Duke Mathematical Journal

The annihilator of the Lefschetz motive

Inna Zakharevich

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this article, we study a spectrum K(Vk) such that π0K(Vk) is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of K0[Vk] and to show that classes in the kernel of multiplication by [A1] can always be represented as [X][Y], where [X][Y], X×A1, and Y×A1 are not piecewise-isomorphic, but [X×A1]=[Y×A1] in K0[Vk]. Along the way, we present a new proof of the result of Larsen–Lunts on the structure on K0[Vk]/([A1]).

Article information

Duke Math. J., Volume 166, Number 11 (2017), 1989-2022.

Received: 29 September 2015
Revised: 20 November 2016
First available in Project Euclid: 28 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19E99: None of the above, but in this section
Secondary: 14E99: None of the above, but in this section

Grothendieck ring of varieties algebraic K-theory birational geometry


Zakharevich, Inna. The annihilator of the Lefschetz motive. Duke Math. J. 166 (2017), no. 11, 1989--2022. doi:10.1215/00127094-0000016X. https://projecteuclid.org/euclid.dmj/1493344842

Export citation


  • [1] M. Barratt and S. Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972), 1–14.
  • [2] A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles (in French), Ann. of Math. (2) 121 (1985), 283–318.
  • [3] J. Blanc and J.-P. Furter, Topologies and structures of the Cremona groups, Ann. of Math. (2) 178 (2013), 1173–1198.
  • [4] J. M. Boardman, “Conditionally convergent spectral sequences” in Homotopy Invariant Algebraic Structures (Baltimore, 1998), Contemp. Math. 239, Amer. Math. Soc., Providence, 1999, 49–84.
  • [5] L. Borisov, Class of the affine line is a zero divisor in the Grothendieck ring, preprint, arXiv:1412.6194v3 [math.AG].
  • [6] S. Cantat, The Cremona group, preprint, https://perso.univ-rennes1.fr/serge.cantat/Articles/Survey-Cremona-SLC.pdf (accessed 9 March 2017).
  • [7] Y. Cornulier, Sofic profile and computability of Cremona groups, Michigan Math. J. 62 (2013), 823–841.
  • [8] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.
  • [9] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197.
  • [10] C. D. Hacon, J. McKernan, and C. Xu, On the birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), 1077–1111.
  • [11] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109–203; II, 205–326.
  • [12] M. Hovey, Model Categories, Math. Surveys Monogr. 63, Amer. Math. Soc., Providence, 1999.
  • [13] A. Ito, M. Miura, S. Okawa, and K. Ueda, The class of the affine line is a zero divisor in the Grothendieck ring: Via $G_{2}$-Grassmannians, preprint, arXiv:1606.04210v2 [math.AG].
  • [14] P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Vol. 2, Oxford Logic Guides 44, Oxford Univ. Press, Oxford, 2002.
  • [15] I. Karzhemanov, On the cut-and-paste property of algebraic varieties, preprint, arXiv:1411.6084v3 [math.AG].
  • [16] S. Lamy and J. Sebag, Birational self-maps and piecewise algebraic geometry, J. Math. Sci. Univ. Tokyo 19 (2012), 325–357.
  • [17] M. Larsen and V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), 85–95, 259.
  • [18] Q. Liu and J. Sebag, The Grothendieck ring of varieties and piecewise isomorphisms, Math. Z. 265 (2010), 321–342.
  • [19] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. Lond. Math. Soc. (3) 82 (2001), 441–512.
  • [20] N. Martin, The class of the affine line is a zero divisor in the Grothendieck ring: An improvement, C. R. Math. Acad. Sci. Paris 354 (2016), 936–939.
  • [21] J. McCleary, A User’s Guide to Spectral Sequences, 2nd ed., Cambridge Stud. Adv. Math. 58, Cambridge Univ. Press, Cambridge, 2001.
  • [22] I. Pan, Une remarque sur la génération du groupe de Cremona, Bull. Braz. Math. Soc. (N.S.) 30 (1999), 95–98.
  • [23] E. Riehl, Categorical Homotopy Theory, New Math. Monogr. 24, Cambridge Univ. Press, Cambridge, 2014.
  • [24] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (accessed 9 March 2017).
  • [25] J. Włodarczyk, “Simple constructive weak factorization” in Algebraic Geometry—Seattle 2005, Part 2, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, 2009, 957–1004.
  • [26] I. Zakharevich, The $K$-theory of assemblers, Adv. Math. 304 (2017), 1176–1218.
  • [27] I. Zakharevich, On $K_{1}$ of an assembler, J. Pure Appl. Algebra 221 (2017), 1867–1898.