Abstract
Let X be a proper scheme over a field k which satisfies Serre’s condition S 2 and G a reductive group over k. We prove that the functor of principal G-bundles, defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Artin M.: Versal deformations and algebraic stacks. Invent. Math. 27, 165–189 (1974)
Aoki M.: HOM stacks. Manuscripta Math. 119(1), 37–56 (2006)
Aoki, M.: Erratum: “Hom stacks” Manuscripta Math. 121(1), 135 (2006)
Baranovsky V.: Algebraization of bundles on non-proper schemes. Trans. Am. Math. Soc. 362, 427–439 (2010)
Bruns, W., Herzog, J.: Cohen–Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, Berlin
Grothendieck, A.: EGA III1, “Étude cohomologique des faiseaux cohérents”. Publ. Math. IHES (11), 5–167 (1961)
Grothendieck, A.: EGA III2, “Étude cohomologique des faiseaux cohérents”. Publ. Math. IHES (17), 17–137 (1963)
Grothendieck, A.: EGA IV 2, “Étude locale des schémas and des morphismes des schémas”. Publ. Math. IHES (24), 5–231 (1965)
Grothendieck, A.: EGA IV 3, “Étude locale des schémas and des morphismes des schémas”. Publ. Math. IHES (28), 5–255 (1966)
Haboush W.J.: Homogeneous vector bundles and reductive subgroups of reductive algebraic groups. Am. J. Math 100(6), 1123–1137 (1978)
Illusie, L.: Complexe Cotangent et Déformations. Part I: LNM 239. Springer, Berlin (1971) (Part II: LNM 283, Springer (1972))
Lieblich M.: Moduli of complexes on a proper morphism. J. Algebr. Geom. 15(1), 175–206 (2006)
Laumon, G., Morer-Bailly, L.: Champs algébriques. Springer, Berlin (2000)
Grothendieck, A., et al.: SGA 2, Cohomologie locale des faiseaux cohérents et théorèmes de Lefschetz locaux et globaux. Advanced Studies in Pure Mathematics, vol. 2, North-Holland, Amsterdam (1968)
Watts C.E.: Intrinsic characterizations of some additive functors. Proc. Am. Math. Soc. 11, 5–8 (1960)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The main result of this paper was conjectured by Vladimir Drinfeld whom the author thanks for useful discussions and encouragement. Thanks are also due to Victor Ginzburg for helping to clarify the role of Serre’s S 2 condition (which is closely related to the concept of a reflexive sheaf), and to the reviewer for pointing out several sloppy statements and gaps in the proofs of the original version of this paper. This work was supported by the Sloan Research Fellowship and a UCI Teaching Relief Grant.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Baranovsky, V. Bundles on non-proper schemes: representability. Sel. Math. New Ser. 16, 297–313 (2010). https://doi.org/10.1007/s00029-010-0021-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-010-0021-3