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Stacks for Everybody

  • Barbara Fantechi
Conference paper
Part of the Progress in Mathematics book series (PM, volume 201)

Abstract

Let G be a category with a Grothendieck topology. A stack over G is a category fibered in groupoids over G, such that isomorphisms form a sheaf and every descent datum is effective. If G is the category of schemes with the étale topology, a stack is algebraic in the sense of Deligne-Mumford (respectively Artin) if it has an étale (resp. smooth) presentation.

I will try to explain the previous definitions so as to make them accessible to the widest possible audience. In order to do this, we will keep in mind one fixed example, that of vector bundles; if you know what pullback of vector bundles is in some geometric context (schemes, complex analytic spaces, but also varieties or manifolds) you should be able to follow this exposition.

Keywords

Vector Bundle Open Covering Fiber Product Frame Bundle Algebraic Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M. Artin, Versai Deformations and Algebraic Stacks, Invent. Math., 27, (1974), 165–189.MathSciNetzbMATHGoogle Scholar
  2. [2]
    M. Artin, Grothendieck Topologies, Cambridge, Mass., 1962, mimeographed lecture notes.zbMATHGoogle Scholar
  3. [3]
    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IRES, 36, (1969), 75–109.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J. Giraud, Cohomologie non-abélienne, Springer-Verlag, Berlin 1971, Grundlehren der mathematischen Wissenschaften, Band 179.zbMATHGoogle Scholar
  5. [5]
    G. Laumon and L. Moret-BaillyChamps algébriques, Springer-Verlag, 1999.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Barbara Fantechi
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly

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