Algebra & Number Theory

Bivariant algebraic cobordism

José González and Kalle Karu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We associate a bivariant theory to any suitable oriented Borel–Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.

Article information

Source
Algebra Number Theory, Volume 9, Number 6 (2015), 1293-1336.

Dates
Received: 28 January 2013
Revised: 21 April 2015
Accepted: 20 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842372

Digital Object Identifier
doi:10.2140/ant.2015.9.1293

Mathematical Reviews number (MathSciNet)
MR3397403

Zentralblatt MATH identifier
1349.14084

Subjects
Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14C15: (Equivariant) Chow groups and rings; motives 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14M25: Toric varieties, Newton polyhedra [See also 52B20] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 57R85: Equivariant cobordism

Keywords
algebraic cobordism bivariant and operational theories operational (equivariant) cobordism operational equivariant cobordism of toric varieties

Citation

González, José; Karu, Kalle. Bivariant algebraic cobordism. Algebra Number Theory 9 (2015), no. 6, 1293--1336. doi:10.2140/ant.2015.9.1293. https://projecteuclid.org/euclid.ant/1510842372


Export citation

References

  • D. Anderson and S. Payne, “Operational $K$-theory”, Doc. Math. 20 (2015), 357–399.
  • A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics 126, Springer, New York, 1991.
  • M. Brion, “Equivariant Chow groups for torus actions”, Transform. Groups 2:3 (1997), 225–267.
  • M. Brion and M. Vergne, “An equivariant Riemann–Roch theorem for complete, simplicial toric varieties”, J. Reine Angew. Math. 482 (1997), 67–92.
  • B. Calmès, V. Petrov, and K. Zainoulline, “Invariants, torsion indices and oriented cohomology of complete flags”, Ann. Sci. Éc. Norm. Supér. $(4)$ 46:3 (2013), 405–448.
  • D. Deshpande, “Algebraic cobordism of classifying spaces”, preprint, 2009.
  • D. Edidin and W. Graham, “Equivariant intersection theory”, Invent. Math. 131:3 (1998), 595–634.
  • W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, 1993.
  • W. Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer, Berlin, 1998.
  • W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31, Amer. Math. Soc., Providence, RI, 1981.
  • H. Gillet, “Homological descent for the $K$-theory of coherent sheaves”, pp. 80–103 in Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982), edited by A. Bak, Lecture Notes in Math. 1046, Springer, Berlin, 1984.
  • J. L. González and K. Karu, “Descent for algebraic cobordism”, J. Algebraic Geom. 24 (2015), 787–804.
  • J. Heller and J. Malagón-López, “Equivariant algebraic cobordism”, J. Reine Angew. Math. 684 (2013), 87–112.
  • S.-i. Kimura, “Fractional intersection and bivariant theory”, Comm. Algebra 20:1 (1992), 285–302.
  • A. Krishna, “Equivariant cobordism of schemes”, Doc. Math. 17 (2012), 95–134.
  • A. Krishna and V. Uma, “The algebraic cobordism ring of toric varieties”, Int. Math. Res. Not. 2013:23 (2013), 5426–5464.
  • M. Levine and F. Morel, Algebraic cobordism, Springer, Berlin, 2007.
  • M. Levine and R. Pandharipande, “Algebraic cobordism revisited”, Invent. Math. 176:1 (2009), 63–130.
  • D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer, Berlin, 1994.
  • S. Payne, “Equivariant Chow cohomology of toric varieties”, Math. Res. Lett. 13:1 (2006), 29–41.
  • B. Totaro, “The Chow ring of a classifying space”, pp. 249–281 in Algebraic $K$-theory (Seattle, WA, 1997), edited by W. Raskind and C. Weibel, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999.
  • G. Vezzosi and A. Vistoli, “Higher algebraic $K$-theory for actions of diagonalizable groups”, Invent. Math. 153:1 (2003), 1–44.
  • S. Yokura, “Oriented bivariant theories, I”, Internat. J. Math. 20:10 (2009), 1305–1334.