Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-03T02:52:32.290Z Has data issue: false hasContentIssue false

Quillen's work in algebraic K-theory

Published online by Cambridge University Press:  11 March 2013

Daniel R. Grayson*
Affiliation:
2409 S. Vine St, Urbana, Illinois 61801, [email protected], http://dangrayson.com/
Get access

Abstract

We survey the genesis and development of higher algebraic K-theory by Daniel Quillen.

Type
Research Article
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. K-theory Preprint Archives: http://k-theory.org/.Google Scholar
2. Front for the arXiv, K-theory: http://front.math.ucdavis.edu/math.KT.Google Scholar
3.Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin, 1971, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par Berthelot, P., Grothendieck, A. et Illusie, L.. Avec la collaboration de Ferrand, D., Jouanolou, J. P., Jussila, O., Kleiman, S., Raynaud, M. et Serre, J. P.. MR 0354655 (50 #7133)Google Scholar
4.Adams, J. F., Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603632. MR 0139178 (25 #2614)Google Scholar
5.Adams, J. F., On the groups J(X). I, Topology 2 (1963), 181195. MR 0159336 (28 #2553)Google Scholar
6.Adams, John Frank, Algebraic topology—a student's guide, Cambridge University Press, London, 1972, London Mathematical Society Lecture Note Series, No. 4. MR 0445484 (56 #3824)Google Scholar
7.Adams, John Frank, Infinite loop spaces, Annals of Mathematics Studies, vol. 90, Princeton University Press, Princeton, N.J., 1978. MR 505692 (80d:55001)Google Scholar
8.Atiyah, M. F., K-theory, Lecture notes by Anderson, D. W., W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0224083 (36 #7130)Google Scholar
9.Atiyah, M. F. and Hirzebruch, F., Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 738. MR 0139181 (25 #2617)Google Scholar
10.Ausoni, Christian and Rognes, John, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), no. 1, 139. MR 1947457 (2004f:19007)Google Scholar
11.Bass, H., K-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. (1964), no. 22, 560. MR 0174604 (30 #4805)Google Scholar
12.Bass, H., Heller, A., and Swan, R. G., The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. (1964), no. 22, 6179. MR 0174605 (30 #4806)Google Scholar
13.Bass, H., Milnor, J., and Serre, J.-P., Solution of the congruence subgroup problem for SLn(n ≥ 3) and Sp2n(n ≥ 2), Inst. Hautes Études Sci. Publ. Math. (1967), no. 33, 59137. MR MR0244257 (39 #5574)Google Scholar
14.Bass, H. and Tate, J., The Milnor ring of a global field, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), Springer, Berlin, 1973, pp. 349446. Lecture Notes in Math. 342. MR MR0442061 (56 #449)Google Scholar
15.Bass, Hyman, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR MR0249491 (40 #2736)Google Scholar
16.Bass, Hyman, Some problems in “classical” algebraic K-theory, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 373. Lecture Notes in Math. 342. MR MR0409606 (53 #13358)Google Scholar
17.Bass, Hyman, Personal reminiscences of the birth of algebraic K-theory, K-Theory 30 (2003), no. 3, 203209, special issue in honor of Hyman Bass on his seventieth birthday. Part III. MR MR2064239 (2005d:19001)Google Scholar
18.Beĭlinson, A. A., Higher regulators and values of L -functions, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181238. MR 760999 (86h:11103)Google Scholar
19.Bloch, Spencer, Algebraic K-theory and classfield theory for arithmetic surfaces, Ann. of Math. (2) 114 (1981), no. 2, 229265. MR 632840 (83m:14025)Google Scholar
20.Borel, Armand, Cohomologie réelle stable de groupes S-arithmétiques classiques, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1700A1702. MR MR0308286 (46 #7400)Google Scholar
21.Borel, Armand and Serre, Jean-Pierre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97136. MR 0116022 (22 #6817)Google Scholar
22.Dundas, B. I., Levine, M., Østvær, P. A., Röndigs, O., and Voevodsky, V., Motivic homotopy theory, Universitext, Springer-Verlag, Berlin, 2007, Lectures from the Summer School held in Nordfjordeid, August 2002. MR 2334212 (2008k:14046)Google Scholar
23.Dundas, Bjørn Ian, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 223242. MR 1607556 (99e:19007)Google Scholar
24.Dwyer, William, Quillen's work on the Adams Conjecture, in this volume.Google Scholar
25.Farrell, F. T. and Hsiang, W.-C., A geometric interpretation of the Künneth formula for algebraic K-theory, Bull. Amer. Math. Soc. 74 (1968), 548553. MR 0224675 (37 #274)Google Scholar
26.Farrell, F. T. and Hsiang, W. C., On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325337. MR 520509 (80g:57043)Google Scholar
27.Friedlander, Eric M. and Grayson, Daniel R. (eds.), Handbook of K-theory. Vol. 1, 2, Springer-Verlag, Berlin, 2005. MR 2182598 (2006e:19001)Google Scholar
