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Higher Algebraic K-Theory (After Quillen, Thomason and Others)

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Abstract

We present an introduction (with a few proofs) to higher algebraic K-theory of schemes based on the work of Quillen, Waldhausen, Thomason and others. Our emphasis is on the application of triangulated category methods in algebraic K-theory.

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References

  1. Adams, J.F.: Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1974)

    Google Scholar 

  2. Mandell, M.A., Blumberg, A.J.: Localization theorems in topological Hochschild homology and topological cyclic homology.arXiv:0802.3938v2 (2008)

    Google Scholar 

  3. Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading (1969)

    MATH  Google Scholar 

  4. Alonso Tarrío, L., JeremíasLópez, A., Lipman, J.: Local homology and cohomology on schemes. Ann. Sci. École Norm. Sup. (4), 30(1), 1–39 (1997)

    Google Scholar 

  5. Alonso Tarrío, L., JeremíasLópez, A., SoutoSalorio, M.J.: Localization in categories of complexes and unbounded resolutions. Can. J. Math. 52(2), 225–247 (2000)

    Google Scholar 

  6. Balmer, P.: Triangular Witt groups. I. The 12-term localization exact sequence. K-Theory 19(4), 311–363 (2000)

    Google Scholar 

  7. Bass, H.: Algebraic K-theory. W.A. Benjamin, New York (1968)

    Google Scholar 

  8. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and topology on singular spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)

    Google Scholar 

  9. Bousfield, A.K., Friedlander, E.M.: Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In: Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II. Lecture Notes in Mathematics, vol. 658, pp. 80–130. Springer, Berlin (1978)

    Google Scholar 

  10. Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. In: Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)

    Google Scholar 

  11. Bloch, S., Kato, K.: p-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. (63), 107–152 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bondal, A.I., Larsen, M., Lunts, V.A.: Grothendieck ring of pretriangulated categories. Int. Math. Res. Not. 29, 1461–1495 (2004)

    Article  MathSciNet  Google Scholar 

  13. Bloch, S.: Algebraic cycles and higher K-theory. Adv. Math. 61(3), 267–304 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86(2), 209–234 (1993)

    MATH  Google Scholar 

  15. Borelli, M.: Divisorial varieties. Pac. J. Math. 13, 375–388 (1963)

    MATH  MathSciNet  Google Scholar 

  16. Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J.Algebra 236(2), 819–834 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Buehler, T.: Exact categories. Expositiones Mathematicae, 28(1), 1–69 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bondal, A., vanden Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36, 258 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Cortiñas, G., Haesemeyer, C., Schlichting, M., Weibel, C.: Cyclic homology, cdh-cohomology and negative K-theory. Ann. Math. (2) 167(2), 549–573 (2008)

    MATH  Google Scholar 

  20. Cortiñas, G., Haesemeyer, C., Weibel, C.: K-regularity, cdh-fibrant Hochschild homology, and a conjecture of Vorst. J. Am. Math. Soc. 21(2), 547–561 (2008)

    Article  MATH  Google Scholar 

  21. Cortiñas, G.: Algebraic v. topological K-theory: a friendly match. In this volume

    Google Scholar 

  22. Cortiñas, G.: The obstruction to excision in K-theory and in cyclic homology. Invent. Math. 164(1), 143–173 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Friedlander, E.M., Grayson, D.R. (eds.): Handbook of K-theory, vol. 1,2. Springer, Berlin (2005)

    Google Scholar 

  25. Fritsch, R., Piccinini, R.A.: Cellular structures in topology. In: Cambridge Studies in Advanced Mathematics, vol. 19. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  26. Franke, J.: On the Brown representability theorem for triangulated categories. Topology 40(4), 667–680 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Friedlander, E.M., Suslin, A.: The spectral sequence relating algebraic K-theory to motivic cohomology. Ann. Sci. École Norm. Sup. (4) 35(6), 773–875 (2002)

    Google Scholar 

  28. Fulton, W.: Intersection theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], second edn., vol.2. Springer, Berlin (1998)

    Google Scholar 

  29. Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)

    MATH  MathSciNet  Google Scholar 

  30. Geisser, T., Hesselholt, L.: On the vanishing of negative k-groups. arXiv:0811.0652 (2008)

    Google Scholar 

  31. Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory. In: Simplicial homotopy theory, vol. 174. Birkhäuser, Basel (1999)

    Google Scholar 

  32. Geisser, T., Levine, M.: The K-theory of fields in characteristic p. Invent. Math. 139(3), 459–493 (2000)

    MATH  MathSciNet  Google Scholar 

  33. Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530, 55–103 (2001)

    MATH  MathSciNet  Google Scholar 

  34. Goodwillie, T.G.: Relative algebraic K-theory and cyclic homology. Ann. Math. (2) 124(2), 347–402 (1986)

    MathSciNet  Google Scholar 

  35. Grayson, D.: Higher algebraic K-theory. II (after Daniel Quillen). In: Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976). Lecture Notes in Mathematics, vol. 551, pp. 217–240. Springer, Berlin (1976)

