Applied Categorical Structures

, Volume 22, Issue 1, pp 269–288 | Cite as

A Coboundary Morphism for the Grothendieck Spectral Sequence

  • David Baraglia


Given an abelian category \(\mathcal{A}\) with enough injectives we show that a short exact sequence of chain complexes of objects in \(\mathcal{A}\) gives rise to a short exact sequence of Cartan-Eilenberg resolutions. Using this we construct coboundary morphisms between Grothendieck spectral sequences associated to objects in a short exact sequence. We show that the coboundary preserves the filtrations associated with the spectral sequences and give an application of these result to filtrations in sheaf cohomology.


Spectral sequence Grothendieck Leray Coboundary Filtration 

Mathematics Subject Classifications (2010)

18G40 18G10 55Txx 


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  1. 1.
    Baraglia, D.: Topological T-duality for torus bundles with monodromy. arXiv:1201.1731v1 (2012)
  2. 2.
    Batanin, M.A.M.: Mappings of spectral sequences and the generalized homotopy axiom. Sib. Mat. Z. 28(5), 22–31, 226 (1987)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bouwknegt, P., Evslin, J., Mathai, V.: T-duality: topology change from H-flux. Commun. Math. Phys. 249(2), 383–415 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. In: Progress in Mathematics, vol. 107, 300 pp. Birkhäuser Boston, Inc., Boston, MA (1993)Google Scholar
  5. 5.
    Bunke, U., Rumpf, P., Schick, T.: The topology of T-duality for T n-bundles. Rev. Math. Phys. 18(10), 1103–1154 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cartan, H., Eilenberg, S.: Homological Algebra, 390 pp. Princeton University Press, Princeton, N. J. (1956)zbMATHGoogle Scholar
  7. 7.
    Engelking, R.: General topology, 2nd edn. In: Sigma Series in Pure Mathematics, vol. 6, 529 pp. Heldermann Verlag, Berlin (1989)Google Scholar
  8. 8.
    Freyd, P.: Abelian categories. In: An Introduction to the Theory of Functors, 164 pp. Harper & Row, New York (1964)Google Scholar
  9. 9.
    Gelfand, S.I., Manin, Y.I.: Methods of homological algebra. In: Translated from the 1988 Russian Original, 372 pp. Springer-Verlag, Berlin (1996)Google Scholar
  10. 10.
    Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9(2), 119–221 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Murray, M.K.: Bundle gerbes. J. Lond. Math. Soc. 54(2), 403–416 (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Stevenson, D.: The Geometry of Bundle Gerbes. PhD thesis, University of Adelaide. math.DG/0004117 (2000)Google Scholar
  13. 13.
    Voisin, C.: Hodge theory and complex algebraic geometry. II. In: Cambridge Studies in Advanced Mathematics, vol. 77, 351 pp. Cambridge University Press, Cambridge (2003)Google Scholar
  14. 14.
    Weibel, C.: An Introduction to Homological Algebra, 450 pp. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia

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