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

Abstract

We study the Hilbert scheme ${\mathrm{Hilb}}_d({\mathbf{A}}^{\infty })$ from an ${\mathbf{A}}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme ${\mathrm{Hilb}}_d({\mathbf{A}}^{\infty })$ is ${\mathbf{A}}^1$-equivalent to the Grassmannian of $(d-1)$-planes in ${\mathbf{A}}^{\infty }$. We then describe the ${\mathbf{A}}^1$-homotopy type of ${\mathrm{Hilb}}_d({\mathbf{A}}^n)$ in a certain range, for $n$ large compared to $d$. For example, we compute the integral cohomology of ${\mathrm{Hilb}}_d({\mathbf{A}}^n)(\mathbf{C})$ in a range. We also deduce that the forgetful map $\mathcal{FF}\mathrm{lat}\to \mathcal{V}\mathrm{ect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an ${\mathbf{A}}^1$-equivalence after group completion. This implies that the moduli stack $\mathcal{FF}\mathrm{lat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic K-theory. 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.