# The motive of the Hilbert scheme of infinite affine space

Research paper by Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson

Indexed on: 27 Feb '20Published on: 26 Feb '20Published in: arXiv - Mathematics - Algebraic Geometry

#### Abstract

We study the Hilbert schemes $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ and $\mathrm{Hilb}_\infty(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint. We show in particular that the rational Voevodsky motive of $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is pure Tate and that $\mathrm{Hilb}_\infty(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-homotopy equivalent to the infinite Grassmannian $\mathrm{Gr}_\infty(\mathbb{A}^\infty)$. We deduce that the forgetful functor $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$-homotopy 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 K-theory.