28.Gersten, S. M., On the functor K2. I, J. Algebra 17 (1971), 212237. MR 0292910 (45 #1992)CrossRefGoogle Scholar
29.Gillet, H. and Soulé, C., Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996), 127176. MR 1409056 (98d:14012)Google Scholar
30.Gillet, Henri, Intersection theory on algebraic stacks and Q-varieties, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), vol. 34, 1984, pp. 193240. MR 772058 (86b:14006)Google Scholar
31.Gillet, Henri, K-theory and intersection theory, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 235293. MR 2181825 (2006h:14013)CrossRefGoogle Scholar
32.Gillet, Henri and Grayson, Daniel R., The loop space of the Q -construction, Illinois J. Math. 31 (1987), no. 4, 574597. MR MR909784 (89h:18012)Google Scholar
33.Goodwillie, Thomas G., Relative algebraic K-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347402. MR 855300 (88b:18008)Google Scholar
34.Grayson, Daniel R., Higher algebraic K-theory. II (after Daniel Quillen), Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 217240. Lecture Notes in Math. 551. MR MR0574096 (58 #28137)Google Scholar
35.Grayson, Daniel R., Projections, cycles, and algebraic K-theory, Math. Ann. 234 (1978), no. 1, 6972. MR MR0491686 (58 #10891)Google Scholar
36.Grayson, Daniel R., Finite generation of K-groups of a curve over a finite field (after Daniel Quillen), Algebraic K-theory, Part I (Oberwolfach, 1980), Lecture Notes in Math. 966, Springer, Berlin, 1982, pp. 6990. MR MR689367 (84f:14018)Google Scholar
37.Grayson, Daniel R., Exact sequences in algebraic K-theory, Illinois J. Math. 31 (1987), no. 4, 598617. MR MR909785 (89c:18011)Google Scholar
38.Grayson, Daniel R., On the K-theory of fields, Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 3155. MR 991975 (90c:18010)Google Scholar
39.Grayson, Daniel R., Weight filtrations in algebraic K-theory, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 207237. MR 1265531 (95a:19006)Google Scholar
40.Grayson, Daniel R., The additivity theorem in algebraic K-theory, Documenta Mathematica 16 (2011), 457464.Google Scholar
41.Grayson, Daniel R., Algebraic K -theory via binary complexes, Preprint, January 6, 2011, K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory/0988/,2011.Google Scholar
42.Haesemeyer, Christian and Weibel, Chuck, Norm varieties and the chain lemma (after Markus Rost), Algebraic topology, Abel Symp., vol. 4, Springer, Berlin, 2009, pp. 95130. MR 2597737 (2011f:19002)Google Scholar
43.Heller, Alex, Some exact sequences in algebraic K-theory, Topology 4 (1965), 389408. MR 0179229 (31 #3477)Google Scholar
44.Hesselholt, Lars and Madsen, Ib, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1113. MR 1998478 (2004k:19003)Google Scholar
45.Hiller, Howard L., λ-rings and algebraic K-theory, J. Pure Appl. Algebra 20 (1981), no. 3, 241266. MR 82e:18016Google Scholar
46.Kahn, Bruno, Algebraic K-theory, algebraic cycles and arithmetic geometry, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 351428. MR 2181827 (2007b:14016)Google Scholar
47.Karoubi, Max and Villamayor, Orlando, Foncteurs Kn en algèbre et en topologie, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A416A419. MR 0251717 (40 #4944)Google Scholar
48.Kervaire, Michel A., Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 6772. MR 0253347 (40 #6562)Google Scholar
49.Lichtenbaum, Stephen, Values of zeta-functions, étale cohomology, and algebraic K-theory, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 489501. Lecture Notes in Math. 342. MR 53 #10765Google Scholar
50.Loday, Jean-Louis, Homotopie des espaces de concordances [d'après F. Waldhausen], Séminaire Bourbaki, 30e année (1977/78), Lecture Notes in Math. 710, Springer, Berlin, 1979, pp. Exp. No. 516, pp. 187205. MR MR554221 (81h:57006)Google Scholar
51.Mazza, Carlo, Voevodsky, Vladimir, and Weibel, Charles, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2, American Mathematical Society, Providence, RI, 2006. MR 2242284 (2007e:14035)Google Scholar
52.McCarthy, Randy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197222. MR 1607555 (99e:19006)Google Scholar
53.Merkur′ev, A. S. and Suslin, A. A., K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 11351136. MR 675529 (84i:12007)Google Scholar
54.Milnor, John, Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J., 1971, Annals of Mathematics Studies 72. MR MR0349811 (50 #2304)Google Scholar
55.Quillen, D., Letter from Quillen to Milnor on Im(πi0 → πiS → KiZ), Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 182188. Lecture Notes in Math. 551. MR MR0482758 (58 #2811)Google Scholar
56.Quillen, Daniel, The Adams conjecture, Topology 10 (1971), 6780. MR 0279804 (43 #5525)Google Scholar
57.Quillen, Daniel, Cohomology of groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 4751. MR MR0488054 (58 #7627a)Google Scholar
