Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-30T21:39:44.790Z Has data issue: false hasContentIssue false
  • English
  • Français

On cocycle bitorsors and gerbes over a Grothendieck topos

Published online by Cambridge University Press:  24 October 2008

K.-H. Ulbrich
K.-H. Ulbrich
Affiliation:
Département de Mathématiques, Université Paris-Nord, C.S.P., Av. J.-B. Clément, 93430 Villetaneuse, France
Get access Rights & Permissions [Opens in a new window]

Extract

The aim of this paper is to give a new description of Giraud's nonabelian cohomology set H2(L), [7], defined for a band L of a Grothendieck topos E as the set of L-equivalence classes of L-gerbes over E. Our description is similar to that for H1 by Čech cohomology, but with cocycles taken from the stack of bitorsors over E. We first continue to study the construction of gerbes by cocycle bitorsors or bouquets [13], originally given in [7] and [4]; Duskin [4, 5] showed that bouquets B and B′ of E give rise to equivalent gerbes if and only if there exist essential equivalences B″ → B and B″ → B′ for another bouquet B″.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 49 - 55
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

‘1’Breen, L.. Bitorseurs et cohomologie nonabélienne. In The Grothendieck Festschrift (ed. Cartier, P. et al. ), (Birkhäuser, 1990).Google Scholar
‘2’Chase, S. U. and Rosenberg, A.. Amitsur cohomology and the Brauer group. Mem. Amer. Math. Soc. 52 (1965), 34–79.Google Scholar
‘3’Deligne, P.. Catégories Tannakiennes. In The Grothendieck Festschrift (ed. Cartier, P. et al. ), (Birkhäuser, 1990).Google Scholar
‘4’Duskin, J.. An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes. Cahiers Topologie Geom. Différentielle Catégoriques 23 (1982), 165–191.Google Scholar
‘5’Duskin, J.. Non-abelian cohomology in a topos (preprint).Google Scholar
‘6’Duskin, J.. An outline of a theory of higher dimensional descent. Bull. Soc. Math. Beig. ser. A 41 (1989), 249–277.Google Scholar
‘7’Giraud, J.. Cohomologie non Abélienne (Springer-Verlag, 1971).CrossRefGoogle Scholar
‘8’Hakim, M.. Topos Annele's et Schémas Relatifs (Springer-Verlag, 1972).CrossRefGoogle Scholar
‘9’Hattori, A.. On groups Hn(S/R) related to the Amitsur cohomology and the Brauer group of commutative rings. Osaka J. Math. 16 (1979), 375–382.Google Scholar
‘10’Raeburn, I. and Taylor, J. L.. The bigger Brauer group and étale cohomology. Pacific J. Math. 119 (1985), 445–462.CrossRefGoogle Scholar
‘11’Rivano, N. Saavedra. Catégories Tannakiennes. Lecture Notes in Math. vol. 265 (Springer-Verlag, 1972).CrossRefGoogle Scholar
‘12’Ulbrich, K.-H.. Group cohomology for Picard categories. J. Algebra 91 (1984), 464–498.CrossRefGoogle Scholar
‘13’Ulbrich, K.-H.. On the correspondence between gerbes and bouquets. Math. Proc. Cambridge Philos. Soc. 108 (1990), 1–5.CrossRefGoogle Scholar
‘14’Villamayor, O. E. and Zelinsky, D.. Brauer groups and Amitsur cohomology for general commutative ring extensions. J. Pure Appl. Algebra 10 (1977), 19–55.CrossRefGoogle Scholar