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DERIVED ALGEBRAIC COBORDISM

Part of: (Co)homology theory Cycles and subschemes Homology and cohomology theories

Published online by Cambridge University Press:  30 October 2014

Parker E. Lowrey  and
Timo Schürg
Parker E. Lowrey
Affiliation:
Department of Mathematics, Middlesex College, The University of Western Ontario, London, ON, Canada (plowrey@uwo.ca)
Timo Schürg
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany (timo_schuerg@operamail.com)
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Abstract

We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.

Keywords

algebraic cobordismvirtual fundamental classesquasi-smooth schemes

MSC classification

Primary: 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Secondary: 14C40: Riemann-Roch theorems 55N22: Bordism and cobordism theories, formal group laws
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 2 , April 2016 , pp. 407 - 443
Copyright
© Cambridge University Press 2014 

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References

Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128(1) (1997), 45–88, MR 1437495 (98e:14022).CrossRefGoogle Scholar
Ciocan-Fontanine, I. and Kapranov, M., Virtual fundamental classes via dg-manifolds, Geom. Topol. 13(3) (2009), 1779–1804, MR 2496057 (2010e:14012).CrossRefGoogle Scholar
Douady, A. and Verdier, J. L., Séminaire de géométrie analytique, Astérisque, Volume 36–37 (Société mathématique de France, 1976).Google Scholar
Fulton, W., Intersection theory, 2nd ed., , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Volume 2, (Springer-Verlag, Berlin, 1998), MR 1644323 (99d:14003).Google Scholar
Joyce, D., D-manifolds and d-orbifolds: a theory of derived differential geometry, 2012, available at http://people.maths.ox.ac.uk/joyce/dmbook.pdf.Google Scholar
Levine, M. and Morel, F., Algebraic Cobordism, Springer Monographs in Mathematics, (Springer, Berlin, 2007), MR 2286826 (2008a:14029).Google Scholar
Levine, M. and Pandharipande, R., Algebraic cobordism revisited, Invent. Math. 176(1) (2009), 63–130, MR 2485880 (2010h:14033).Google Scholar
Li, J. and Tian, G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11(1) (1998), 119–174, MR 1467172 (99d:14011).Google Scholar
Lowrey, P. and Schürg, T., Grothendieck–Riemann–Roch for derived schemes. Arxiv e-prints.Google Scholar
Lurie, J., Quasi-coherent sheaves and tannaka duality theorems, 2011, available athttp://www.math.harvard.edu/∼lurie/papers/DAG-VIII.pdf.Google Scholar
Manolache, C., Virtual pull-backs, J. Algebraic Geom. 21 (2012), 201–245.CrossRefGoogle Scholar
Quillen, D., On the (co-) homology of commutative rings, in Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pp. 65–87 (American Mathematical Society, Providence, RI, 1970), MR 0257068 (41 #1722).Google Scholar
Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7 (1971), 29–56, MR 0290382 (44 #7566).Google Scholar
Serre, J.-P., Local Algebra (Springer, 2000).Google Scholar
Spivak, D. I., Derived smooth manifolds, Duke Math. J. 153(1) (2010), 55–128, MR 2641940 (2012a:57043).Google Scholar