Equivariant connective $K$-theory
Authors:
Nikita A. Karpenko and Alexander S. Merkurjev
Journal:
J. Algebraic Geom. 31 (2022), 181-204
DOI:
https://doi.org/10.1090/jag/773
Published electronically:
October 28, 2021
MathSciNet review:
4372412
Full-text PDF
Abstract |
References |
Additional Information
Abstract: For separated schemes of finite type over a field with an action of an affine group scheme of finite type, we construct the bi-graded equivariant connective $K$-theory mapping to the equivariant $K$-homology of Guillot and the equivariant algebraic $K$-theory of Thomason. It has all the standard basic properties as the homotopy invariance and localization. We also get the equivariant version of the Brown-Gersten-Quillen spectral sequence and study its convergence.
References
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- Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2019.
- R. W. Thomason, Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563. MR 921490
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References
- M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64. MR 148722
- Shuang Cai, Algebraic connective $K$-theory and the niveau filtration, J. Pure Appl. Algebra 212 (2008), no. 7, 1695–1715. MR 2400737, DOI 10.1016/j.jpaa.2007.12.002
- Shouxin Dai and Marc Levine, Connective algebraic $K$-theory, J. K-Theory 13 (2014), no. 1, 9–56. MR 3177817, DOI 10.1017/is013012007jkt249
- Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634. MR 1614555, DOI 10.1007/s002220050214
- Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530, DOI 10.1090/coll/056
- Pierre Guillot, Geometric methods for cohomological invariants, Doc. Math. 12 (2007), 521–545. MR 2365912
- Jeremiah Heller and José Malagón-López, Equivariant algebraic cobordism, J. Reine Angew. Math. 684 (2013), 87–112. MR 3181557, DOI 10.1515/crelle-2011-0004
- N. A. Karpenko and A. S. Merkurjev, Chow filtration on representation rings of algebraic groups, Int. Math. Res. Not. IMRN (2021), no. 9, 6691–6717. MR 4251288
- Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
- M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
- Martin Olsson, Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, vol. 62, American Mathematical Society, Providence, RI, 2016. MR 3495343, DOI 10.1090/coll/062
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129
- Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393. MR 1418952
- Clayton Sherman, Some theorems on the $K$-theory of coherent sheaves, Comm. Algebra 7 (1979), no. 14, 1489–1508. MR 541048, DOI 10.1080/00927877908822414
- Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2019.
- R. W. Thomason, Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563. MR 921490
- R. W. Thomason, Equivariant algebraic vs. topological $K$-homology Atiyah-Segal-style, Duke Math. J. 56 (1988), no. 3, 589–636. MR 948534, DOI 10.1215/S0012-7094-88-05624-4
- Burt Totaro, The Chow ring of a classifying space, Algebraic $K$-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 249–281. MR 1743244, DOI 10.1090/pspum/067/1743244
- Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104. MR 2223406
Additional Information
Nikita A. Karpenko
Affiliation:
Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada
MR Author ID:
290050
Email:
karpenko@ualberta.ca
Alexander S. Merkurjev
Affiliation:
Department of Mathematics, University of California, Los Angeles, California
MR Author ID:
191878
ORCID:
0000-0002-4447-1838
Email:
merkurev@math.ucla.edu
Received by editor(s):
October 31, 2019
Received by editor(s) in revised form:
June 9, 2020, and August 1, 2020
Published electronically:
October 28, 2021
Additional Notes:
A part of the work by the first author was done during his visit of the Institut des Hautes Etudes Scientifiques in September-October 2019. The second author was supported by the NSF grant DMS #1801530.
Article copyright:
© Copyright 2021
University Press, Inc.
