Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T20:59:14.120Z Has data issue: false hasContentIssue false
  • English
  • Français

HIGHER $K$-THEORY OF FORMS I. FROM RINGS TO EXACT CATEGORIES

Part of: Forms and linear algebraic groups Higher algebraic $K$-theory $K$-theory of forms

Published online by Cambridge University Press:  02 August 2019

Marco Schlichting Open the ORCID record for Marco Schlichting [Opens in a new window]
Marco Schlichting*
Affiliation:
Marco Schlichting, Mathematics Institute, Zeeman Building, University of Warwick, CoventryCV4 7AL, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the analog for the $K$-theory of forms of the $Q=+$ theorem in algebraic $K$-theory. That is, we show that the $K$-theory of forms defined in terms of an $S_{\bullet }$-construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter.

Keywords

higher K-theory of formsgroup completion

MSC classification

Primary: 19G38: Hermitian $K$-theory, relations with $K$-theory of rings
Secondary: 19D06: $Q$- and plus-constructions 11E70: $K$-theory of quadratic and Hermitian forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 4 , July 2021 , pp. 1205 - 1273
Check for updates
Copyright
© The Author(s), 2019. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bak, A., K-theory of Forms, Annals of Mathematics Studies, Volume 98 (Princeton University Press, Princeton, NJ, 1981). University of Tokyo Press, Tokyo.Google Scholar
Bass, H., Algebraic K-theory (W. A. Benjamin, Inc., New York–Amsterdam, 1968).Google Scholar
Baues, H. J., Quadratic functors and metastable homotopy, J. Pure Appl. Algebra 91(1–3) (1994), 49–107.CrossRefGoogle Scholar
Bouc, S., Green Functors and G-sets, Lecture Notes in Mathematics, Volume 1671 (Springer, Berlin, 1997).CrossRefGoogle Scholar
Bourbaki, N., Éléments de mathématique, Algèbre. Chapitre 9 (Springer, Berlin, 2007). Reprint of the 1959 original.Google Scholar
Bousfield, A. K. and Friedlander, E. M., Homotopy theory of 𝛤-spaces, spectra, and bisimplicial sets, in Geometric Applications of Homotopy Theory (Proc. Conf., Evanston, IL, 1977), II, Lecture Notes in Mathematics, Volume 658, pp. 80–130 (Springer, Berlin, 1978).CrossRefGoogle Scholar
Charney, R. and Lee, R., On a theorem of Giffen, Michigan Math. J. 33(2) (1986), 169–186.CrossRefGoogle Scholar
Dotto, E. and Ogle, C., K-theory of Hermitian Mackey functors, real traces, and assembly, Ann. K-theory. in press.Google Scholar
Elmendorf, A. D. and Mandell, M. A., Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205(1) (2006), 163–228.CrossRefGoogle Scholar
Grayson, D., Higher algebraic K-theory. II (after Daniel Quillen), in Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, IL, 1976), Lecture Notes in Mathematics, Volume 551, pp. 217–240 (Springer, Berlin, 1976).Google Scholar
Hesselholt, L. and Madsen, I., Real algebraic $K$ -theory, http://www.math.ku.dk/larsh/papers/s05/, 2015.Google Scholar
Karoubi, M., Périodicité de la K-théorie hermitienne. (French), in Algebraic K-theory, III: Hermitian K-theory and Geometric Applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, Volume 343, pp. 301–411 (Springer, Berlin, 1973).Google Scholar
Karoubi, M., Le théorème fondamental de la K-théorie hermitienne, Ann. of Math. (2) 112(2) (1980), 259–282.CrossRefGoogle Scholar
Karoubi, M., Théorie de Quillen et homologie du groupe orthogonal, Ann. of Math. (2) 112(2) (1980), 207–257.CrossRefGoogle Scholar
Karoubi, M., K-theory, Classics in Mathematics (Springer, Berlin, 2008). An introduction, Reprint of the 1978 edition, With a new postface by the author and a list of errata.Google Scholar
Lurie, J., Course on algebraic $L$ -theory and surgery, http://www.math.harvard.edu/lurie/287x.html, 2013.Google Scholar
MacLane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics, Volume 5 (Springer, New York-Berlin, 1971).Google Scholar
Milnor, J. and Husemoller, D., Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73 (Springer, New York–Heidelberg, 1973).CrossRefGoogle Scholar
Moerdijk, I., Bisimplicial sets and the group-completion theorem, in Algebraic K-theory: Connections with Geometry and Topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Volume 279, pp. 225–240 (Kluwer Acad. Publ., Dordrecht, 1989).Google Scholar
Nesterenko, Yu. P. and Suslin, A. A., Homology of the general linear group over a local ring, and Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53(1) (1989), 121–146.Google Scholar
Quillen, D., Higher algebraic K-theory. I, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, Volume 341, pp. 85–147 (Springer, Berlin, 1973).Google Scholar
Quillen, D., Characteristic classes of representations, in Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, IL, 1976), Lecture Notes in Mathematics, Volume 551, pp. 189–216 (Springer, Berlin, 1976).Google Scholar
Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 270 (Springer, Berlin, 1985).CrossRefGoogle Scholar
Schlichting, M., Higher $K$ -theory of forms II. From exact categories to chain complexes. In preparation.Google Scholar
Schlichting, M., Higher $K$ -theory of forms III. From chain complexes to derived categories. In preparation.Google Scholar
Schlichting, M., Symplectic and orthogonal $K$ -groups of the integers. In preparation.Google Scholar
Schlichting, M., Hermitian K-theory. On a theorem of Giffen, K-Theory 32(3) (2004), 253–267.CrossRefGoogle Scholar
Schlichting, M., Hermitian K-theory of exact categories, J. K-Theory 5(1) (2010), 105–165.CrossRefGoogle Scholar
Schlichting, M., The Mayer–Vietoris principle for Grothendieck–Witt groups of schemes, Invent. Math. 179(2) (2010), 349–433.CrossRefGoogle Scholar
Schlichting, M., Euler class groups and the homology of elementary and special linear groups, Adv. Math. 320 (2017), 1–81.CrossRefGoogle Scholar
Schlichting, M., Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra 221(7) (2017), 1729–1844.CrossRefGoogle Scholar
Waldhausen, F., Algebraic K-theory of spaces, in Algebraic and Geometric Topology (New Brunswick, NJ, 1983), Lecture Notes in Mathematics, Volume 1126, pp. 318–419 (Springer, Berlin, 1985).CrossRefGoogle Scholar