Acta Mathematica

Algebraic geometry of topological spaces I

Abstract

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case $M = \mathbb{N}_0^n$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case $M = {\mathbb{Z}^n}$. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C*-algebras, and for a homology theory of commutative algebras to vanish on C*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

Article information

Source
Acta Math., Volume 209, Number 1 (2012), 83-131.

Dates
Received: 18 March 2010
Revised: 19 August 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892647

Digital Object Identifier
doi:10.1007/s11511-012-0082-6

Mathematical Reviews number (MathSciNet)
MR2979510

Zentralblatt MATH identifier
1266.19003

Rights
2012 © Institut Mittag-Leffler

Citation

Cortiñas, Guillermo; Thom, Andreas. Algebraic geometry of topological spaces I. Acta Math. 209 (2012), no. 1, 83--131. doi:10.1007/s11511-012-0082-6. https://projecteuclid.org/euclid.acta/1485892647

References

• B ass, H., Some problems in “classical” algebraic K-theory, in Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Seattle, WA, 1972), Lecture Notes in Math., 342, pp. 3–73. Springer, Berlin–Heidelberg, 1973.
• B asu, S., P ollack, R. & R oy, M. F., Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, 10. Springer, Berlin–Heidelberg, 2003.
• B eĭlinson, A. A., Higher regulators and values of L-functions, in Current Problems in Mathematics, Vol. 24, Itogi Nauki i Tekhniki, pp. 181–238. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984 (Russian).
• B ourbaki, N., Éléments de mathématique. Espaces vectoriels topologiques. Chapitres 1 à 5. Masson, Paris, 1981.
• B rumfiel, G. W., Quotient spaces for semialgebraic equivalence relations. Math. Z., 195 (1987), 69–78.
• C alder, A. & S iegel, J., Homotopy and Kan extensions, in Categorical Topology (Mannheim, 1975), Lecture Notes in Math., 540, pp. 152–163. Springer, Berlin–Heidelberg, 1976.
• — Kan extensions of homotopy functors. J. Pure Appl. Algebra, 12 (1978), 253–269.
• C ortiñas, G., H aesemeyer, C., W alker, M. E. & W eibel, C., Bass’ NK groups and cdh-fibrant Hochschild homology. Invent. Math., 181 (2010), 421–448.
• — A negative answer to a question of Bass. Proc. Amer. Math. Soc., 139 (2011), 1187–1200.
• C ortiñas, G. & T hom, A., Comparison between algebraic and topological K-theory of locally convex algebras. Adv. Math., 218 (2008), 266–307.
• D avis, J. F., Some remarks on Nil groups in algebraic K-theory. Preprint, 2008.
• D rinfeld, V., Infinite-dimensional vector bundles in algebraic geometry: an introduction, in The Unity of Mathematics, Progr. Math., 244, pp. 263–304. Birkhäuser, Boston, MA, 2006.
• F eĭgin, B. L. & T sygan, B. L., Additive K-theory, in K-Theory, Arithmetic and Geometry (Moscow, 1984–1986), Lecture Notes in Math., 1289, pp. 67–209. Springer, Berlin–Heidelberg, 1987.
• F rei, A., Kan extensions along full functors: Kan and Čech extensions of homotopy in-variant functors. J. Pure Appl. Algebra, 17 (1980), 285–292.
• F riedlander, E. M. & W alker, M. E., Comparing K-theories for complex varieties. Amer. J. Math., 123 (2001), 779–810.
• G eller, S. C. & W eibel, C. A., Hodge decompositions of Loday symbols in K-theory and cyclic homology. K-Theory, 8 (1994), 587–632.
• G ersten, S. M., On the spectrum of algebraic K-theory. Bull. Amer. Math. Soc., 78 (1972), 216–219.
• — Some exact sequences in the higher K-theory of rings, in Algebraic K-theory, I: Higher K-theories (Seattle, WA, 1972), Lecture Notes in Math., 341, pp. 211–243. Springer, Berlin–Heidelberg, 1973.
• G ubeladze, J., The Anderson conjecture and projective modules over monoid algebras. Soobshch. Akad. Nauk Gruzin. SSR, 125 (1987), 289–291 (Russian).
• — The Anderson conjecture and a maximal class of monoids over which projective modules are free. Mat. Sb., 135 (177) (1988), 169–185, 271 (Russian); English translation in Math. USSR-Sb., 63 (1989), 165–180.
• — On Bass’ question for finitely generated algebras over large fields. Bull. Lond. Math. Soc., 41 (2009), 36–40.
• H ardt, R. M., Semi-algebraic local-triviality in semi-algebraic mappings. Amer. J. Math., 102 (1980), 291–302.