    Google Scholar 

  36. Grayson, D.R.: Localization for flat modules in algebraic K-theory. J. Algebra 61(2), 463–496 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  37. Grothendieck, A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960)

    Google Scholar 

  38. Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. 11, 167 (1961)

    Google Scholar 

  39. Haesemeyer, C.: Descent properties of homotopy K-theory. Duke Math. J. 125(3), 589–620 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  40. Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  41. Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  42. Hsiang, W.C.: Geometric applications of algebraic K-theory. In: Proceedings of the International Congress of Mathematicians, (Warsaw, 1983), vol.1,2, pp. 99–118. PWN, Warsaw (1984)

    Google Scholar 

  43. Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)

    MATH  MathSciNet  Google Scholar 

  44. Kapranov, M.M.: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92(3), 479–508 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  45. Karoubi, M.: Foncteurs dérivés et K-théorie. Catégories filtrées. C. R. Acad. Sci. Paris Sér. A-B 267, A328–A331 (1968)

    MathSciNet  Google Scholar 

  46. Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67(4), 379–417 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  47. Keller, B.: Derived categories and universal problems. Commun. Algebra 19(3), 699–747 (1991)

    Article  MATH  Google Scholar 

  48. Keller, B.: Derived categories and their uses. In: Handbook of algebra, vol.1, pp. 671–701. North-Holland, Amsterdam (1996)

    Google Scholar 

  49. Keller, B.: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  50. Keller, B.: On differential graded categories. In: International Congress of Mathematicians, vol. II, pp. 151–190. European Mathematical Society, Zürich (2006)

    Google Scholar 

  51. Kurihara, M.: Some remarks on conjectures about cyclotomic fields and K-groups of Z. Compos. Math. 81(2), 223–236 (1992)

    MATH  MathSciNet  Google Scholar 

  52. Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218(5), 1340–1369 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  53. Levine, M.: Bloch’s higher Chow groups revisited. In: K-theory (Strasbourg, 1992). Astérisque 226, 10, 235–320 (1994)

    Google Scholar 

  54. Levine, M.: K-theory and motivic cohomology of schemes. K-theory archive preprint 336 (1999)

    Google Scholar 

  55. Levine, M.: The homotopy coniveau tower. J. Topol. 1(1), 217–267 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  56. Lück, W., Reich, H.: The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory. In: Handbook of K-theory, vol. 1,2, pp. 703–842. Springer, Berlin (2005)

    Chapter  Google Scholar 

  57. Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, second edn., vol.8. Cambridge University Press, Cambridge (1989) Translated from the Japanese by M. Reid.

    Google Scholar 

  58. Peter May, J.: Simplicial objects in algebraic topology. In: Van Nostrand Mathematical Studies, vol. 11. D. Van Nostrand Co., Inc., Princeton (1967)

    Google Scholar 

  59. Milnor, J.: Introduction to algebraic K-theory. In: Annals of Mathematics Studies, vol.72. Princeton University Press, Princeton (1971)

    Google Scholar 

  60. Mitchell, S.A.: Hypercohomology spectra and Thomason’s descent theorem. In: Algebraic K-theory (Toronto, ON, 1996). Fields Institute Communications, vol.16, pp.221–277. American Mathematical Society, Providence, RI (1997)

    Google Scholar 

  61. MacLane, S.: Categories for the working mathematician. In: Graduate Texts in Mathematics, second edn., vol.5. Springer, New York (1998)

    Google Scholar 

  62. Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. Clay Mathematics Monographs, vol.2. American Mathematical Society, Providence, RI (2006)

    MATH  Google Scholar 

  63. Neeman, A.: Some new axioms for triangulated categories. J. Algebra 139(1), 221–255 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  64. Neeman, A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25(5), 547–566 (1992)

    Google Scholar 

  65. Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  66. Neeman, A.: Triangulated categories. In: Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton, NJ (2001)

    Google Scholar 

  67. Neeman, A., Ranicki, A.: Noncommutative localisation in algebraic K-theory. I. Geom. Topol. 8, 1385–1425 (electronic) (2004)

    Google Scholar 

  68. Nesterenko, Y.P., Suslin, A.A.: Homology of the general linear group over a local ring, and Milnor’s K-theory. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 121–146 (1989)

    MATH  MathSciNet  Google Scholar 

  69. Panin, I.A.: The equicharacteristic case of the Gersten conjecture. Tr. Mat. Inst. Steklova, 241(Teor. Chisel, Algebra i Algebr. Geom.) pp. 169–178 (2003)

    Google Scholar 

  70. Popescu, N.: Abelian categories with applications to rings and modules. London Mathematical Society Monographs, vol. 3. Academic Press, London (1973)

    Google Scholar 

  71. Pedrini, C., Weibel, C.: The higher K-theory of complex varieties. K-Theory. 21(4), 367–385 (2000) Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part V

    Google Scholar 

  72. Quillen, D.: On the cohomology and K-theory of the general linear groups over a finite field. Ann. Math. (2) 96, 552–586 (1972)

