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  • Marco Schlichting (a1)


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.



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1. Bak, A., K-theory of Forms, Annals of Mathematics Studies, Volume 98 (Princeton University Press, Princeton, NJ, 1981). University of Tokyo Press, Tokyo.
2. Bass, H., Algebraic K-theory (W. A. Benjamin, Inc., New York–Amsterdam, 1968).
3. Baues, H. J., Quadratic functors and metastable homotopy, J. Pure Appl. Algebra 91(1–3) (1994), 49107.
4. Bouc, S., Green Functors and G-sets, Lecture Notes in Mathematics, Volume 1671 (Springer, Berlin, 1997).
5. Bourbaki, N., Éléments de mathématique, Algèbre. Chapitre 9 (Springer, Berlin, 2007). Reprint of the 1959 original.
6. 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. 80130 (Springer, Berlin, 1978).
7. Charney, R. and Lee, R., On a theorem of Giffen, Michigan Math. J. 33(2) (1986), 169186.
8. Dotto, E. and Ogle, C., K-theory of Hermitian Mackey functors, real traces, and assembly, Ann. K-theory. in press.
9. Elmendorf, A. D. and Mandell, M. A., Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205(1) (2006), 163228.
10. 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. 217240 (Springer, Berlin, 1976).
11. Hesselholt, L. and Madsen, I., Real algebraic $K$ -theory,, 2015.
12. 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. 301411 (Springer, Berlin, 1973).
13. Karoubi, M., Le théorème fondamental de la K-théorie hermitienne, Ann. of Math. (2) 112(2) (1980), 259282.
14. Karoubi, M., Théorie de Quillen et homologie du groupe orthogonal, Ann. of Math. (2) 112(2) (1980), 207257.
15. 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.
16. Lurie, J., Course on algebraic $L$ -theory and surgery,, 2013.
17. MacLane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics, Volume 5 (Springer, New York-Berlin, 1971).
18. Milnor, J. and Husemoller, D., Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73 (Springer, New York–Heidelberg, 1973).
19. 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. 225240 (Kluwer Acad. Publ., Dordrecht, 1989).
20. 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), 121146.
21. 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. 85147 (Springer, Berlin, 1973).
22. Quillen, D., Characteristic classes of representations, in Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, IL, 1976), Lecture Notes in Mathematics, Volume 551, pp. 189216 (Springer, Berlin, 1976).
23. Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 270 (Springer, Berlin, 1985).
24. Schlichting, M., Higher $K$ -theory of forms II. From exact categories to chain complexes. In preparation.
25. Schlichting, M., Higher $K$ -theory of forms III. From chain complexes to derived categories. In preparation.
26. Schlichting, M., Symplectic and orthogonal $K$ -groups of the integers. In preparation.
27. Schlichting, M., Hermitian K-theory. On a theorem of Giffen, K-Theory 32(3) (2004), 253267.
28. Schlichting, M., Hermitian K-theory of exact categories, J. K-Theory 5(1) (2010), 105165.
29. Schlichting, M., The Mayer–Vietoris principle for Grothendieck–Witt groups of schemes, Invent. Math. 179(2) (2010), 349433.
30. Schlichting, M., Euler class groups and the homology of elementary and special linear groups, Adv. Math. 320 (2017), 181.
31. Schlichting, M., Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra 221(7) (2017), 17291844.
32. Waldhausen, F., Algebraic K-theory of spaces, in Algebraic and Geometric Topology (New Brunswick, NJ, 1983), Lecture Notes in Mathematics, Volume 1126, pp. 318419 (Springer, Berlin, 1985).
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  • Marco Schlichting (a1)


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