• Corpus ID: 221704154

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

@article{Hoyois2020TheHS,
title={The Hilbert scheme of infinite affine space and algebraic K-theory},
author={Marc Hoyois and Joachim Jelisiejew and Denis Nardin and Burt Totaro and Maria Yakerson},
journal={arXiv: Algebraic Geometry},
year={2020}
}
• Published 26 February 2020
• Mathematics
• arXiv: Algebraic Geometry
We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-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…
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