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.