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Hermitian K-theory of exact categories

Published online by Cambridge University Press:  13 November 2009

Marco Schlichting
Marco Schlichting
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA, [email protected]
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Abstract

We study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, dévissage and localization theorems – preparing the ground for the theory of higher Grothendieck-Witt groups of schemes as developed in [Sch08a] and [Sch08b]. No assumption on the characteristic is being made.

Keywords

hermitian K-theoryQuillen K-theoryQ-constructionexact categories with duality
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 1 , February 2010 , pp. 105 - 165
Copyright
Copyright © ISOPP 2010

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References

Bal01.Balmer, Paul. Triangular Witt groups. II. From usual to derived. >Math. Z. 236(2):351–382, 2001Google Scholar
CL86.Charney, Ruth and Lee, Ronnie. On a theorem of Giffen. >Michigan Math. J. 33(2):169–186, 1986Google Scholar
CP97.Cárdenas, M. and Pedersen, E. K.. On the Karoubi filtration of a category. K-Theory 12(2):165–191, 1997Google Scholar
GJ99.Goerss, Paul G. and Jardine, John F.. Simplicial homotopy theory, Progress in Mathematics 174. Birkhäuser Verlag, Basel, 1999Google Scholar
Gra79.Grayson, Daniel R.. Localization for flat modules in algebraic K-theory. >J. Algebra 61(2):463–496, 1979Google Scholar
Hor02.Hornbostel, Jens. Constructions and dévissage in Hermitian K-theory. K-Theory 26(2):139–170, 2002Google Scholar
Hor05.Hornbostel, Jens. A 1-representability of Hermitian K-theory and Witt groups. >Topology 44(3):661–687, 2005Google Scholar
HS04.Hornbostel, Jens and Schlichting, Marco. Localization in Hermitian K-theory of rings. J. London Math. Soc. (2) 70(1):77–124, 2004Google Scholar
Kar74.Karoubi, Max. Localisation de formes quadratiques. I. Ann. Sci. École Norm. Sup. (4) 7:359–403 (1975), 1974Google Scholar
Kar80a.Karoubi, Max. Le théorème fondamental de la K-théorie hermitienne. Ann. of Math. (2) 112(2):259–282, 1980Google Scholar
Kar80b.Karoubi, Max. Théorie de Quillen et homologie du groupe orthogonal. Ann. of Math. (2) 112(2):207–257, 1980Google Scholar
Kel90.Keller, Bernhard. Chain complexes and stable categories. >Manuscripta Math. 67(4):379–417, 1990Google Scholar
Kel96.Keller, Bernhard. Derived categories and their uses. In Handbook of algebra 1, pages 671–701. North-Holland, Amsterdam, 1996Google Scholar
Kne77.Knebusch, Manfred. Symmetric bilinear forms over algebraic varieties. In Conference on Quadratic Forms—1976 (Proc. Conf., Queen's Univ., Kingston, Ont., 1976), pages 103–283. Queen's Papers in Pure and Appl. Math. 46. Queen's Univ., Kingston, Ont., 1977Google Scholar
Knu91.Knus, Max-Albert. Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 294. Springer-Verlag, Berlin, 1991. With a foreword by I. BertuccioniGoogle Scholar
Pop73.Popescu, N.. Abelian categories with applications to rings and modules. London Mathematical Society Monographs 3. Academic Press, London, 1973.Google Scholar
PW89.Pedersen, Erik K. and Weibel, Charles A.. K-theory homology of spaces. In Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math. 1370, pages 346–361. Springer, Berlin, 1989Google Scholar
QSS79.Quebbemann, H.-G., Scharlau, W., and Schulte, M.. Quadratic and Hermitian forms in additive and abelian categories. J. Algebra 59(2):264–289, 1979Google Scholar
Qui71.Quillen, Daniel. The Adams conjecture. Topology 10:67–80, 1971Google Scholar
Qui73.Quillen, Daniel. Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85–147. Lecture Notes in Math. 341. Springer, Berlin, 1973Google Scholar
Ran92.Ranicki, Andrew. Lower K- and L-theory, London Mathematical Society Lecture Note Series 178. Cambridge University Press, Cambridge, 1992Google Scholar
Sch85.Scharlau, Winfried. Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 270. Springer-Verlag, Berlin, 1985Google Scholar
Sch04a.Schlichting, Marco. Delooping the K-theory of exact categories. >Topology 43(5):1089–1103, 2004Google Scholar
Sch04b.Schlichting, Marco. Hermitian K-theory on a theorem of Giffen. K-Theory 32(3):253–267, 2004Google Scholar
Sch06.Schlichting, Marco. Negative K-theory of derived categories. >Math. Z. 253(1):97–134, 2006Google Scholar
Sch08a.Schlichting, Marco. The Mayer-Vietoris principle for higher Grothendieck-Witt groups of schemes. Invent. Math., to appear.Google Scholar
Sch08b.Schlichting, Marco. Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem. in preparation, 2008Google Scholar
TT90.Thomason, R. W. and Trobaugh, Thomas. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift, Vol. III, Progr. Math. 88, pages 247–435. Birkhäuser Boston, Boston, MA, 1990Google Scholar
Uri90.Uridia, M.. U-theory of exact categories. In K-theory and homological algebra (Tbilisi, 1987–88), Lecture Notes in Math. 1437, pages 303–313. Springer, Berlin, 1990Google Scholar
Wag72.Wagoner, J. B.. Delooping classifying spaces in algebraic K-theory. Topology 11:349–370, 1972CrossRefGoogle Scholar
Wal85.Waldhausen, Friedhelm. Algebraic K-theory of spaces. In Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math. 1126, pages 318–419. Springer, Berlin, 1985Google Scholar
Wal03.Walter, Charles. Grothendieck-Witt groups of triangulated categories. preprint, 2003. www.math.uiuc.edu/K-theory/Google Scholar