Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T03:32:11.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic and topological aspects of the schematization functor

Part of: Homotopy theory Compact analytic spaces Cycles and subschemes

Published online by Cambridge University Press:  01 May 2009

L. Katzarkov ,
T. Pantev  and
B. Toën
L. Katzarkov
Affiliation:
Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, USA (email: l.katzarkov@math.miami.edu)
T. Pantev
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA (email: tpantev@math.upenn.edu)
B. Toën
Affiliation:
Laboratoire Emile Picard, Université Paul Sabatier Bat 1R2, 31062 Toulouse Cedex 9, France (email: bertrand.toen@math.univ-toulouse.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types, namely cosimplicial Hopf alegbras and equivariant cosimplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor that are helpful for describing the schematization of smooth projective complex varieties. In a companion paper, these results are used in the construction of a non-abelian Hodge structure on the schematic homotopy type of a smooth projective variety.

Keywords

schematic homotopy typeshomotopy theorynon-abelian Hodge theory

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 32J27: Compact Kähler manifolds 55P62: Rational homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 633 - 686
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Anderson, M. P., Exactness properties of profinite completion functors, Topology 13 (1974), 229–239.CrossRefGoogle Scholar
[2]Artin, M., Grothendieck, A. and Verdier, J. L., Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, vol. 269 (Springer, Berlin, 1972).Google Scholar
[3]Beilinson, A., On the derived category of perverse sheaves, in K -theory, arithmetic and geometry, Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), 240–264.Google Scholar
[4]Beke, T., Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), 447–475.CrossRefGoogle Scholar
[5]Blander, B., Local projective model structure on simplicial presheaves, K-theory 24 (2001), 283–301.CrossRefGoogle Scholar
[6]Bousfield, A. K. and Gugenheim, V. K. A. M., On PL DeRham theory and rational homotopy type, Mem. Amer. Math. Soc. 179 (1976).Google Scholar
[7]Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Mathematics, vol. 304 (Springer, Berlin, New York, 1972).CrossRefGoogle Scholar
[8]Brown, E. H. and Szczarba, R. H., Rational and real homotopy theory with arbitrary fundamental groups, Duke Math. J. 71 (1993), 299–316.CrossRefGoogle Scholar
[9]Demazure, M. and Gabriel, P., Groupes algébriques, I (Masson/North-Holland, Paris, 1970).Google Scholar
[10]Demazure, M. and Grothendieck, A., Schémas en groupes I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, vol. 151 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[11]Goerss, P. and Jardine, J. F., Simplicial homotopy theory, Progress in Mathematics, vol. 174 (Birkhäuser, Basel, 1999).CrossRefGoogle Scholar
[12]Hinich, V. A. and Schechtman, V. V., Homotopy limits of homotopy algebras in K -theory, arithmetic and geometry, Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), 240–264.Google Scholar
[13]Hirschhorn, P., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99 (American Mathematical Society, Providence, RI, 2003).Google Scholar
[14]Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1998).Google Scholar
[15]Jardine, J. F., Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), 35–87.CrossRefGoogle Scholar
[16]Katzarkov, L., Pantev, T. and Toën, B., Schematic homotopy types and non-abelian Hodge theory, Compos. Math. 144 (2008), 582–632.CrossRefGoogle Scholar
[17]Kriz, I., p-adic homotopy theory, Topol. Appl. 52 (1993), 279–308.CrossRefGoogle Scholar
[18]Maclane, S., Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5 (Springer, New York, Berlin, 1971).Google Scholar
[19]May, J. P., Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, vol. 11 (D. Van Nostrand Co., Princeton, Toronto, London, 1967).Google Scholar
[20]Pridham, J. P., Pro-algebraic homotopy types, Proc. Lond. Math. Soc. (3) 97 (2008), 273–338.CrossRefGoogle Scholar
[21]Pridham, J. P., Galois actions on homotopy groups, Preprint (2007), arXiv:0712.0928.Google Scholar
[22]Pridham, J. P., Non-abelian real Hodge theory for proper varieties, Preprint (2006), arXiv:math/0611686.Google Scholar
[23]Quillen, D., Homotopical algebra, Lecture Notes in Mathematics, vol. 43 (Springer, Berlin, 1967).CrossRefGoogle Scholar
[24]Serre, J. P., Cohomologie Galoisienne, Lecture Notes in Mathematics, fifth edition, vol. 5 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[25]Simpson, C., Secondary Kodaira-Spencer classes and non-abelian Dolbeault cohomology, Preprint (1997), arXiv:alg-geom/9712020.Google Scholar
[26]Schwede, S. and Shipley, B., Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), 491–511.CrossRefGoogle Scholar
[27]Sullivan, D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331.CrossRefGoogle Scholar
[28]Tanré, D., Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, vol. 1025 (Springer, Berlin, 1983).CrossRefGoogle Scholar
[29]Toën, B., Champs affines, Selecta Math. (N.S.) 12 (2006), 39–135.CrossRefGoogle Scholar
[30]Toën, B., Vers une interprétation Galoisienne de la théorie de l’homotopie, Cahiers de Top. et Géom. Diff. Cat. 43 (2002), 257–312.Google Scholar
[31]Toën, B. and Vezzosi, G., Homotopy algebraic geometry I: Topos theory, Adv. Math. 193 (2005), 257–372.CrossRefGoogle Scholar