## 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

## Additional Information

**Jonathan A. Campbell**- Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240
- Email: j.campbell@vanderbilt.edu
- Received by editor(s): January 9, 2017
- Received by editor(s) in revised form: February 14, 2018, February 28, 2018, and March 23, 2018
- Published electronically: February 14, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 7845-7884 - MSC (2010): Primary 19D99
- DOI: https://doi.org/10.1090/tran/7648
- MathSciNet review: 3955537