Bivariant derived algebraic cobordism
Author:
Toni Annala
Journal:
J. Algebraic Geom. 30 (2021), 205-252
DOI:
https://doi.org/10.1090/jag/754
Published electronically:
June 12, 2020
MathSciNet review:
4233182
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic $K^0$ analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.
References
- T. Annala, Ample line bundles, global generation and $K_0$ on quasi-projective derived schemes, preprint, 2019.
- Dave Anderson and Sam Payne, Operational $K$-theory, Doc. Math. 20 (2015), 357–399. MR 3398716
- Frédéric Déglise, Bivariant theories in motivic stable homotopy, Doc. Math. 23 (2018), 997–1076. MR 3874952
- William Fulton, 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], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 404257
- William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI https://doi.org/10.1090/memo/0243
- Joseph Gubeladze, Toric varieties with huge Grothendieck group, Adv. Math. 186 (2004), no. 1, 117–124. MR 2065508, DOI https://doi.org/10.1016/j.aim.2003.07.009
- José Luis González and Kalle Karu, Bivariant algebraic cobordism, Algebra Number Theory 9 (2015), no. 6, 1293–1336. MR 3397403, DOI https://doi.org/10.2140/ant.2015.9.1293
- Moritz Kerz, Florian Strunk, and Georg Tamme, Algebraic $K$-theory and descent for blow-ups, Invent. Math. 211 (2018), no. 2, 523–577. MR 3748313, DOI https://doi.org/10.1007/s00222-017-0752-2
- A. Khan and D. Rydh, Virtual Cartier divisors and blow-ups, arXiv:math/1802.05702v2[math.AG], 2019.
- Marc Levine, Comparison of cobordism theories, J. Algebra 322 (2009), no. 9, 3291–3317. MR 2567421, DOI https://doi.org/10.1016/j.jalgebra.2009.03.032
- Marc Levine, Intersection theory in algebraic cobordism, J. Pure Appl. Algebra 221 (2017), no. 7, 1645–1690. MR 3614972, DOI https://doi.org/10.1016/j.jpaa.2016.12.022
- M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
- M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR 2485880, DOI https://doi.org/10.1007/s00222-008-0160-8
- P. Lowrey and T. Schürg, Grothendieck-Riemann-Roch for derived schemes, preprint, 2012.
- Parker E. Lowrey and Timo Schürg, Derived algebraic cobordism, J. Inst. Math. Jussieu 15 (2016), no. 2, 407–443. MR 3466543, DOI https://doi.org/10.1017/S1474748014000334
- Cristina Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), no. 2, 201–245. MR 2877433, DOI https://doi.org/10.1090/S1056-3911-2011-00606-1
- J. Lurie, Spectral algebraic geometry, preprint, 2018.
- Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633, DOI https://doi.org/10.1090/memo/0902
- Shoji Yokura, Oriented bivariant theories. I, Internat. J. Math. 20 (2009), no. 10, 1305–1334. MR 2574317, DOI https://doi.org/10.1142/S0129167X09005777
- Shoji Yokura, Oriented bivariant theory, II: Algebraic cobordism of $S$-schemes, Internat. J. Math. 30 (2019), no. 6, 1950031, 40. MR 3977282, DOI https://doi.org/10.1142/S0129167X19500319
References
- T. Annala, Ample line bundles, global generation and $K_0$ on quasi-projective derived schemes, preprint, 2019.
- Dave Anderson and Sam Payne, Operational $K$-theory, Doc. Math. 20 (2015), 357–399. MR 3398716
- Frédéric Déglise, Bivariant theories in motivic stable homotopy, Doc. Math. 23 (2018), 997–1076. MR 3874952
- William Fulton, 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], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 0404257
- William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI https://doi.org/10.1090/memo/0243
- Joseph Gubeladze, Toric varieties with huge Grothendieck group, Adv. Math. 186 (2004), no. 1, 117–124. MR 2065508, DOI https://doi.org/10.1016/j.aim.2003.07.009
- José Luis González and Kalle Karu, Bivariant algebraic cobordism, Algebra Number Theory 9 (2015), no. 6, 1293–1336. MR 3397403, DOI https://doi.org/10.2140/ant.2015.9.1293
- Moritz Kerz, Florian Strunk, and Georg Tamme, Algebraic $K$-theory and descent for blow-ups, Invent. Math. 211 (2018), no. 2, 523–577. MR 3748313, DOI https://doi.org/10.1007/s00222-017-0752-2
- A. Khan and D. Rydh, Virtual Cartier divisors and blow-ups, arXiv:math/1802.05702v2[math.AG], 2019.
- Marc Levine, Comparison of cobordism theories, J. Algebra 322 (2009), no. 9, 3291–3317. MR 2567421, DOI https://doi.org/10.1016/j.jalgebra.2009.03.032
- Marc Levine, Intersection theory in algebraic cobordism, J. Pure Appl. Algebra 221 (2017), no. 7, 1645–1690. MR 3614972, DOI https://doi.org/10.1016/j.jpaa.2016.12.022
- M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
- M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR 2485880, DOI https://doi.org/10.1007/s00222-008-0160-8
- P. Lowrey and T. Schürg, Grothendieck-Riemann-Roch for derived schemes, preprint, 2012.
- Parker E. Lowrey and Timo Schürg, Derived algebraic cobordism, J. Inst. Math. Jussieu 15 (2016), no. 2, 407–443. MR 3466543, DOI https://doi.org/10.1017/S1474748014000334
- Cristina Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), no. 2, 201–245. MR 2877433, DOI https://doi.org/10.1090/S1056-3911-2011-00606-1
- J. Lurie, Spectral algebraic geometry, preprint, 2018.
- Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633, DOI https://doi.org/10.1090/memo/0902
- Shoji Yokura, Oriented bivariant theories. I, Internat. J. Math. 20 (2009), no. 10, 1305–1334. MR 2574317, DOI https://doi.org/10.1142/S0129167X09005777
- Shoji Yokura, Oriented bivariant theory, II: Algebraic cobordism of $S$-schemes, Internat. J. Math. 30 (2019), no. 6, 1950031, 40. MR 3977282, DOI https://doi.org/10.1142/S0129167X19500319
Additional Information
Toni Annala
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T1Z2, Canada
Email:
tannala@math.ubc.ca
Received by editor(s):
November 25, 2018
Received by editor(s) in revised form:
December 8, 2018, and August 25, 2019
Published electronically:
June 12, 2020
Article copyright:
© Copyright 2020
University Press, Inc.
