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Towards Vorst's conjecture in positive characteristic

Part of: Higher algebraic $K$-theory

Published online by Cambridge University Press:  20 May 2021

Moritz Kerz ,
Florian Strunk  and
Georg Tamme
Moritz Kerz
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany moritz.kerz@ur.de
Florian Strunk
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany florian.strunk@ur.de
Georg Tamme
Affiliation:
Institut für Mathematik, Fachbereich 08, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany georg.tamme@uni-mainz.de
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Abstract

Vorst's conjecture relates the regularity of a ring with the $\mathbb {A}^{1}$-homotopy invariance of its $K$-theory. We show a variant of this conjecture in positive characteristic.

Keywords

Cartier isomorphismVorst's conjecturevaluation ringK-regularity

MSC classification

Primary: 19D35: Negative $K$-theory, NK and Nil
Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 6 , June 2021 , pp. 1143 - 1171
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Copyright
© The Author(s) 2021

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Footnotes

The authors are supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg).

References

Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.CrossRefGoogle Scholar
Borger, J., The basic geometry of Witt vectors, I: The affine case, Algebra Number Theory 5 (2011), 231–285; MR 2833791.CrossRefGoogle Scholar
Bourbaki, N., Commutative algebra. Chapters 1–7, in Elements of Mathematics (Springer, Berlin, 1989); translated from the French, reprint of the 1972 edition.Google Scholar
Cisinski, D.-C., Descente par éclatements en $K$théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425–448.CrossRefGoogle Scholar
Clausen, D., Mathew, A. and Morrow, M., $K$-theory and topological cyclic homology of henselian pairs, J. Amer. Math. Soc. 34 (2021), 411–473.Google Scholar
Cortiñas, G., Haesemeyer, C. and Weibel, C., $K$-regularity, $cdh$-fibrant Hochschild homology, and a conjecture of Vorst, J. Amer. Math. Soc. 21 (2008), 547–561.CrossRefGoogle Scholar
Datta, R. and Smith, K. E., Excellence in prime characteristic, in Local and global methods in algebraic geometry, Contemporary Mathematics, vol. 712 (American Mathematical Society, Providence, RI, 2018), 105–116.CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967).CrossRefGoogle Scholar
Fujiwara, K. and Kato, F., Foundations of rigid geometry. I, EMS Monographs in Mathematics (European Mathematical Society (EMS), Zürich, 2018); MR 3752648.CrossRefGoogle Scholar
Gabber, O., Letter to M. Kerz dated 5 July 2018.Google Scholar
Gabber, O. and Orgogozo, F., Sur la $p$-dimension des corps, Invent. Math. 174 (2008), 47–80.CrossRefGoogle Scholar
Gabber, O. and Ramero, L., Almost ring theory, Lecture Notes in Mathematics, vol. 1800 (Springer, Berlin, 2003).CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., The de Rham-Witt complex and $p$-adic vanishing cycles, J. Amer. Math. Soc. 19 (2006), 1–36; MR 2169041.CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., On the $K$-theory of complete regular local $\mathbb{F}_p$-algebras, Topology 45 (2006), 475–493.CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., On a conjecture of Vorst, Math. Z. 270 (2012), 445–452.CrossRefGoogle Scholar
Geisser, T. and Levine, M., The $K$-theory of fields in characteristic $p$, Invent. Math. 139 (2000), 459–493.CrossRefGoogle Scholar
Greco, S., Two theorems on excellent rings, Nagoya Math. J. 60 (1976), 139–149.CrossRefGoogle Scholar
Hesselholt, L., On the $p$-typical curves in Quillen's $K$-theory, Acta Math. 177 (1996), 1–53.CrossRefGoogle Scholar
Hesselholt, L., The big de Rham-Witt complex, Acta Math. 214 (2015), 135–207; MR 3316757.CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the $K$-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–101.CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the de Rham-Witt complex in mixed characteristic, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 1–43.CrossRefGoogle Scholar
Hiller, H., $\lambda$-rings and algebraic $K$-theory, J. Pure Appl. Algebra 20 (1981), 241–266; MR 604319.CrossRefGoogle Scholar
Illusie, L., Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239 (Springer, Berlin, 1971).CrossRefGoogle Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.CrossRefGoogle Scholar
Katz, N., Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math. Inst. Hautes Études Sci. 39 (1970), 175–232.CrossRefGoogle Scholar
Kelly, S. and Morrow, M., $K$-theory of valuation rings, Compos. Math. 157 (2021), 1121–1142.CrossRefGoogle Scholar
Kerz, M., Strunk, F. and Tamme, G., Algebraic $K$-theory and descent for blow-ups, Invent. Math. 211 (2018), 523–577.CrossRefGoogle Scholar
Kratzer, C., $\lambda$-structure en $K$-théorie algébrique, Comment. Math. Helv. 55 (1980), 233–254.CrossRefGoogle Scholar
Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, NJ, 2009).CrossRefGoogle Scholar
Lurie, J., Spectral algebraic geometry, Preprint (2018), available online at www.math.ias.edu/~lurie.Google Scholar
Matsumura, H., Commutative algebra, second edition, Mathematics Lecture Note Series, vol. 56 (Benjamin/Cummings, Reading, MA, 1980).Google Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986); translated from the Japanese by Reid, M..Google Scholar
Merkurjev, A. S. and Suslin, A. A., $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136.Google Scholar
Serre, J.-P., Galois cohomology, English edition, Springer Monographs in Mathematics (Springer, Berlin, 2002); translated from the French by Patrick Ion and revised by the author.Google Scholar
The Stacks Project Authors, The Stacks Project (2019), https://stacks.math.columbia.edu.Google Scholar
Thomason, R. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift. Vol. III, Progress in Mathematics, vol. 88 (Birkhäuser, Boston, MA, 1990), 247–435.CrossRefGoogle Scholar
van der Kallen, W., The $K_{2}$ of rings with many units, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 473–515.CrossRefGoogle Scholar
van der Kallen, W., Descent for the $K$-theory of polynomial rings, Math. Z. 191 (1986), 405–415; MR 824442.CrossRefGoogle Scholar
Vorst, T., Localization of the $K$-theory of polynomial extensions, Math. Ann. 244 (1979), 33–53, with an appendix by van der Kallen, Wilberd.CrossRefGoogle Scholar
Weibel, C., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
Weibel, C., An introduction to algebraic K-theory, in The K-book, Graduate Studies in Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 2013).Google Scholar