Hoyois, Marc and Jelisiejew, Joachim and Nardin, Denis and Totaro, Burt and Yakerson, Maria (2025) The Hilbert scheme of infinite affine space and algebraic K-theory. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 27 (5). pp. 2037-2071. ISSN 1435-9855
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We study the Hilbert scheme Hilbd (A1/ from an A1-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme Hilbd (A1/ is A1-equivalent to the Grassmannian of (d-1/-planes in A1. We then describe the A1-homotopy type of Hilbd (An/ in a certain range, for n large compared to d. For example, we compute the integral cohomology of Hilbd (An/(C/ in a range. We also deduce that the forgetful map FFlat-+/- Vect from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an A1-equivalence after group completion. This implies that the moduli stack FFlat, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum kgl representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the kgl-homology of smooth proper schemes over a perfect field.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | COMMUTATIVE ALGEBRAS; INVARIANT; HOMOLOGY; DESCENT; Hilbert scheme; K -theory; pure Tate motive |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Marc Hoyois |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 18 Jun 2026 08:34 |
| Last Modified: | 18 Jun 2026 08:34 |
| URI: | https://pred.uni-regensburg.de/id/eprint/66536 |
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