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K-theory of valuation rings

Part of: Higher algebraic $K$-theory (Co)homology theory

Published online by Cambridge University Press:  20 May 2021

Shane Kelly  and
Matthew Morrow
Shane Kelly
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo152-8551, [email protected]
Matthew Morrow
Affiliation:
CNRS & IMJ-PRG, SU – 4 place Jussieu, Case 247, 75252Paris, [email protected]
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Abstract

We prove several results showing that the algebraic $K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.

Keywords

K-theoryvaluation rings

MSC classification

Primary: 19D50: Computations of higher $K$-theory of rings
Secondary: 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 6 , June 2021 , pp. 1121 - 1142
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Copyright
© The Author(s) 2021

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References

Antieau, B., Gepner, D. and Heller, J., $K$-theoretic obstructions to bounded $t$-structures, Invent. Math. 216 (2019), 241–300.10.1007/s00222-018-00847-0CrossRefGoogle Scholar
Bhatt, B., Lurie, J. and Mathew, A., Revisiting the de Rham–Witt complex. Preprint (2018), arXiv:1805.05501.Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397.10.1007/s10240-019-00102-zCrossRefGoogle Scholar
Bhatt, B., Morrow, M. and Scholze, P., Topological Hochschild homology and integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310.10.1007/s10240-019-00106-9CrossRefGoogle Scholar
Bhatt, B. and Scholze, P., Projectivity of the Witt vector affine Grassmannian, Invent. Math. 209 (2017), 329–423.10.1007/s00222-016-0710-4CrossRefGoogle Scholar
Cisinski, D.-C., Descente par éclatements en $K$-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425–448.10.4007/annals.2013.177.2.2CrossRefGoogle 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
Elmanto, E., Hoyois, M., Iwasa, R. and Kelly, S., Cdh descent, cdarc descent, and Milnor excision. Math. Ann. 379 (2021), 1011–1045.10.1007/s00208-020-02083-5CrossRefGoogle Scholar
Gabber, O., K-theory of Henselian local rings and Henselian pairs, in Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemporary Mathematics, vol. 126 (American Mathematical Society, Providence, RI, 1992), 59–70.Google Scholar
Gabber, O. and Kelly, S., Points in algebraic geometry, J. Pure Appl. Algebra 219 (2015), 4667–4680.10.1016/j.jpaa.2015.03.001CrossRefGoogle Scholar
Gabber, O. and Ramero, L., Almost ring theory, Lecture Notes in Mathematics, vol. 1800 (Springer, Berlin, 2003).10.1007/b10047CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., Topological cyclic homology of schemes, in Algebraic K-theory (Seattle, WA, 1997), Proceedings of Symposia in Pure Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1999), 41–87.10.1090/pspum/067/1743237CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., On the $K$-theory of complete regular local $\Bbb F_p$-algebras, Topology 45 (2006), 475–493.10.1016/j.top.2005.09.002CrossRefGoogle Scholar
Geisser, T. and Levine, M., The $K$-theory of fields in characteristic $p$, Invent. Math. 139 (2000), 459–493.10.1007/s002220050014CrossRefGoogle Scholar
Gersten, S. M., $K$-theory of free rings, Comm. Algebra 1 (1974), 39–64.10.1080/00927877408548608CrossRefGoogle Scholar
Glaz, S., Commutative coherent rings, Lecture Notes in Mathematics, vol. 1371 (Springer, Berlin, 1989).10.1007/BFb0084570CrossRefGoogle Scholar
Goodwillie, T. G. and Lichtenbaum, S., A cohomological bound for the $h$-topology, Amer. J. Math. 123 (2001), 425–443.10.1353/ajm.2001.0016CrossRefGoogle Scholar
Haesemeyer, C., Descent properties of homotopy $K$-theory, Duke Math. J. 125 (2004), 589–620.10.1215/S0012-7094-04-12534-5CrossRefGoogle Scholar
Hesselholt, L., On the $p$-typical curves in Quillen's $K$-theory, Acta Math. 177 (1996), 1–53.10.1007/BF02392597CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the $K$-theory of local fields, Ann. of Math. (2) 158 (2003), 1–113.10.4007/annals.2003.158.1CrossRefGoogle Scholar
