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

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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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*Equivariant algebraic cobordism*, J. Reine Angew. Math. **684** (2013), 87–112. MR **3181557**, DOI 10.1515/crelle-2011-0004
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*Algebraic spaces*, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR **0302647**, DOI 10.1007/BFb0059750
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*Algebraic spaces and stacks*, American Mathematical Society Colloquium Publications, vol. 62, American Mathematical Society, Providence, RI, 2016. MR **3495343**, DOI 10.1090/coll/062
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*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**
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*Chow groups with coefficients*, Doc. Math. **1** (1996), No. 16, 319–393. MR **1418952**
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*Some theorems on the $K$-theory of coherent sheaves*, Comm. Algebra **7** (1979), no. 14, 1489–1508. MR **541048**, DOI 10.1080/00927877908822414
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*Stacks Project*, https://stacks.math.columbia.edu, 2019.
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*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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*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
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*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
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*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**

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.