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Derived moduli of schemes and sheaves

Published online by Cambridge University Press:  08 December 2011

J.P. Pridham
J.P. Pridham*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, [email protected]
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Abstract

We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

Keywords

Derived moduli
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 1 , August 2012 , pp. 41 - 85
Copyright
Copyright © ISOPP 2011

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References

Art.Artin, M.. Versal deformations and algebraic stacks. Invent. Math. 27:165–189, 1974.Google Scholar
Ber.Bergner, Julia E.. A model category structure on the category of simplicial categories. Trans. Amer. Math. Soc. 359(5):2043–2058 (electronic), 2007.CrossRefGoogle Scholar
Bre.Breen, Lawrence. Extensions of abelian sheaves and Eilenberg-MacLane algebras. Invent. Math. 9:15–44, 1969/1970.Google Scholar
Car.Cartan, Henri. Sur les groupes d'Eilenberg-Mac Lane. II. Proc. Nat. Acad. Sci. U. S. A. 40:704–707, 1954.CrossRefGoogle ScholarPubMed
CFK1.Ciocan-Fontanine, Ionuţ and Kapranov, Mikhail. Derived Quot schemes. Ann. Sci. École Norm. Sup. (4), 34(3):403–440, 2001.Google Scholar
CFK2.Ciocan-Fontanine, Ionuţ and Kapranov, Mikhail M.. Derived Hilbert schemes. J. Amer. Math. Soc. 15(4):787–815 (electronic), 2002.Google Scholar
CR.Cegarra, A. M. and Remedios, Josué. The relationship between the diagonal and the bar constructions on a bisimplicial set. Topology Appl. 153(1):21–51, 2005.Google Scholar
DK.Dwyer, W. G. and Kan, D. M.. Homotopy theory and simplicial groupoids. Nederl. Akad. Wetensch. Indag. Math. 46(4):379–385, 1984.Google Scholar
GJ.Goerss, Paul G. and Jardine, John F.. Simplicial homotopy theory, Progress in Mathematics 174. Birkhäuser Verlag, Basel, 1999.Google Scholar
Hüt.Hüttemann, Thomas. On the derived category of a regular toric scheme. Geom. Dedicata 148:175–203, 2010. arXiv:0805.4089v2 [math.KT].CrossRefGoogle Scholar
Ill1.Illusie, Luc. Complexe cotangent et déformations. II. Lecture Notes in Math. 283. Springer-Verlag, Berlin, 1972.Google Scholar
Ill2.Illusie, Luc. Cotangent complex and deformations of torsors and group schemes. In Toposes, algebraic geometry and logic (Conf., Dalhousie Univ., Halifax, N.S., 1971), pages 159–189. Lecture Notes in Math. 274. Springer, Berlin, 1972.Google Scholar
Lur1.Lurie, J.. Derived Algebraic Geometry. PhD thesis, M.I.T., 2004. www.math.harvard.edu/~lurie/papers/DAG.pdf or http://hdl.handle.net/1721.1/30144.Google Scholar
Lur2.Lurie, Jacob. A survey of elliptic cohomology. Available at: http://www-math.mit.edu/~lurie/papers/survey.pdf, 2007.Google Scholar
Pri1.Pridham, J. P.. Presenting higher stacks as simplicial schemes. arXiv:0905.4044v2 [math.AG], submitted, 2009.Google Scholar
Pri2.Pridham, J. P.. Constructing derived moduli stacks. arXiv:1101.3300v1 [math.AG], 2010.Google Scholar
Pri3.Pridham, J. P.. Representability of derived stacks. arXiv:1011.2189v2 [math.AG], submitted, 2010.Google Scholar
Toë1.Toën, Bertrand. Higher and derived stacks: a global overview. arXiv math/0604504v3, 2006.Google Scholar
Toë2.Toën, Bertrand. Flat descent for Artin n-stacks. arXiv:0911.3554v2 [math.AG], 2009.Google Scholar
TV1.Toën, Bertrand and Vezzosi, Gabriele. Segal topoi and stacks over segal categories. arXiv:math/0212330v2 [math.AG], 2002.Google Scholar
TV2.Toën, Bertrand and Vezzosi, Gabriele. Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc. 193(902):x+224, 2008. arXiv math.AG/0404373 v7.Google Scholar
Wei.Weibel, Charles A.. An introduction to homological algebra. Cambridge University Press, Cambridge, 1994Google Scholar