Dans cet article, nous prouvons que le complexe tangent de la K-théorie, en termes de problèmes de déformations formels et sur un corps de caractéristique nulle, n’est autre que l’homologie cyclique sur . Cette équivalence est de plus compatible aux -opérations. Nous démontrons également que le morphisme tangent du morphisme canonique est homotope au morphisme de trace généralisée de Loday-Quillen et Tsygan. La démonstration s’appuie sur des résultats de Goodwillie, à l’aide du théorème d’excision pour l’homologie cyclique de Wodzicki et de la théorie des déformations formelles à la Lurie-Pridham.
We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic field , is the cyclic homology (over ). This equivalence is compatible with -operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field of characteristic . We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map . The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham.
Accepté le :
Publié le :
Keywords: K-theory, cyclic homology, formal moduli problem
Mot clés : K-théorie, homologie cyclique, espace de module formel
@article{JEP_2021__8__895_0, author = {Hennion, Benjamin}, title = {The tangent complex of {K-theory}}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {895--932}, publisher = {Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.161}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.161/} }
Hennion, Benjamin. The tangent complex of K-theory. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 895-932. doi : 10.5802/jep.161. http://www.numdam.org/articles/10.5802/jep.161/
[Bei87] On the derived category of perverse sheaves, -theory, arithmetic and geometry (Moscow, 1984–1986) (Lect. Notes in Math.), Volume 1289, Springer, Berlin, 1987, pp. 27-41 | DOI | MR | Zbl
[Bei14] Relative continuous -theory and cyclic homology, Münster J. Math., Volume 7 (2014) no. 1, pp. 51-81 | MR | Zbl
[BKP18] Generators in formal deformations of categories, Compositio Math., Volume 154 (2018) no. 10, pp. 2055-2089 | DOI | MR | Zbl
[Blo73] On the tangent space to Quillen -theory, Algebraic -theory, I: Higher -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (Lect. Notes in Math.), Volume 341, Springer, 1973, pp. 205-210 | MR | Zbl
[Bur86] Cyclic homology and the algebraic -theory of spaces. I, Applications of algebraic -theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) (Contemp. Math.), Volume 55, American Mathematical Society, Providence, RI, 1986, pp. 89-115 | DOI | MR | Zbl
[Cat91] -structures in algebraic -theory and cyclic homology, -Theory, Volume 4 (1990/91) no. 6, pp. 591-606 | DOI | MR
[CHW09] Infinitesimal cohomology and the Chern character to negative cyclic homology, Math. Ann., Volume 344 (2009) no. 4, pp. 891-922 | DOI | MR
[CW09] Relative Chern characters for nilpotent ideals, Algebraic topology (Abel Symp.), Volume 4, Springer, Berlin, 2009, pp. 61-82 | DOI | MR
[DL14] The excision theorems in Hochschild and cyclic homologies, Proc. Roy. Soc. Edinburgh Sect. A, Volume 144 (2014) no. 2, pp. 305-317 | DOI | MR
[FHK19] Higher Kac-Moody algebras and moduli spaces of -bundles, Adv. Math., Volume 346 (2019), pp. 389-466 | DOI | MR
[FT87] Additive -theory, -theory, arithmetic and geometry (Moscow, 1984–1986) (Lect. Notes in Math.), Volume 1289, Springer, Berlin, 1987, pp. 67-209 | DOI | MR
[GG96] The theorem of excision for Hochschild and cyclic homology, J. Pure Appl. Algebra, Volume 106 (1996) no. 1, pp. 57-60 | DOI | MR
[Goo86] Relative algebraic -theory and cyclic homology, Ann. of Math. (2), Volume 124 (1986) no. 2, pp. 347-402 | DOI | MR | Zbl
[Lod92] Cyclic homology, Grundlehren Math. Wiss., 301, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl
[LQ83] Homologie cyclique et homologie de l’algèbre de Lie des matrices, C. R. Acad. Sci. Paris Sér. I Math., Volume 296 (1983) no. 6, pp. 295-297 | Zbl
[Lur09] Higher topos theory, Annals of Math. Studies, 170, Princeton University Press, Princeton, NJ, 2009, xviii+925 pages | DOI | MR
[Lur11] Derived algebraic geometry X: Formal moduli problems (2011) (available at https://www.math.ias.edu/~lurie/papers/DAG-X.pdf)
[Lur16] Higher algebra (2016) (available at https://www.math.ias.edu/~lurie/papers/HA.pdf)
[Mal49] Nilpotent torsion-free groups, Izv. Akad. Nauk SSSR Ser. Mat., Volume 13 (1949), pp. 201-212 | MR
[Pri10] Unifying derived deformation theories, Adv. Math., Volume 224 (2010) no. 3, pp. 772-826 Corrigendum: Ibid. 228 (2011), no. 4, p. 2554–2556 | DOI | MR
[Pri16] Smooth functions on algebraic K-theory, 2016 | arXiv
[SS03] Equivalences of monoidal model categories, Algebraic Geom. Topol., Volume 3 (2003), pp. 287-334 | DOI | MR
[Sus81] On the equivalence of -theories, Comm. Algebra, Volume 9 (1981) no. 15, pp. 1559-1566 | DOI | MR
[SW92] Excision in algebraic -theory, Ann. of Math. (2), Volume 136 (1992) no. 1, pp. 51-122 | DOI | MR
[Tam18] Excision in algebraic -theory revisited, Compositio Math., Volume 154 (2018) no. 9, pp. 1801-1814 | DOI | MR | Zbl
[Tsy83] Homology of matrix Lie algebras over rings and the Hochschild homology, Uspehi Mat. Nauk, Volume 38 (1983) no. 2(230), pp. 217-218 | MR | Zbl
[TV08] Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc., 193, no. 902, American Mathematical Society, Providence, RI, 2008 | DOI
[Vol71] Algebraic -theory as an extraordinary homology theory on the category of associative rings with a unit, Izv. Akad. Nauk SSSR Ser. Mat., Volume 35 (1971), pp. 844-873
[Wal85] Algebraic -theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) (Lect. Notes in Math.), Volume 1126, Springer, Berlin, 1985, pp. 318-419 | DOI | MR
[Wei97] The Hodge filtration and cyclic homology, -Theory, Volume 12 (1997) no. 2, pp. 145-164 | DOI | MR
[Wod89] Excision in cyclic homology and in rational algebraic -theory, Ann. of Math. (2), Volume 129 (1989) no. 3, pp. 591-639 | DOI | MR | Zbl
Cité par Sources :