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Cohomology Theory on Schemes

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A Royal Road to Algebraic Geometry

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Abstract

This chapter opens with a short reminder of some homological algebra, used to treat derived functors and Grothendieck cohomology. This is complemented by the Čech cohomology for abelian sheaves on a topological space.

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Notes

  1. 1.

    Inspired by [7].

  2. 2.

    We give several references for this verification: Hartshorne provides a proof in [18], on pp. 214–215. Grothendieck has provided a proof in [16], and J.M. Campbell has given a very understandable, elementary proof in the spirit of [35], in [5].

References

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  2. Campbell, J.M.: A note on flasque sheaves. Bull. Aust. Math. Soc. 2, 229–232 (1970)

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  3. Chenevert, G., Kassaei, P.: Sheaf Cohomology. Paper on the web (2003). http://www.math.mcgill.ca/goren/SeminarOnCohomology/SheafCohomology.pdf

  4. Garrett, P.: Sheaf Cohomology. Note available at the authors home page www.math.umn.edu/garret

  5. Grothendieck, A.: Éléments de géométrie algébrique. i–iv. Publ. Math. I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32

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  6. Grothendieck, A.: Local Cohomology. Lecture Notes in Mathematics, vol. 41. Springer, Berlin (1967)

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  7. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics. Springer, Berlin (1977)

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  8. Kosters, M.: Injective modules and the injective hull of a module. Notes from Leiden University available at www.math.leidenuniv.nl

  9. Lam, T.Y.: Lectures on Modules and Rings. Springer, Berlin (1999)

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  10. Macdonald, I.G.: Algebraid Geometry. Introduction to Schemes. Mathematics Lecture Notes Series. W.A. Benjamin, New York (1968)

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008, Bergen, Norway

    Audun Holme

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  1. Audun Holme
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Correspondence to Audun Holme .

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Holme, A. (2012). Cohomology Theory on Schemes. In: A Royal Road to Algebraic Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19225-8_17

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