# 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} }

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…

## 7 Citations

Torus actions, Morse homology, and the Hilbert scheme of points on
affine space

- MathematicsÉpijournal de Géométrie Algébrique
- 2021

We formulate a conjecture on actions of the multiplicative group in motivic
homotopy theory. In short, if the multiplicative group G_m acts on a
quasi-projective scheme U such that U is attracted as…

The very effective covers of KO and KGL over Dedekind schemes

- Mathematics
- 2022

We answer a question of Hoyois–Jelisiejew–Nardin–Yakerson regarding framed models of motivic connective K-theory spectra over Dedekind schemes. 1. Statement of results Let S be a scheme. The category…

Cohomology of the moduli stack of algebraic vector bundles

- Mathematics
- 2021

A BSTRACT . Let V ect n be the moduli stack of vector bundles of rank n on derived schemes. We prove that, if E is a Zariski sheaf of ring spectra which is equipped with ﬁnite quasi-smooth transfers…

Hermitian K-theory via oriented Gorenstein algebras

- Mathematics
- 2021

We show that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along finite Gorenstein morphisms with trivialized dualizing sheaf. As an application, we…

The Motivic Segal-Becker Theorem

- Mathematics
- 2021

The present paper is a continuation of earlier work by Gunnar Carlsson and the first author on a motivic variant of the classical Becker-Gottlieb transfer and an additivity theorem for such a…

Twisted K-theory in motivic homotopy theory

- Mathematics
- 2021

In this paper, we study twisted algebraic K-theory from a motivic viewpoint. For a smooth variety X over a field of characteristic zero and an Azumaya algebra A over X, we construct the A-twisted…

Cancellation theorem for motivic spaces with finite flat transfers

- Mathematics
- 2020

We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In…

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Torus actions, Morse homology, and the Hilbert scheme of points on
affine space

- MathematicsÉpijournal de Géométrie Algébrique
- 2021

We formulate a conjecture on actions of the multiplicative group in motivic
homotopy theory. In short, if the multiplicative group G_m acts on a
quasi-projective scheme U such that U is attracted as…