## Geometry & Topology

### Noncommutative localisation in algebraic $K$–theory I

#### Abstract

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic $K$–theory. The main result goes as follows. Let $A$ be an associative ring and let $A→B$ be the localisation with respect to a set $σ$ of maps between finitely generated projective $A$–modules. Suppose that $TornA(B,B)$ vanishes for all $n>0$. View each map in $σ$ as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes $Dperf(A)$. Denote by $〈σ〉$ the thick subcategory generated by these complexes. Then the canonical functor $Dperf(A)→Dperf(B)$ induces (up to direct factors) an equivalence $Dperf(A)∕〈σ〉→Dperf(B)$. As a consequence, one obtains a homotopy fibre sequence

$K ( A , σ ) → K ( A ) → K ( B )$

(up to surjectivity of $K0(A)→K0(B)$) of Waldhausen $K$–theory spectra.

In subsequent articles [??] we will present the $K$– and $L$–theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of $TornA(B,B)$, we also assume that every map in $σ$ is a monomorphism, then there is a description of the homotopy fiber of the map $K(A)→K(B)$ as the Quillen $K$–theory of a suitable exact category of torsion modules.

#### Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1385-1425.

Dates
Revised: 1 September 2004
Accepted: 11 October 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883471

Digital Object Identifier
doi:10.2140/gt.2004.8.1385

Mathematical Reviews number (MathSciNet)
MR2119300

Zentralblatt MATH identifier
1083.18007

#### Citation

Neeman, Amnon; Ranicki, Andrew. Noncommutative localisation in algebraic $K$–theory I. Geom. Topol. 8 (2004), no. 3, 1385--1425. doi:10.2140/gt.2004.8.1385. https://projecteuclid.org/euclid.gt/1513883471

#### References

• Hyman Bass, Algebraic ${K}$–theory, W. A. Benjamin, Inc., New York–Amsterdam (1968)
• Alexander A Beilinson, Joseph Bernstein, Pierre Deligne, Analyse et topologie sur les éspaces singuliers, Astérisque 100, Soc. Math. France (1982)
• George M Bergman, Warren Dicks, Universal derivations and universal ring constructions, Pacific J. Math. 79 (1978) 293–337
• Marcel Bökstedt, Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993) 209–234
• AK Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133–150
• AK Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
• Paul M Cohn, Free rings and their relations, Academic Press, London (1971), London Mathematical Society Monographs, No. 2
• Warren Dicks, Mayer–Vietoris presentations over colimits of rings, Proc. London Math. Soc. (3) 34 (1977) 557–576
• Michael Farber, Andrew Ranicki, The Morse–Novikov theory of circle-valued functions and noncommutative localization, Tr. Mat. Inst. Steklova 225 (1999) 381–388
• Michael Farber, Pierre Vogel, The Cohn localization of the free group ring, Math. Proc. Cambridge Philos. Soc. 111 (1992) 433–443
• Werner Geigle, Helmut Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991) 273–343
• Henri Gillet, Riemann–Roch theorems for higher algebraic ${K}$–theory, Adv. in Math. 40 (1981) 203–289
• Daniel R Grayson, Higher algebraic $K$–theory. II (after Daniel Quillen), from: “Algebraic $K$–theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976)”, Lecture Notes in Mathematics 551, Springer, Berlin (1976) 217–240
• Daniel R Grayson, ${K}$–theory and localization of noncommutative rings, J. Pure Appl. Algebra 18 (1980) 125–127
• Daniel R Grayson, Exact sequences in algebraic $K$–theory, Ill. J. Math. 31 (1987) 598–617
• Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994) 63–102
• Bernhard Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994) 193–198
• Henning Krause, Smashing subcategories and the telescope conjecture –- an algebraic approach, Invent. Math. 139 (2000) 99–133
• Henning Krause, Cohomological quotients and smashing localizations, http://www.math.upb.de/~hkrause/publications.html
• Marc Levine, Localization on singular varieties, Inv. Math. 91 (1988) 423–464
• Amnon Neeman, The chromatic tower for $D(R)$, Topology 31 (1992) 519–532
• Amnon Neeman, The connection between the $K$–theory localisation theorem of Thomason, Trobaugh and Yao, and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Normale Supérieure 25 (1992) 547–566
• Amnon Neeman, Triangulated Categories, volume 148 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ (2001)
• Amnon Neeman, A non-commutative generalisation of Thomason's localisation theorem, from: “Proceedings of the 2002 Edinburgh ICMS conference on noncommutative localization in algebra and topology” (to appear)
• Amnon Neeman, Andrew Ranicki, Noncommutative localization and chain complexes I. Algebraic $K$– and $L$–theory.
• Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic $K$–theory II, http://www.maths.anu.edu.au/~neeman/preprints.html
• Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic $L$–theory, http://www.maths.anu.edu.au/~neeman/preprints.html
• Amnon Neeman, Andrew Ranicki, Aidan Schofield, Representations of algebras as universal localizations, Math. Proc. Cambridge Philos. Soc. 136 (2004) 105–117
• Daniel Quillen, Higher algebraic ${K}$–theory. I, from: “Algebraic $K$–theory, I: Higher $K$–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)”, Lecture Notes in Mathematics 341, Springer, Berlin (1973) 85–147
• Andrew Ranicki, High-dimensional knot theory, Springer–Verlag, New York (1998)
• Andrew Ranicki, Noncommutative localization in topology, to appear in Proc. 2002 ICMS conference on Noncommutative Localization in Algebra and Topology.
• Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (1989) 436–456
• Aidan H Schofield, Representation of rings over skew fields, volume 92 of London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge (1985)
• Nicolas Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988) 121–154
• Robert W Thomason, Thomas F Trobaugh, Higher algebraic $K$–theory of schemes and of derived categories, from: “The Grothendieck Festschrift”, volume 3, Birkhäuser (1990) 247–435
• Pierre Vogel, Localization in algebraic ${L}$–theory, from: “Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979)”, Springer, Berlin (1980) 482–495
• Pierre Vogel, On the obstruction group in homology surgery, Inst. Hautes Études Sci. Publ. Math. 55 (1982) 165–206
• Friedhelm Waldhausen, Algebraic ${K}$–theory of spaces, from: “Algebraic and geometric topology (New Brunswick, N.J., 1983)”, Lecture Notes in Mathematics 1126, Springer, Berlin (1985) 318–419
• Charles A Weibel, Negative ${K}$–theory of varieties with isolated singularities, J. Pure Appl. Algebra 34 (1984) 331–342
• Charles A Weibel, A Brown–Gersten spectral sequence for the ${K}$–theory of varieties with isolated singularities, Adv. in Math. 73 (1989) 192–203
• Charles A Weibel, Dongyuan Yao, Localization for the ${K}$–theory of noncommutative rings, from: “Algebraic $K$–theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989)”, Contemp. Math. 126, Amer. Math. Soc., Providence, RI (1992) 219–230
• Dongyuan Yao, Higher algebraic ${K}$–theory of admissible abelian categories and localization theorems, J. Pure Appl. Algebra 77 (1992) 263–339