# 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 '20**Published on: **26 Feb '20**Published in: **arXiv - Mathematics - Algebraic Geometry**

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#### 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.