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The mathematical enterprises of Robert Thomason. (English) Zbl 0861.19001

Summary: During his career, Bob Thomason was involved in an interesting and varied group of mathematical endeavors. This is a retrospective survey of his contributions.

MSC:

19-03 History of \(K\)-theory
01A70 Biographies, obituaries, personalia, bibliographies
18-03 History of category theory
55-03 History of algebraic topology

Biographic References:

Thomason, Robert
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[1] Kenneth S. Brown and Stephen M. Gersten, Algebraic \?-theory as generalized sheaf cohomology, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 266 – 292. Lecture Notes in Math., Vol. 341.
[2] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. · Zbl 0259.55004
[3] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. · Zbl 0285.55012
[4] William G. Dwyer and Eric M. Friedlander, Étale \?-theory and arithmetic, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 453 – 455. · Zbl 0494.18009
[5] William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thomason, Algebraic \?-theory eventually surjects onto topological \?-theory, Invent. Math. 66 (1982), no. 3, 481 – 491. · Zbl 0501.14013
[6] Eric M. Friedlander, Étale \?-theory. I. Connections with etale cohomology and algebraic vector bundles, Invent. Math. 60 (1980), no. 2, 105 – 134. , https://doi.org/10.1007/BF01405150 Eric M. Friedlander, Étale \?-theory. II. Connections with algebraic \?-theory, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 231 – 256. · Zbl 0519.14010
[7] Daniel Grayson, Higher algebraic \?-theory. II (after Daniel Quillen), Algebraic \?-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 217 – 240. Lecture Notes in Math., Vol. 551.
[8] Henri A. Gillet and Robert W. Thomason, The \?-theory of strict Hensel local rings and a theorem of Suslin, Proceedings of the Luminy conference on algebraic \?-theory (Luminy, 1983), 1984, pp. 241 – 254. · Zbl 0577.13009
[9] M. Harada, A proof of the Riemann-Roch theorem, Ph.D. thesis, Johns Hopkins University, Baltimore, 1987.
[10] Dana May Latch, Robert W. Thomason, and W. Stephen Wilson, Simplicial sets from categories, Math. Z. 164 (1979), no. 3, 195 – 214. · Zbl 0408.55019
[11] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. · Zbl 0285.55012
[12] J. P. May and R. Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978), no. 3, 205 – 224. · Zbl 0391.55007
[13] Erik K. Pedersen and Charles A. Weibel, \?-theory homology of spaces, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 346 – 361.
[14] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. · Zbl 0168.20903
[15] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205 – 295. · Zbl 0191.53702
[16] Daniel Quillen, Higher algebraic \?-theory. I, Algebraic \?-theory, I: Higher \?-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85 – 147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004
[17] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293 – 312. · Zbl 0284.55016
[18] M. Artin, A. Grothendieck and J.-L. Verdier, Théorie de Topos et Cohomologie Étale des Schémas (SGA 4), Lecture Notes in Math., vols. 269,270,305, Springer-Verlag, 1972-73.
[19] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966 – 1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre.
[20] A. Suslin, Algebraic \(K\)-theory and Motivic Cohomology, Proc. International Congress of Mathematicians, Zürich 1994, vol. 1, Birkhäuser, 1995, pp. 342-351. · Zbl 0841.19003
[21] R. W. Thomason, A note on spaces with normal product with some compact space, Proc. Amer. Math. Soc. 44 (1974), 509 – 510. · Zbl 0298.54006
[22] R.W. Thomason, Homotopy colimits in \(\mathbf {Cat} \), with applications to algebraic \(K\)-theory and loop space theory, Ph.D. thesis, 124 pages, Princeton University, 1977, available from University Microfilms, Ann Arbor, MI 48104.
[23] R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91 – 109. · Zbl 0392.18001
[24] R. W. Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979), no. 2, 217 – 252. · Zbl 0413.55012
[25] R. W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21 (1980), no. 3, 305 – 324. · Zbl 0473.18012
[26] R. W. Thomason, First quadrant spectral sequences in algebraic \?-theory, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 332 – 355.
[27] Robert W. Thomason, First quadrant spectral sequences in algebraic \?-theory via homotopy colimits, Comm. Algebra 10 (1982), no. 15, 1589 – 1668. · Zbl 0502.55012
