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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, fornlarge compared tod. 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.
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This content will become publicly available on July 11, 2026
Perfect even modules and the even filtration
Inspired by the work of Hahn–Raksit–Wilson, we introduce a variant of the even filtration which is naturally defined on\mathbf{E}_{1}-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn–Raksit–Wilson filtration on ring spectra of arithmetic interest, showing that various “motivic” filtrations are in fact invariants of the\mathbf{E}_{1}-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective\mathbf{E}_{1}-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights.
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- Award ID(s):
- 1926686
- PAR ID:
- 10625766
- Publisher / Repository:
- European Mathematical Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- ISSN:
- 1435-9855
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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