Transfer maps and projection formulas

Tabuada, Gonçalo

doi:10.1090/S0002-9939-2011-11169-9

Transfer maps and projection formulas
HTML articles powered by AMS MathViewer

by Gonçalo Tabuada PDF

Proc. Amer. Math. Soc. 140 (2012), 2589-2597 Request permission

Abstract:

Transfer maps and projection formulas are undoubtedly one of the key tools in the development and computation of (co)homology theories. In this paper we develop a unified treatment of transfer maps and projection formulas in the non-commutative setting of dg categories. As an application, we obtain transfer maps and projection formulas in algebraic $K$-theory, cyclic homology, topological cyclic homology, and other scheme invariants.

References

A. Blumberg and M. Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology. Available at arXiv:$0802.3938$.
M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic $K$-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133, DOI 10.1007/BF01231296
Francis Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. Categories and structures. MR 1313497
D.-C. Cisinski and G. Tabuada, Non-connective $K$-theory via universal invariants. Compositio Mathematica 147 (2011), 1281–1320.
—, Symmetric monoidal structure on non-commutative motives. Available at arXiv:$1001.0228\textrm {v}2$. To appear in J. of $K$-Theory.
Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004), no. 2, 643–691. MR 2028075, DOI 10.1016/j.jalgebra.2003.05.001
A. Grothendieck, Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Lecture Notes in Mathematics 225 (1971).
Lars Hesselholt and Ib Madsen, On the $K$-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. MR 1998478, DOI 10.4007/annals.2003.158.1
Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
Bernhard Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), no. 1, 1–56. MR 1667558, DOI 10.1016/S0022-4049(97)00152-7
Bernhard Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math. 3 (1998), 231–259. MR 1647519
Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
Valery A. Lunts and Dmitri O. Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), no. 3, 853–908. MR 2629991, DOI 10.1090/S0894-0347-10-00664-8
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Christian Schlichtkrull, Transfer maps and the cyclotomic trace, Math. Ann. 336 (2006), no. 1, 191–238. MR 2242623, DOI 10.1007/s00208-006-0783-2
Marco Schlichting, Negative $K$-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134. MR 2206639, DOI 10.1007/s00209-005-0889-3
Gonçalo Tabuada, Higher $K$-theory via universal invariants, Duke Math. J. 145 (2008), no. 1, 121–206. MR 2451292, DOI 10.1215/00127094-2008-049
Gonçalo Tabuada, Invariants additifs de DG-catégories, Int. Math. Res. Not. 53 (2005), 3309–3339 (French). MR 2196100, DOI 10.1155/IMRN.2005.3309
Gonçalo Tabuada, On Drinfeld’s dg quotient, J. Algebra 323 (2010), no. 5, 1226–1240. MR 2584954, DOI 10.1016/j.jalgebra.2009.12.020
Gonçalo Tabuada, Generalized spectral categories, topological Hochschild homology and trace maps, Algebr. Geom. Topol. 10 (2010), no. 1, 137–213. MR 2580431, DOI 10.2140/agt.2010.10.137
—, Products, multiplicative Chern characters, and finite coefficients via non-commutative motives. Available at arXiv:1101.0731.
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Bertrand Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667. MR 2276263, DOI 10.1007/s00222-006-0025-y
C. Weibel, The $K$-book: An introduction to algebraic $K$-theory. A graduate textbook in progress. Available at http://www.math.rutgers.edu/weibel/Kbook.html.