The Hilbert scheme of infinite affine space and algebraic K-theory

  • Marc Hoyois

    Universität Regensburg, Germany
  • Joachim Jelisiejew

    University of Warsaw, Poland
  • Denis Nardin

    Universität Regensburg, Germany
  • Burt Totaro

    UCLA, Los Angeles, USA
  • Maria Yakerson

    ETH Zürich, Switzerland
The Hilbert scheme of infinite affine space and algebraic K-theory cover

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Abstract

We study the Hilbert scheme from an -homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme is -equivalent to the Grassmannian of -planes in . We then describe the -homotopy type of in a certain range, for large compared to . For example, we compute the integral cohomology of in a range. We also deduce that the forgetful map from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an -equivalence after group completion. This implies that the moduli stack , viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the -homology of smooth proper schemes over a perfect field.

Cite this article

Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Burt Totaro, Maria Yakerson, The Hilbert scheme of infinite affine space and algebraic K-theory. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1340