäten, Wiss. Beitr. Martin-Luther-Univ. Halle-Wittenberg M 6 Beitr. Algebra Geom. 4 (1975), 29–46 (German). MR 0450277
Oscar Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc. 5 (1951), 90. MR 41487