Abstract
We announce the development of a theory of algebraic De Rham cohomology and homology for arbitrary schemes over a field of characteristic zero. Over the complex numbers, this theory is equivalent to singular cohomology. Applications include generalizations of theorems of Lefschetz and Barth on the cohomology of projective varieties.
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ANDREOTTI, A., and FRANKEL, T.: The Lefschetz theorem on hyperplane sections. Ann. Math. 69, 713–717 (1959).
ANDREOTTI, A., and GRAUERT, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France 90, 193–259 (1962).
ATIYAH, M., and HODGE, W. V. D.: Integrals of the second kind on an algebraic variety. Ann. Math. 62, 56–91 (1955).
BARTH, W.: Transplanting cohomology classes in complex-projective space. Amer. J. Math. 92, 951–967 (1970).
BOREL, A., and MOORE, J.: Homology theory for locally compact spaces. Mich. Math. J. 7, 137–159 (1960).
BOTT, R.: On a theorem of Lefschetz. Mich. Math. J. 6, 211–216 (1959).
CHEVALLEY, C.: Anneaux de Chow et applications. Séminaire Chevalley: Paris 1958.
GROTHENDIECK, A.: On the De Rham cohomology of algebraic varieties. Publ. Math. I.H.E.S. 29, 95–103 (1966).
HARTSHORNE, R.: Ample vector bundles. Publ. Math. I. H. E. S. 29, 63–94 (1966).
HARTSHORNE, R.: Residues and duality. Springer Lecture Notes 20 (1966).
: Cohomological dimension of algebraic varieties. Ann. Math. 88, 403–450 (1968).
HARTSHORNE, R.: Ample subvarieties of algebraic varieties. Springer Lecture Notes 156 (1970).
: Cohomology of non-complete algebraic varieties. Compos. Math. 23, 257–264 (1971).
HIRONAKA, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964).
LÊ DŨNG TRANG: Singularités isolées des intersections complètes. Séminaire Weishu Shih, I.H.E.S. (1969–70).
LIEBERMAN, D., and HERRERA, M.: Duality and the De Rham cohomology of infinitesimal neighborhoods. Invent. Math. 13, 97–124 (1971).
MILNOR, J.: Singular points of complex hypersurfaces. Ann. Math. Studies 61: Princeton 1968.
OGUS, A.: Local cohomological dimensions of algebraic varieties, to appear.
SERRE, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6, 1–42 (1955–56).
ZARISKI, O.: Some open questions in the theory of singularities. Bull. A.M.S. 77, 481–491 (1971).
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Hartshorne, R. Algebraic De Rham cohomology. Manuscripta Math 7, 125–140 (1972). https://doi.org/10.1007/BF01679709
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DOI: https://doi.org/10.1007/BF01679709