• H ironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math., 79 (1964), 109–203, 205–326.
• H ochschild, G., K ostant, B. & R osenberg, A., Differential forms on regular affine algebras. Trans. Amer. Math. Soc., 102 (1962), 383–408.
• J ouanolou, J. P., Une suite exacte de Mayer–Vietoris en K-théorie algébrique, in Algebraic K-theory, I: Higher K-theories (Seattle, WA, 1972), Lecture Notes in Math., 341, pp. 293–316. Springer, Berlin–Heidelberg, 1973.
• K ahn, B., Algebraic K-theory, algebraic cycles and arithmetic geometry, in Handbook of K-Theory. Vol. 1, pp. 351–428. Springer, Berlin–Heidelberg, 2005.
• K assel, C. & S letsjøe, A. B., Base change, transitivity and Künneth formulas for the Quillen decomposition of Hochschild homology. Math. Scand., 70 (1992), 186–192.
• K ratzer, C., λ-structure en K-théorie algébrique. Comment. Math. Helv., 55 (1980), 233–254.
• L oday, J.-L., Cyclic Homology. Grundlehren der Mathematischen Wissenschaften, 301. Springer, Berlin–Heidelberg, 1998.
• L oday, J.-L. & Q uillen, D., Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv., 59 (1984), 569–591.
• L ück, W. & R eich, H., The Baum–Connes and the Farrell–Jones conjectures in K- and L-theory, in Handbook of K-theory. Vol. 2, pp. 703–842. Springer, Berlin–Heidelberg, 2005.
• P edersen, E. K. & W eibel, C. A., A nonconnective delooping of algebraic K-theory, in Algebraic and Geometric Topology (New Brunswick, NJ, 1983), Lecture Notes in Math., 1126, pp. 166–181. Springer, Berlin–Heidelberg, 1985.
• K-theory homology of spaces, in Algebraic Topology (Arcata, CA, 1986), Lecture Notes in Math., 1370, pp. 346–361. Springer, Berlin–Heidelberg, 1989.
• Q uillen, D., Higher algebraic K-theory. I, in Algebraic K-Theory, I: Higher K-Theories (Seattle, WA, 1972), Lecture Notes in Math., 341, pp. 85–147. Springer, Berlin–Heidelberg, 1973.
• — Projective modules over polynomial rings. Invent. Math., 36 (1976), 167–171.
• R osenberg, J., K and KK: topology and operator algebras, in Operator Theory: Operator Algebras and Applications(Durham, NH, 1988), Proc. Sympos. Pure Math., 51, Part 1, pp. 445–480. Amer. Math. Soc., Providence, RI, 1990.
• — The algebraic K-theory of operator algebras. K-Theory, 12 (1997), 75–99.
• — Comparison between algebraic and topological K-theory for Banach algebras and C*-algebras, in Handbook of K-Theory. Vol. 2, pp. 843–874. Springer, Berlin–Heidelberg, 2005.
• S erre, J.-P., Faisceaux algébriques cohérents. Ann. of Math., 61 (1955), 197–278.
• S oulé, C., Opérations en K-théorie algébrique. Canad. J. Math., 37 (1985), 488–550.
• S uslin, A. A., Projective modules over polynomial rings are free. Dokl. Akad. Nauk SSSR, 229 (1976), 1063–1066 (Russian); English translation in Soviet Math. Dokl., 17 (1976), 1160–1164.
• — On the K-theory of algebraically closed fields. Invent. Math., 73 (1983), 241–245.
• S uslin, A. A. & W odzicki, M., Excision in algebraic K-theory and Karoubi’s conjecture. Proc. Nat. Acad. Sci. USA, 87:24 (1990), 9582–9584.
• — Excision in algebraic K-theory. Ann. of Math., 136 (1992), 51–122.
• S wan, R. G., Projective modules over Laurent polynomial rings. Trans. Amer. Math. Soc., 237 (1978), 111–120.
• — Gubeladze’s proof of Anderson’s conjecture, in Azumaya Algebras, Actions, and Modules (Bloomington, IN, 1990), Contemp. Math., 124, pp. 215–250. Amer. Math. Soc., Providence, RI, 1992.
• S witzer, R. M., Algebraic Topology—Homotopy and Homology. Classics in Mathematics. Springer, Berlin–Heidelberg, 2002.
• T hom, A. B., Connective E-Theory and Bivariant Homology. Ph.D. Thesis, Universität Münster, Münster, 2003.
• T homason, R. W. & T robaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III, Progr. Math., 88, pp. 247–435. Birkhäuser, Boston, MA, 1990.
• W agoner, J. B., Delooping classifying spaces in algebraic K-theory. Topology, 11 (1972), 349–370.
• W eibel, C. A., Homotopy algebraic K-theory, in Algebraic K-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., 83, pp. 461–488. Amer. Math. Soc., Providence, RI, 1989.
• W odzicki, M., Excision in cyclic homology and in rational algebraic K-theory. Ann. of Math., 129 (1989), 591–639.
• — Homological properties of rings of functional-analytic type. Proc. Nat. Acad. Sci. USA, 87:13 (1990), 4910–4911.