Thomason’s theorem for varieties over algebraically closed fields

Walker, Mark

doi:10.1090/S0002-9947-03-03479-2

Thomason’s theorem for varieties over algebraically closed fields
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Trans. Amer. Math. Soc. 356 (2004), 2569-2648 Request permission

Abstract:

We present a novel proof of Thomason’s theorem relating Bott inverted algebraic $K$-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic $K$-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic $K$-homology and algebraic $K$-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

References

Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Spencer Bloch and Steven Lichtenbaum. A spectral sequence for motivic cohomology. Preprint. Available at http://www.math.uiuc.edu/K-theory/0062/.
A. K. Bousfield, On the homology spectral sequence of a cosimplicial space, Amer. J. Math. 109 (1987), no. 2, 361–394. MR 882428, DOI 10.2307/2374579
A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR 513569
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020, DOI 10.1007/BF02698644
Eric M. Friedlander and Andrei Suslin. The spectral sequence relating algebraic $K$-theory to motivic cohomology. Ann. Ec. Norm. Sup., 35:773–875, 2002.
Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander, Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, NJ, 2000. MR 1764197
William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI 10.1090/memo/0243
Daniel Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 217–240. MR 0574096
Daniel R. Grayson, Projections, cycles, and algebraic $K$-theory, Math. Ann. 234 (1978), no. 1, 69–72. MR 491686, DOI 10.1007/BF01409341
Daniel R. Grayson, Weight filtrations in algebraic $K$-theory, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 207–237. MR 1265531
Daniel R. Grayson, Weight filtrations via commuting automorphisms, $K$-Theory 9 (1995), no. 2, 139–172. MR 1340843, DOI 10.1007/BF00961457
Daniel R. Grayson and Mark E. Walker, Geometric models for algebraic $K$-theory, $K$-Theory 20 (2000), no. 4, 311–330. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV. MR 1803641, DOI 10.1023/A:1026506218989
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Marc Levine. $K$-theory and motivic cohomology of schemes. Preprint. Available at http: [3]//www.math.uiuc.edu/K-theory/0336/, February 1999.
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI 10.1007/BF01390094
Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468, DOI 10.1007/978-3-662-21576-0
Andrei A. Suslin, On the $K$-theory of local fields, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 301–318. MR 772065, DOI 10.1016/0022-4049(84)90043-4
Andrei Suslin and Vladimir Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), no. 1, 61–94. MR 1376246, DOI 10.1007/BF01232367
Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander, Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, NJ, 2000. MR 1764197
Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander, Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, NJ, 2000. MR 1764197
R. W. Thomason, Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR 826102, DOI 10.24033/asens.1495
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander, Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, NJ, 2000. MR 1764197
Friedhelm Waldhausen, Algebraic $K$-theory of generalized free products. I, II, Ann. of Math. (2) 108 (1978), no. 1, 135–204. MR 498807, DOI 10.2307/1971165
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
Mark E. Walker. Motivic Complexes and the $K$-theory of Automorphisms. Ph.D. thesis, University of Illinois, Urbana–Champaign, 1996. Available at http://www.math.unl.edu/ [3]$\sim$mwalker.
Mark E. Walker, The primitive topology of a scheme, J. Algebra 201 (1998), no. 2, 656–685. MR 1612355, DOI 10.1006/jabr.1997.7289
Mark E. Walker, Adams operations for bivariant $K$-theory and a filtration using projective lines, $K$-Theory 21 (2000), no. 2, 101–140. MR 1804538, DOI 10.1023/A:1007896730627
Mark E. Walker, Weight zero motivic cohomology and the general linear group of a simplicial ring, J. Pure Appl. Algebra 147 (2000), no. 3, 311–319. MR 1747445, DOI 10.1016/S0022-4049(98)00141-8
Mark E. Walker, Semi-topological $K$-homology and Thomason’s theorem, $K$-Theory 26 (2002), no. 3, 207–286. MR 1936355, DOI 10.1023/A:1020649830539
Charles A. Weibel, Homotopy algebraic $K$-theory, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 461–488. MR 991991, DOI 10.1090/conm/083/991991
Charles A. Weibel, Negative $K$-theory of varieties with isolated singularities, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 331–342. MR 772067, DOI 10.1016/0022-4049(84)90045-8