    Google Scholar 

  73. Quillen, D.: Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., vol. 341, pp. 85–147. Springer, Berlin (1973)

    Google Scholar 

  74. Ranicki, A.: Noncommutative localization in topology. In: Non-commutative localization in algebra and topology. London Mathematics Society Lecture Note Series, vol. 330, pp. 81–102. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  75. Reid, L.: N-dimensional rings with an isolated singular point having nonzero K N . K-Theory. 1(2), 197–205 (1987)

    Google Scholar 

  76. Samokhin, A.: Some remarks on the derived categories of coherent sheaves on homogeneous spaces. J. Lond. Math. Soc. (2) 76(1), 122–134 (2007)

    Google Scholar 

  77. Schlichting, M.: Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem. In preparation

    Google Scholar 

  78. Schlichting, M.: Witt groups of singular varieties. In preparation.

    Google Scholar 

  79. Schwede, S.: Book project about symmetric spectra. www.math.uni-bonn.de/people/schwede

  80. Schlichting, M.: A note on K-theory and triangulated categories. Invent. Math. 150(1), 111–116 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  81. Schlichting, M.: Delooping the K-theory of exact categories. Topology 43(5), 1089–1103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  82. Schlichting, M.: Negative K-theory of derived categories. Math. Z. 253(1), 97–134 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  83. Schlichting, M.: The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes. Invent. Math. 179(2), 349–433 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  84. Serre, J.-P.: Modules projectifs et espaces fibrés à fibre vectorielle. In: Séminaire, P., Dubreil, M.-L., Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, p.18. Secrétariat mathématique, Paris (1958)

    Google Scholar 

  85. Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de Ferrand, D., Jouanolou, J.P., Jussila, O., Kleiman, S., Raynaud, M. et Serre, J.P. Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)

    Google Scholar 

  86. Sherman, C.: Group representations and algebraic K-theory. In: Algebraic K-theory, Part I (Oberwolfach, 1980). Lecture Notes in Mathematics, vol. 966, pp. 208–243. Springer, Berlin (1982)

    Google Scholar 

  87. Spaltenstein, N.: Resolutions of unbounded complexes. Compos. Math. 65(2), 121–154 (1988)

    MATH  MathSciNet  Google Scholar 

  88. Suslin, A., Voevodsky, V.: Bloch-Kato conjecture and motivic cohomology with finite coefficients. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998). NATO Science Series C: Mathematical and Physical Sciences, vol. 548, pp. 117–189. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  89. Swan, R.G.: K-theory of quadric hypersurfaces. Ann. Math. (2), 122(1), 113–153 (1985)

    Google Scholar 

  90. Thomason, R.W.: Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire. Invent. Math. 112(1):195–215 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  91. Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  92. Toen, B.: Lectures on DG-categories. In this volume

    Google Scholar 

  93. Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory. K-Theory 6(2), 177–189 (1992)

    Google Scholar 

  94. Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. In: The Grothendieck Festschrift, Vol.III. Progress in Mathematics, vol.88, pp. 247–435. Birkhäuser, Boston, MA (1990)

    Google Scholar 

  95. Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque 239, pp. xii+253 (1996) With a preface by Illusie, L., edited and with a note by Maltsiniotis,G

    Google Scholar 

  96. Voevodsky, V.: Motivic cohomology with Z∕2-coefficients. Publ. Math. Inst. Hautes Études Sci. (98), 59–104 (2003)

    Google Scholar 

  97. Vorst, T.: Localization of the K-theory of polynomial extensions. Math. Ann. 244(1), 33–53 (1979) With an appendix by Wilberd van der Kallen

    Google Scholar 

  98. Voevodsky, V., Suslin, A., Friedlander, E.M.: In: Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, vol. 143. Princeton University Press, Princeton, NJ (2000)

    Google Scholar 

  99. Waldhausen, F.: Algebraic K-theory of generalized free products. I, II. Ann. Math. (2) 108(1), 135–204 (1978)

    Google Scholar 

  100. Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and geometric topology (New Brunswick, N.J., 1983). Lecture Notes in Mathematics, vol. 1126, pp. 318–419. Springer, Berlin (1985)

    Google Scholar 

  101. Washington, L.C.: Introduction to cyclotomic fields. Graduate Texts in Mathematics, vol.83. Springer, New York (1982)

    Google Scholar 

  102. Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, vol.38. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  103. Weibel, C.: Algebraic K-theory of rings of integers in local and global fields. In: Handbook of K-theory, vol. 1,2, pp. 139–190. Springer, Berlin (2005)

    Google Scholar 

  104. Whitehead, G.W.: In: Elements of homotopy theory. Graduate Texts in Mathematics, vol.61. Springer, New York (1978)

    Google Scholar 

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The author acknowledges support from NSF and MPIM-Bonn.

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Schlichting, M. (2011). Higher Algebraic K-Theory (After Quillen, Thomason and Others). In: Topics in Algebraic and Topological K-Theory. Lecture Notes in Mathematics(), vol 2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15708-0_4

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