The 𝐾-theory spectrum of varieties

Campbell, Jonathan

doi:10.1090/tran/7648

The $K$-theory spectrum of varieties
HTML articles powered by AMS MathViewer

by Jonathan A. Campbell PDF

Trans. Amer. Math. Soc. 371 (2019), 7845-7884 Request permission

Abstract:

We produce an $E_\infty$-ring spectrum $K(\mathbf {Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (\mathbf {Var}_{/k})$. This is achieved by slightly modifying Waldhausen categories and the Waldhausen $S_\bullet$-construction. As an application, we produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausen’s $K$-theory of spaces $A(\ast )$ and to $K(\mathbf {Q})$.

References

Michael Barratt and Stewart Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972), 1–14. MR 314940, DOI 10.1007/BF02566785
Clark Barwick, On the algebraic $K$-theory of higher categories, J. Topol. 9 (2016), no. 1, 245–347. MR 3465850, DOI 10.1112/jtopol/jtv042
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada, A universal characterization of higher algebraic $K$-theory, Geom. Topol. 17 (2013), no. 2, 733–838. MR 3070515, DOI 10.2140/gt.2013.17.733
Andrew J. Blumberg and Michael A. Mandell, Derived Koszul duality and involutions in the algebraic $K$-theory of spaces, J. Topol. 4 (2011), no. 2, 327–342. MR 2805994, DOI 10.1112/jtopol/jtr003
Andrew J. Blumberg and Michael A. Mandell, The homotopy groups of the algebraic k-theory of the sphere spectrum, 2014.
Andrew J. Blumberg and Michael A. Mandell, Tate-Poitou duality and the fiber of the cyclotomic trace for the sphere spectrum, 2015.
Alexey I. Bondal, Michael Larsen, and Valery A. Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 29 (2004), 1461–1495. MR 2051435, DOI 10.1155/S1073792804140385
Denis-Charles Cisinski, personal communication.
A. D. Elmendorf and M. A. Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006), no. 1, 163–228. MR 2254311, DOI 10.1016/j.aim.2005.07.007
Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, Algebraic $K$-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 41–87. MR 1743237, DOI 10.1090/pspum/067/1743237
E. Getzler, Resolving mixed Hodge modules on configuration spaces, Duke Math. J. 96 (1999), no. 1, 175–203. MR 1663927, DOI 10.1215/S0012-7094-99-09605-9
Henri Gillet and Daniel R. Grayson, The loop space of the $Q$-construction, Illinois J. Math. 31 (1987), no. 4, 574–597. MR 909784
Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
John R. Isbell, On coherent algebras and strict algebras, J. Algebra 13 (1969), 299–307. MR 249484, DOI 10.1016/0021-8693(69)90076-3
Michael Larsen and Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259 (English, with English and Russian summaries). MR 1996804, DOI 10.17323/1609-4514-2003-3-1-85-95
Qing Liu and Julien Sebag, The Grothendieck ring of varieties and piecewise isomorphisms, Math. Z. 265 (2010), no. 2, 321–342. MR 2609314, DOI 10.1007/s00209-009-0518-7
Eduard Looijenga, Motivic measures, Astérisque 276 (2002), 267–297. Séminaire Bourbaki, Vol. 1999/2000. MR 1886763
Jacob Lurie, Higher algebra, 2015, unpublished.
M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR 1806878, DOI 10.1112/S0024611501012692
Randy McCarthy, On fundamental theorems of algebraic $K$-theory, Topology 32 (1993), no. 2, 325–328. MR 1217072, DOI 10.1016/0040-9383(93)90023-O
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
Dan Petersen, Is there a higher Grothendieck ring of varieties?, MathOverflow. https://mathoverflow.net/q/179022 (version: 2014-08-21). https://mathoverflow.net/q/179022.
Daniel Quillen, Higher algebraic $K$-theory: I [MR0338129], Cohomology of groups and algebraic $K$-theory, Adv. Lect. Math. (ALM), vol. 12, Int. Press, Somerville, MA, 2010, pp. 413–478. MR 2655184
John Rognes, A spectrum level rank filtration in algebraic $K$-theory, Topology 31 (1992), no. 4, 813–845. MR 1191383, DOI 10.1016/0040-9383(92)90012-7
Karl Schwede, Gluing schemes and a scheme without closed points, Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, 2005, pp. 157–172. MR 2182775, DOI 10.1090/conm/386/07222
Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2015.
Ross E. Staffeldt, On fundamental theorems of algebraic $K$-theory, $K$-Theory 2 (1989), no. 4, 511–532. MR 990574, DOI 10.1007/BF00533280
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
Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/gsm/145
Inna Zakharevich, The annihilator of the Lefschetz motive, Duke Math. J. 166 (2017), no. 11, 1989–2022. MR 3694563, DOI 10.1215/00127094-0000016X
Inna Zakharevich, The $K$-theory of assemblers, Adv. Math. 304 (2017), 1176–1218. MR 3558230, DOI 10.1016/j.aim.2016.08.045