Hoyois, M., A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula, Algebr. Geom. Topol. 14 (2014), 3603–3658.10.2140/agt.2014.14.3603CrossRefGoogle Scholar
Huber, A. and Kelly, S., Differential forms in positive characteristic, II: cdh-descent via functorial Riemann-Zariski spaces, Algebra Number Theory 12 (2018), 649–692.CrossRefGoogle Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.10.24033/asens.1374CrossRefGoogle Scholar
Illusie, L., Laszlo, Y. and Orgogozo, F. (eds), Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents (Séminaire à l’École Polytechnique 2006–2008), Astérisque, vol. 363–364 (Société Mathématique de France, Paris, 2014), with the collaboration of Frédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, Joël Riou, Benoît Stroh, Michael Temkin and Weizhe Zheng.Google Scholar
Kelly, S., Vanishing of negative $K$-theory in positive characteristic, Compos. Math. 150 (2014), 1425–1434.CrossRefGoogle Scholar
Kelly, S., Voevodsky motives and $l$dh-descent, Astérisque 391 (2017).Google Scholar
Kelly, S., A better comparison of cdh- and $l$dh-cohomologies, Nagoya Math. J. 236 (2019), 183–213.CrossRefGoogle Scholar
Kerz, M. and Strunk, F., On the vanishing of negative homotopy $K$-theory, J. Pure Appl. Algebra 221 (2017), 1641–1644.CrossRefGoogle Scholar
Kerz, M., Strunk, F. and Tamme, G., Algebraic $K$-theory and descent for blow-ups, Invent. Math. 211 (2018), 523–577.10.1007/s00222-017-0752-2CrossRefGoogle Scholar
Kratzer, C., $\lambda$-structure en $K$-théorie algébrique, Comment. Math. Helv. 55 (1980), 233–254.CrossRefGoogle Scholar
Land, M. and Tamme, G., On the $K$-theory of pullbacks, Ann. of Math. (2) 190 (2019), 877–930.10.4007/annals.2019.190.3.4CrossRefGoogle Scholar
Matsumura, H., Commutative algebra, second edition, Mathematics Lecture Note Series, vol. 56 (Benjamin/Cummings, Reading, MA, 1980).Google Scholar
Morrow, M., Pro cdh-descent for cyclic homology and $K$-theory, J. Inst. Math. Jussieu 15 (2016), 539–567.10.1017/S1474748014000413CrossRefGoogle Scholar
Morrow, M., $K$-theory and logarithmic Hodge–Witt sheaves of formal schemes in characteristic $p$, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), 1537–1601.CrossRefGoogle Scholar
Nikolaus, T. and Scholze, P., On topological cyclic homology, Acta Math. 221 (2018), 203–409.10.4310/ACTA.2018.v221.n2.a1CrossRefGoogle Scholar
Osofsky, B. L., Global dimension of valuation rings, Trans. Amer. Math. Soc. 127 (1967), 136–149.CrossRefGoogle Scholar
Osofsky, B. L., The subscript of $\aleph_{n}$, projective dimension, and the vanishing of $\varprojlim^{(n)}$, Bull. Amer. Math. Soc. 80 (1974), 8–26.CrossRefGoogle Scholar
Quillen, D., Higher algebraic K-theory: I, in Higher K-theories, ed. H. Bass, Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, Heidelberg, 1973).CrossRefGoogle Scholar
Raynaud, M. and Gruson, L., Critères de platitude et de projectivité. Techniques de “platification” d'un module, Invent. Math. 13 (1971), 1–89.CrossRefGoogle Scholar
Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548 (Kluwer Academic, Dordrecht, 2000), 117–189.Google Scholar
Tamme, G., Excision in algebraic $K$-theory revisited, Compos. Math. 154 (2018), 1801–1814.CrossRefGoogle Scholar
Temkin, M., Inseparable local uniformization, J. Algebra 373 (2013), 65–119.CrossRefGoogle Scholar
Voevodsky, V., Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra 214 (2010), 1384–1398.CrossRefGoogle Scholar
Voevodsky, V., Unstable motivic homotopy categories in Nisnevich and cdh-topologies, J. Pure Appl. Algebra 214 (2010), 1399–1406.CrossRefGoogle Scholar
Weibel, C. A., Homotopy algebraic K-theory, in Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Contemporary Mathematics, vol. 83 (American Mathematical Society, Providence, RI, 1989), 461–488.10.1090/conm/083/991991CrossRefGoogle Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar