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Determinant functors on triangulated categories

Published online by Cambridge University Press:  21 July 2010

Manuel Breuning
Manuel Breuning
Affiliation:
King's College London, Department of Mathematics, Strand, London WC2R 2LS, United Kingdom, manuel.breuning@kcl.ac.uk
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Abstract

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small triangulated category Τ we give a natural definition of groups K0(Τ) and K1(Τ) in terms of the universal determinant functor on Τ, and we show that Ki(Τ) ≅ Ki(ε) for i = 0 and 1 if Τ has a non-degenerate bounded t-structure with heart ε.

Keywords

Determinant functorTriangulated categoryPicard categoryK-theory

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Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 2 , October 2011 , pp. 251 - 291
Copyright
Copyright © ISOPP 2010

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References

1. Beilinson, A.-A., Bernstein, J., Deligne, P., Analyse et topologie sur les espaces singuliers I., Astérisque 100 (1982).Google Scholar
2. Breuning, M., Determinants of perfect complexes and Euler characteristics in relative K0-groups, preprint 2008, arXiv:0812.1556.Google Scholar
3. Breuning, M., Burns, D., Additivity of Euler characteristics in relative algebraic K-groups, Homology, Homotopy and Applications 7 no. 3 (2005), 11–36.CrossRefGoogle Scholar
4. Burns, D., Equivariant Whitehead torsion and refined Euler characteristics, in: Number theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc., Providence, RI, 2004, pp. 35–59.Google Scholar
5. Burns, D., Flach, M., Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570.CrossRefGoogle Scholar
6. Deligne, P., Le déterminant de la cohomologie, in: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177.Google Scholar
7. Eriksson, D., A Deligne-Riemann-Roch isomorphism I: Preliminaries on virtual categories, preprint 2009, arXiv:0904.4059.Google Scholar
8. Fukaya, T., Kato, K., A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society, Vol. XII, Amer. Math. Soc. Transl. Ser. 2, 219, Amer. Math. Soc., Providence, RI, 2006, pp. 1–85.Google Scholar
9. Knudsen, F. F., Determinant functors on exact categories and their extensions to categories of bounded complexes, Michigan Math. J. 50 (2002), no. 2, 407–444.CrossRefGoogle Scholar
10. Knudsen, F. F., Mumford, D., The projectivity of the moduli space of stable curves. I: Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55.CrossRefGoogle Scholar
11. Muro, F., Tonks, A., The 1-type of a Waldhausen K-theory spectrum, Adv. Math. 216 (2007), 178–211.CrossRefGoogle Scholar
12. Neeman, A., The K-theory of triangulated categories, in: Friedlander, E. M., Grayson, D. R. (eds.), Handbook of K-theory, Volume 2, Springer-Verlag, 2005, pp. 1011–1078.CrossRefGoogle Scholar
13. Rivano, N. Saavedra, Catégories Tannakiennes, Lecture Notes in Mathematics 265, Springer-Verlag, 1972.CrossRefGoogle Scholar
14. Schlichting, M., A note on K-theory and triangulated categories, Invent. Math. 150 (2002), no. 1, 111–116.CrossRefGoogle Scholar
15. Vaknin, A., Determinants in triangulated categories, K-Theory 24 (2001), 57–68.CrossRefGoogle Scholar
16. Verdier, J. L., Catégories dérivées, état 0, in: SGA 41/2, Lecture Notes in Mathematics 569, Springer-Verlag, 1977.Google Scholar
17. Witte, M., Noncommutative Iwasawa main conjectures for varieties over finite fields, PhD-thesis, Leipzig, 2008.Google Scholar