The Hilbert scheme of infinite affine space and algebraic Ktheory
Abstract
We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$homotopical viewpoint and obtain applications to algebraic Ktheory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$equivalent to the Grassmannian of $(d1)$planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$equivalence after group completion. This implies that the moduli stack $\mathrm{FFlat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic Ktheory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the $\mathrm{kgl}$homology of smooth proper schemes over a perfect field.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.11439
 Bibcode:
 2020arXiv200211439H
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology
 EPrint:
 24 pages. Comments welcome! v3: minor change v2: new title, new author, and many new results