[28] R. W. Thomason, Beware the phony multiplication on Quillen’s \?\(^{-}\)\textonesuperior \?, Proc. Amer. Math. Soc. 80 (1980), no. 4, 569 – 573. · Zbl 0469.18008
[29] R. W. Thomason, Algebraic \?-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437 – 552. R. W. Thomason, Erratum: ”Algebraic \?-theory and étale cohomology” [Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437 – 552; MR0826102 (87k:14016)], Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 675 – 677 (French). · Zbl 0596.14012
[30] R. W. Thomason, The Lichtenbaum-Quillen conjecture for \?/\?_{\ast }[\?\(^{-}\)\textonesuperior ], Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 117 – 139.
[31] R. W. Thomason, Riemann-Roch for algebraic versus topological \?-theory, J. Pure Appl. Algebra 27 (1983), no. 1, 87 – 109. · Zbl 0545.14007
[32] R. W. Thomason, Bott stability in algebraic \?-theory, Applications of algebraic \?-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 389 – 406.
[33] R. W. Thomason, The homotopy limit problem, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, R.I., 1983, pp. 407 – 419. · Zbl 0528.55008
[34] R. W. Thomason, Absolute cohomological purity, Bull. Soc. Math. France 112 (1984), no. 3, 397 – 406 (English, with French summary). · Zbl 0584.14007
[35] R. W. Thomason, Algebraic \?-theory of group scheme actions, Algebraic topology and algebraic \?-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539 – 563.
[36] R. W. Thomason, Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), no. 3, 515 – 543. · Zbl 0653.14005
[37] R. W. Thomason, Comparison of equivariant algebraic and topological \?-theory, Duke Math. J. 53 (1986), no. 3, 795 – 825. · Zbl 0614.14002
[38] R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16 – 34. · Zbl 0624.14025
[39] R. W. Thomason, Equivariant algebraic vs. topological \?-homology Atiyah-Segal-style, Duke Math. J. 56 (1988), no. 3, 589 – 636. · Zbl 0655.55002
[40] R. W. Thomason, The finite stable homotopy type of some topoi, J. Pure Appl. Algebra 47 (1987), no. 1, 89 – 104. · Zbl 0626.18004
[41] R. W. Thomason, A finiteness condition equivalent to the Tate conjecture over \?_{\?}, Algebraic \?-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 385 – 392.
[42] R. W. Thomason, Survey of algebraic vs. étale topological \?-theory, Algebraic \?-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 393 – 443.
[43] Robert W. Thomason, The local to global principle in algebraic \?-theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 381 – 394. · Zbl 0759.19004
[44] R. W. Thomason, Une formule de Lefschetz en \?-théorie équivariante algébrique, Duke Math. J. 68 (1992), no. 3, 447 – 462 (French). · Zbl 0813.19002
[45] R. W. Thomason, Le principe de scindage et l’inexistence d’une \?-theorie de Milnor globale, Topology 31 (1992), no. 3, 571 – 588 (French). · Zbl 0768.55003
[46] R. W. Thomason, Les \?-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math. 112 (1993), no. 1, 195 – 215 (French). · Zbl 0816.19004
[47] R.W. Thomason, Les \(K\)-groupes d’un fibré projectif, Algebraic \(K\)-theory and algebraic topology, NATO ASI Series C, vol. 407, Kluwer, 1993, pp. 243-248. · Zbl 0910.19002
[48] R.W. Thomason, The classification of triangulated subcategories, preprint (1995), Compositio Math. (1996), to appear. · Zbl 0873.18003
[49] R.W. Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categories 1 (1995), 78-118, (electronic journal http://www.tac.mta.ca/tac/). · Zbl 0876.55009
[50] R. W. Thomason and Thomas Trobaugh, Higher algebraic \?-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247 – 435. · Zbl 0731.14001
[51] Robert W. Thomason and Thomas F. Trobaugh, Le théorème de localisation en \?-théorie algébrique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 16, 829 – 831 (French, with English summary). · Zbl 0697.18004
[52] Robert W. Thomason and W. Stephen Wilson, Hopf rings in the bar spectral sequence, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 124, 507 – 511. · Zbl 0449.55018
[53] Friedhelm Waldhausen, Algebraic \?-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318 – 419. · Zbl 0579.18006
[54] C. Weibel, Robert W. Thomason (1952-1995), Notices of the AMS 43 (1996), 860-862. · Zbl 1044.01547
[55] D. Yao, Higher algebraic \(K\)-theory of admissible abelian categories and localization theorems, Ph.D. thesis, Johns Hopkins University, Baltimore, 1990.
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