58.Quillen, Daniel, A letter from Quillen to Segal, dated July 25, 1972, This letter is about: (1) the image of J, the stable homotopy groups of spheres, and the K-groups of the integers; (2) the Q-construction and the S-construction. K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory/1003/., 1972.Google Scholar
59.Quillen, Daniel, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552586. MR MR0315016 (47 #3565)Google Scholar
60.Quillen, Daniel, Finite generation of the groups Ki of rings of algebraic integers, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 179198. Lecture Notes in Math. 341. MR MR0349812 (50 #2305)Google Scholar
61.Quillen, Daniel, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85147. Lecture Notes in Math. 341. MR MR0338129 (49 #2895)Google Scholar
62.Quillen, Daniel, Higher K-theory for categories with exact sequences, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London, 1974, pp. 95103. London Math. Soc. Lecture Note Ser., No. 11. MR MR0335604 (49 #384)Google Scholar
63.Quillen, Daniel, Higher algebraic K-theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 171176. MR MR0422392 (54 #10382)Google Scholar
64.Quillen, Daniel, Characteristic classes of representations, Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 189216. Lecture Notes in Math. 551. MR MR0578500 (58 #28215)Google Scholar
65.Quillen, Daniel, On the group completion of a simplicial monoid, Mem. Amer. Math. Soc. 110 (1994), no. 529, 89105, appeared in preprint form June 11, 1971. Published as Appendix Q to the paper Filtrations on the homology of algebraic varieties by Eric Friedlander and and Barry Mazur. MR MR1211371 (95a:14023)Google Scholar
66.Rapoport, M., Schappacher, N., and Schneider, P. (eds.), Beilinson's conjectures on special values of L -functions, Perspectives in Mathematics 4, Academic Press Inc., Boston, MA, 1988. MR 944987 (89a:14002)Google Scholar
67.Schlichting, Marco, Higher algebraic K-theory, Topics in algebraic and topological K-theory, Lecture Notes in Math. 2008, Springer, Berlin, 2011, pp. 167241. MR 2762556 (2012a:19001)Google Scholar
68.Segal, Graeme, Categories and cohomology theories, Topology 13 (1974), 293312, preprint released in 1969. MR 50 #5782CrossRefGoogle Scholar
69.Serre, J.-P., Modules projectifs et espaces fibrés à fibre vectorielle, Dubreil, P. Séminaire, Dubreil-Jacotin, M.-L. et Pisot, C., 1957/1958, Fasc. 2, Exposé 23, Secrétariat mathématique, Paris, 1958, p. 18. MR 0177011 (31 #1277)Google Scholar
70.Soulé, C., K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), no. 3, 251295. MR 553999 (81i:12016)CrossRefGoogle Scholar
71.Staffeldt, Ross E., On fundamental theorems of algebraic K-theory, K-Theory 2 (1989), no. 4, 511532. MR 990574 (90g:18009)Google Scholar
72.Suslin, A., On the K-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241245. MR MR714090 (85h:18008a)Google Scholar
73.Swan, Richard G., Nonabelian homological algebra and K-theory, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 88123. MR 0257185 (41 #1839)Google Scholar
74.Thomason, R. W. and Trobaugh, Thomas, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247435. MR MR1106918 (92f:19001)Google Scholar
75.Voevodsky, Vladimir, Suslin, Andrei, and Friedlander, Eric M., Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, 143, Princeton University Press, Princeton, NJ, 2000. MR 1764197 (2001d:14026)Google Scholar
76.Volodin, I. A., Algebraic K-theory as an extraordinary homology theory on the category of associative rings with a unit, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 844873. MR 0296140 (45 #5201)Google Scholar
77.Wagoner, J. B., Delooping classifying spaces in algebraic K-theory, Topology 11 (1972), 349370. MR 0354816 (50 #7293)Google Scholar
78.Waldhausen, Friedhelm, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math. 1126, Springer, Berlin, 1985, pp. 318419. MR MR802796 (86m:18011)Google Scholar
79.Waldhausen, Friedhelm, Jahren, Bjørn, and Rognes, John, Spaces of PL manifolds and categories of simple maps, to appear in Annals of Mathematics Studies. Available at http://folk.uio.no/rognes/papers/plmf.pdf, May 14, 2008.Google Scholar
80.Weibel, Charles, Algebraic K-theory of rings of integers in local and global fields, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 139190. MR 2181823 (2006g:11232)Google Scholar
81.Weibel, Charles A., The development of algebraic K-theory before 1980, Algebra, K-theory, groups, and education (New York, 1997), Contemp. Math. 243, Amer. Math. Soc., Providence, RI, 1999, pp. 211238. MR 1732049 (2000m:19001)Google Scholar
82.Weiss, Michael and Williams, Bruce, Automorphisms of manifolds, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 165220. MR 1818774 (2002a:57041)Google Scholar
83.Whitehead, J. H. C., Simple homotopy types, Amer. J. Math. 72 (1950), 157.Google Scholar